Sunday, December 30, 2012

1. The 12 new (qualitative) laws of the new millennium physics





THE CURRENT BLOG (ONLINE BOOK) HAS MAINLY SPECULATIVE CHARACTER AND IS  INTRODUCING ONLY VERY PARTIALLY NEW QUANTITATIVE FORMULAE OF PHYSICAL PHENOMENA. IT IS MAINLY A CHANGE OF PERCEPTION OF THE PHYSICAL REALITY AS WE KNOW IT, TO ONE THAT HAS ALSO A 2ND  MICROSCOPIC LAYER BASED ON THE ANALOGUE OF THE TRIAD OF FREE AND PERMANENT PARTICLES OF PROTON ELECTRON AND NEUTRON BUT ON A SMALLER SCALE AS MICRO-TRIAD OF FREE AND PERMANENT PARTICLES OF MICRO-PROTON MICRO-ELECTRON AND MICRO-NEUTRON (BUT ALSO MACROSCOPIC LAYER) , THAT CURRENTLY WE KNOW PARTIALLY AS FIELDS LIKE GRAVITATIONAL FIELD, ELECTROMAGNETIC FIELD OR QUANTUM VACUUM. THE MAIN PROOF OF THE  EXISTENCE OF MICRO-PROTON , MICRO-ELECTRON AND MICRO-NEUTRON  IS THE FAMOUS IN QUANTUM MECHANICS, 2-SLIT ELECTRON EXPERIMENT WHERE IT IS PHOTOGRAPHED NOT ONLY THE ELECTRON THIS HITS THE PHOTOGRAPHIC PLATE BUT ALSO THE WAVE OF CHARGE BY SUCH MICRO-PARTICLES THAT THE ELECTRON MOTION CREATES. WE LEAVE OPEN THE MANY DIFFERENT WAYS THAT THE MATTER OF PROTONS/NEUTRONS/ELECTRONS CAN COUPLE WITH THE 2ND LAYER MATTER OF MICRO-PROTINS/MICRO-NEUTRONS/MICRO-ELECTRONS IN OTHER WORDS THE GRAVITATIONAL FIELD THE ELECTROMAGNETIC FIELD AND THE "QUANTUM VACUUM".
THE ONLY NEW SPECULATION OF QUANTITATIVE DESCRIPTION IS THE APPLICATION OF THE WELL KNOWN NAVIER-STOKES EQUATIONS FOR THIS 2ND MICROSCOPIC FLUID LAYER IN THE ABSENCE OF COUPLING WITH THE VISIBLE MATTER (OF ELECTRONS PROTONS NEUTRONS OR PLANETS ). THEREFORE A QUANTITATIVE FORMULATION WHICH IS OF APPLICATIONS TO A RATHER  MACROSCOPIC SCALE E.G. SOLAR SYSTEM SCALE.
THIS WORK DOES THE BEST ONE CAN DO TO DISCOVER THE TRUTH BEHIND THE STANDARD EQUATIONS OF GRAVITATION AND ELECTROMAGNETISM AS FAR AS A 2ND MATERIAL LAYER UNIFICATION IS  INTENDED, WHEN NO LABORATORIES EXPERIMENTS AND LARGE GROUPS OF SCIENTISTS WITH  WELL FUNDED RESEARCH IS POSSIBLE.  THE MAIN 3 CONTRIBUTIONS ARE 
1) THE REVEALING OF THE TRUE MEANING OF THE POISSON EQUATION OF NEWTON'S SPHERICAL GRAVITATIONAL POTENTIAL IN RELATION TO THE POISSON EQUATION OF SPHERICAL HEAT PROPAGATION IS THE KEY SHIFT IN THE CORRECT PHYSICAL INTERPRETATION OF GRAVITATION WHICH ALLOWS FOR  DISCOVERING SOLAR RENEWABLE ENERGY STORED IN IT. 
2) THE REVEALING INTERPRETATION OF MAXWELL'S  ELECTROMAGNETIC MAGITUTES IN RELATION TO AN IONIZED OR CHANGED 2ND MATERIAL LAYER SUBSTRATUM  AND 
3) THE PRELIMINARY UNIFYING ROLE OF NAVIER-STOKES EQUATIONS OF THE SUBSTRATUM MATERIAL FLUID FOR  GRAVITATION AND ELECTROMAGNETISM  WHICH ALLOWS FOR THE EXISTENCE OF NEW ELECTROMAGNETIC PROPULSION FOR TRANSPORTATIONS. 

 IT IS ALSO SPECULATED  HOW BY CHANGING THE PERCEPTION OF MATERIAL REALITY FROM A SINGLE LAYER (OR FREQUENCY) OF THE STANDARD ABOVE FREE AND PERMANENT TRIAD OF PARTICLES TO AT LEAST A DOUBLE LAYER (OR FREQUENCIES) PHYSICAL REALITY WITH BOTH THE STANDARD TRIAD OF FREE AND PERMANENT PARTICLES BUT ALSO MICRO-TRIAD  OF FREE AND PERMANENT PARTICLES WE COULD DO THE NEXT:
A) REFORMULATE EQUATIONS OF GRAVITATION WHICH INCLUDE THOSE OF I. NEWTON AND ARE MORE EXACT AND PHYSICALLY MEANINGFUL AND REALISTIC THAN THOSE OF A. EINSTEIN. 
B) REFORMULATE AND PROVE AGAIN THE SPECIAL RELATIVITY FORMULAE AS A KIND OF "LINEARIZED WAVED INERTIA" OF E.G. AN ELECTRON ON THE SUBSTRATUM 2ND LAYER PHYSICAL REALITY, BUT UNDER TOTALLY DIFFERENT AXIOMS THAT DO NOT INVOLVE THAT NOTHING GOES FASTER THAN LIGHT.
C) REFORMULATE NON-LINEAR EQUATIONS FOR THE CLASSICAL ELECTROMAGNETISM WHICH INVOLVE THOUGH MAGNITUDES OF GRAVITATION TOO, FOR  LARGE SCALE PHENOMENA E.G. SOLAR SYSTEM SCALE PHENOMENA . THE CLASSICAL ELECTROMAGNETISM IS ONLY THE LINEAR COUPLING OF THE ELECTROMAGNETIC FIELD WHICH IS MADE FROM THE MICRO-TRIAD OF MICRO-PROTONS MICRO-NEUTRONS AND MICRO-ELECTRONS, WITH THE MATTER OF ELECTRONS/PROTONS/NEUTRONS AND IS ACCEPTABLE APPROXIMATELY CORRECT AT SMALL LABORATORY SCALE.  
D) AFTER THE JOIN REFORMULATION OF CLASSICAL GRAVITATION AND CLASSICAL ELECTROMAGNETISM IN THE NEXT DECADES PREDICT  THE EXISTENCE OF ELECTROMAGNETIC DEVICES THAT MAY EXTRACT RENEWABLE SOLAR ENERGY STORED IN THE GRAVITATIONAL FIELD (FREE ENERGY). AND THE EXISTENCE OF ELECTROMAGNETIC PROPULSION FLYING VEHICLES IN VARIOUS SHAPES, INCLUDING DISC-SHAPES, THAT MAY USE THE ABOVE ENERGY FOR FLYING .
E) DERIVE THE SCHRODINGER WAVE MECHANICS OR THE EQUIVALENT HEISENBERG MATRIX MECHANICS FROM A LINEAR COUPLING OF THE  SUBSTRATUM  LAYER NEUTRAL FLUID OF MICRO-TRIAD OF FREE AND PERMANENT PARTICLES (CALLED ERRONEOUSLY QUANTUM VACUUM) OR GRAVITATIONAL FIELD WITH THE MATTER OF ELECTRONS/PROTONS/NEUTRONS. 

IN SUMMARY MORE THAN 80% THE CONTENT OF THE CURRENT WORKS IS TO CHANGE DEEPLY AND IN A PROFOUND WAY OUR QUALITATIVE PERCEPTIONS ABOUT THE PHYSICAL REALITY WHICH CAN LEAD IN  THE FUTURE DECADES TO A MORE DETAILED  UNIFIED QUANTITATIVE APPROACH TO GRAVITATION AND ELECTROMAGNETISM. IN LESS THAN 20% OF THE SPECULATIONS, THE ONLY QUANTITATIVE SET OF FORMULAE (THOSE OF NAVIER-STOKES) THAT ARE INTRODUCED ARE ONLY A PRELIMINARY STEP THAT CAN BE DONE IN RATHER EASIER WAY FOR THIS UNIFICATION, AND HAS APPLICATIONS MAINLY TO LARGER SCALE PHENOMENA E.G.  SOLAR SYSTEM SCALE GRAVITATION AND ELECTROMAGNETISM.


  

Here is a list of past Physics assumptions that I consider that the 21th century and the new millennium physics has already started and will eventually turn all of them false
1) The inertial mass of bodies ( of constant amount of molecular matter) at low
 speed (non-relativistic) cannot be decreased below the inertia of the rest mass.
2) All matter starts with protons, neutrons, electrons. In other words, there are not smaller permanent particles (Other known Quantum particles are excluded as they are not permanent or free)
3) Nothing goes faster than photons
4) All macroscopic electromagnetic interactions are described with the linear equations of Maxwell.
5) All forces acting on laboratory macroscopic objects at low speed (non-relativistic) are of the next 5 types a) Inertial, b) by contact with other material bodies made from protons, neutrons, electrons, c) Newtonian gravitation forces d) Maxwell's electromagnetic forces e) no other type of forces.

6) When a body like a meteorite is attracted from the gravitational field of earth and acquires kinetic energy falling on earth which is converted to thermal energy, this energy is not subtracted from the gravitational field of earth.
7) When a small ball of  iron is attracted from a magnet and acquires kinetic energy falling on the magnet which is converted to thermal energy, this energy is not subtracted from the magnetic energy of the magnet.

The more correct attitude is that
1) It is possible under special conditions to have radical decrease of the inertia of the rest mass of a body in slow motion
2) Aether is a 2nd material reality made from permanent free particles of positive negative and neutral charge, transcendentally finer than protons-neutrons-electrons and the electromagnetic and gravitational field is aspects of the functions of aether.
3) To say than nothing goes faster than light in aether, is like saying than no airplane can go faster than the sound, in air.
4) The linear Maxwell equations of electromagnetism are correct only for a limited realm of experiments those discovered at the end of the 19th century, not all laboratory scale experiments that we now know. They need revision, and their correct version include parameters of gravitation too, and are non-linear.

5) Besides, inertial forces, classical electromagnetic forces and Newtonian and Einsteinian gravitational forces, and forces by contact of bodies,  exist also a 5th type of macroscopic laboratory scale forces on bodies from the rest of the gravitational field that we do not know. (In underground physics, this unknown field is called anti-gravity or the classical gravity is called aether-statics and this field, aether-dynamic field of the neutral aether) 

6) When a body like a meteorite is attracted from the gravitational field of earth and acquires kinetic energy falling on earth which is converted to thermal energy, this energy is indeed subtracted from the gravitational field of earth which nevertheless is renewed from solar energy stored in the gravitational field.
7) When a small ball of  iron is attracted from a magnet and acquires kinetic energy falling on the magnet which is converted to thermal energy, this energy is indeed  subtracted from the magnetic energy of the magnet which nevertheless is renewed by energy from  the gravitational field of earth.  

As A. Einstein took the Nobel prize when he proved the existence of atoms by the almost one century old known experimental fact the Brownian motion that can be derived by the influence of atoms in matter , in the same way the wave mechanics or matrix mechanics formulation of Quantum physics is a way of random behavior of a proton or an electron or a neutron when the move and the proof of the existence of finer free and permanent atoms that make the electromagnetic and gravitational field (the same particles) or aether charged or neutral . This was highlighted by the nobel prize winner Dirac. Also von-Neumann mentioned that the absence of aether (of Dirac) is equivalent to the abandoning of the scientific principle of sufficient physical causes for observable phenomena like the motion of an electron. Instead of Brownian motion we have here the Shrowndiger or De Broglie motion. Of course to be able to derive mathematically that a proton or an electron or a neutron by their spin motion alone create waves in the aether which make them move randomly in the way that matrix mechanics describes or wave mechanics by a probability wave of motion of them is my far more difficult than the 5-6 pages paper of mathematical derivation of Brownian motion of A. Einstein. It requires non-linear random flow and Navier-stokes formulation of the aether, then linearization e.g. to a Dalambertian wave equation due to spin of e.g. an electron. Thus motion , momentum etc of the electron will correspond to an operator in a Hilbert space of aether material waves which is isomorphic with a corresponding Hilbert space of probability waves (Shroendiger waves) of the motion of electron and then back derivation of the Heisenberg matrix formulation of the motion of electron as von-Neumann had proved as equivalent formulation. Such a work though would makes us wake up about the 2nd frequency or 2nd resolution material reality (aether and liquid and solid versions of such matter also, that might make whole planet rather invisible ) which although we do know as electromagnetic waves or gravitational field we have not realized that we are talking about a whole new higher frequency physical reality with a new triad of free and permanent atoms the 1st micro electron , 1st micro-proton and 1st micro-neutron (to discriminate ot from 2nd 3rd aether etc).


Other beliefs that we have in Cosmology and Astronomy that seem not to be correct are discussed below.

1) Those referring to , the physical reality, and cosmology.

1.1) The universe was not created by a Big Bang 13-14 billion years ago. It is by far more reasonable and probable that there is a swirl motion of galaxies and stars that increases and decreases the world entropy periodically , and that it may not be improbable that the universes age is about 21 trillion years, as many galactic civilization 3-4 thousand years more advanced that us, believe. 

For a form of cosmic motion of the stars as a Newtonian isotropic, compressible fluid we state here a well known solution of the Navier-Stokes equations.

This overall vortex motion of the Universe, gives rise as a turbulent flow of matter (in many frequencies as we shall see) to smaller cortices (similar to the known phenomenon of vortex stretching and splitting of a larger vortex to smaller, governed by the Navier-Stokes equations) , which become the galaxies, and in their turn the vortex motion of galaxies gives rise to smaller vortex motion of matter that become the stars, black-holes and planets. 
A three-dimensional steady-state vortex solution
File:Hopfkeyrings.jpg

Some of the flow lines along a Hopf fibration.
A nice steady-state example with no singularities comes from considering the flow along the lines of a Hopf fibration. Let r be a constant radius to the inner coil. One set of solutions is given by:
{\displaystyle {\begin{aligned}\rho (x,y,z)&={\frac {3B}{r^{2}+x^{2}+y^{2}+z^{2}}}\\p(x,y,z)&={\frac {-A^{2}B}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{3}}}\\\mathbf {u} (x,y,z)&={\frac {A}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{2}}}{\begin{pmatrix}2(-ry+xz)\\2(rx+yz)\\r^{2}-x^{2}-y^{2}+z^{2}\end{pmatrix}}\\g&=0\\\mu &=0\end{aligned}}}


for arbitrary constants A and B. This is a solution in a non-viscous gas (compressible fluid) whose density, velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium problem because that refers to incompressible fluids where \rho is a constant, neither does it deal with the uniqueness of the Navier–Stokes equations with respect to any turbulence properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the Pythagorean quadruple parametrization. Other choices of density and pressure are possible with the same velocity field:



For this  periodic motion see also e.g. 

http://www.youtube.com/watch?v=EKtevjrZOGs

1.2) The physical reality is not exhausted by the matter that we know , which is made from protons neutrons and electrons (the only permanent and free particles, anti-matter not included). On the contrary there are many more similar layers of matter , made from such triads of smaller free and permanent particles. Each such layer having a frequency. Therefore although the layer of matter that we know (made from protons neutron electrons, and which is only one frequency of material reality) may have an age of 21 billion years (as as many galactic civilization 3-4 thousand years, more advanced than us believe) the universe as a whole, having more such frequencies-layers, should be older. The advanced civilizations in the galaxy report at least 10 such material physical frequencies-layers. The last one the 10th created recently during the 90s. The next (second) frequency-layer material reality after the one that we know, has sometimes being called aether in the history of the human civilization. The term frequency, in the matter we know, refers to the average frequency of the spin of protons, neutrons, and electrons. 
(For the numbering of the frequencies of physical material realities see current post below) 

1.3) The interior of the earth is not full completely from magma and high temperature matter. It does hold that from the center and till the 2/3 of the radius of earth it is hollow with inner atmosphere and it has a sun-like gaseous fiery ball at the center. In order to understand this we should have more advanced scientific theory of gravitation than the one we have. This fact of hollow planet, holds for almost all planets and stars in the galaxies.

Here is an 3D animation video, which gives the above assumed concept (in the video the width of the shell is shown half of what mentioned here)

http://www.youtube.com/watch?v=ubKFmIDBMjU



 Inspired by the  12 universal laws  from  Milanovich and McCunes book "The Light Shall Set You Free"(1998) I resume some of the principles of my research in physics  but without the mathematical equations and details,   to 12 universal physical laws for the new millennium physics. Six of the laws 1,5,7,8,9,12 deal with the vertical interplay of the levels, and the other six 2,3,4,6,10,11 deal with horizontal interplay at each level.



1. The Law of levelled physical material reality

1.0 What we consider as physical material reality is subject and conditioned to our development in consciousness our hidden beliefs and our scientific operational competence and development. E.g. in ancient Rome the protons. neutrons, and electrons could not  stand a chance to be considered as physical reality or existent at all. What was considered as physical reality was only gross pieces and aggregations of matter made by protons, neutrons, and electrons. Anything else would be shear philosophical or metaphysical. 
1.1 After the 19th century we are universally mature enough to accept that any form of material existence consists of elementary indivisibles. 

1.2 The content of this law is that all mater does not starts from protons, neutrons and electrons only. What we call traditionally as gravitational field, electromagnetic field, etc is accepted to have energy density in space even inertial density in space (there are classical college physics experiments with the momentum and inertia of the electromagnetic field, and similar in Einstein’s physics for the gravitational field). Therefore these  fields have material existence, but they do not consist from protons electrons and neutrons! We would contradict ourselves if we would  not assume that they consists too from permanent indivisible particles. Such particles have to be permanent and extremely small (e.g. 10^(-36) times smaller than the electron) otherwise quantum and nuclear physics would have pinned them down and would know them. (e.g. the neutrino that can cross the earth and not hit a single earth’s atom is 10^(-6) times only smaller than the electron. All particles of nuclear physics and quantum mechanics besides the proton, neutron, electron are of very short life duration and not permanent). This material reality of the fields we call the 4th material density or resolution, or 1st aether, or 2nd order  micro-atomic material reality. There exist also in a similar way the 5th material density.  The content of this law is also that the 3rd material density as well as the 4th and 5th material densities consists from particles. Also that all the material physical reality consists of superposition of such material densities. The sun, and the stars do not exist only in one level or frequency of physical material reality, but in many. Our planet too, has a tonal frequency of material reality (the material reality of proton;s neutrons, electrons) but has also "harmonic" frequencies material realities (4th, 5th etc), although life exists massively only in the tonal frequency. Nevertheless there are planets and stars, that their tonal frequency is the 4th density-frequency  or 5th density-frequency material reality. They are therefore invisible to our current state of technology. The material particles themselves of protons, neutrons ,electrons, (and those of higher densities-frequencies) exist also in many frequencies ("harmonics")  physical material realities! That is why it is not simple to understand them in our quantum physics. 

1.3 As far as I know our experiments go up to the 4th material density and barely only go up to the 5th material density. But they may as well exist 6th or more. Some say it exists till 11 (or 9 if you strat counting levels from the 3rd density) but as we mentioned at the beginning, this depends on the evolution of the civilisation and the consciousness of the human beings in it. An obvious question is why we start counting from 3. This is so, as we are counting the matter that is granulated in to the planets and stars in aggregation as 2nd material density. The next question would be what is the 1st material density. We consider the overall world of galaxies and everywhere that classical  light can go as one celestial body or…..particle assuming that there more such particles creating the 1st material density. 



2. The Law of Vibration
This Law states that everything in the Universe (e.g. protons, neutrons, electrons of the 3rd material density) vibratesAll  electrons, protons and neutrons were simultaneously created through a vibration from finer physical realities.  The spin of protons, electrons, and neutrons has a vibration frequency. The same with an atom that may have many frequencies like a musical chord (which is also the basis of  smelling: Our sensing cells of smelling detect the atom’s chord-vibrations ). This holds also for the basic particles of the 4th and 5th the material density. So each material density has its own vibration characteristic frequency. Depending on the vibration source, the vibration wave may propagate in many different material densities with different speeds. In the 4th material density there is also “sound” as compression wave, but it may have the speed of light.

3. The Law of Action
This law in the 3rd material density is known and analysed as the 3 Newtonian laws:
a) The conservation of momentum a) The law of force and acceleration, c) the law of action reaction.  Later than Newton scientists proved based on the previous 3 Newtonian laws, the law of conservation of energy. Lagrange and Hamilton were also able to derive the momentum and energy conservation from
the  law of stationary action

4. The Law of Correspondence
This Law states that the known classical principles and laws of physics that explain the physical world  at the 3rd density- energy, waves, vibration, and motion - have their corresponding principles in the 4th and 5th material density. "As above, so below" was the ancient quote. As in the 3rd material density there are the 3 material states , gaseous, liquid, and solid, so there are in the 4th and 5th. E.g. the gaseous and liquid 4th density mater can go through the gaseous, liquid and solid 3rd density mater (e.g. the 4th density gaseous electromagnetic waves of a mobile telephone device can go through air, water, and sold walls). But the 4th density solid mater will hit hard the 3rd  solid mater and would not go through. It seems though that it is difficult technology to isolate or produce such 4th or 5th density solid state mater.

5. The Law of Cause and Effect
The contact causalities or particle collision causalities or horizontal causalities that we know in the 3rd material density  (e.g. law of action and reaction) hold also in the 4th and 5th material density.

6. The Law of Compensation and Deeper causalities
Besides the horizontal contact causalities within a material layer hold also the vertical causalities among material layers. The top-down flow (from 5th density to 4th, to 3rd density) is relevant to our (in-out) creative abilities.

7. The Law of Attraction
This law is essentially what Newton started studying in his universal attraction (later called gravitation) and his inverse square law. This law can only be fully revealed and the inverses square law explained only if more than the 3rd material density is included. This is the process of discovering the Unified Field Theory which is nothing else than the equations of the gaseous 4th material density. Nevertheless in the 4th material density exist also liquid and solid states. The 0th key to start understanding the mechanism behind the inverse square law of Newton, in universal attraction or gravitation is the identification: The Newtonian gravitational scalar potential φ is proportional to the aether temperature (or gaseous 4th density matter temperature)     
Both the presence of matter and infrared solar radiation contribute to the creation of aether heat that in its turn creates gravity. But the factor of infrared solar radiation seems to be the major. 

8. The Law of Perpetual Transmutation of Energy
The material densities (3rd, 4th, 5th) are in continuous energy exchange through contact, friction, vibration, etc. There is a natural flow of energy  from the 3rd density to the 4th density through friction of the spinning and rotating protons, neutrons and electrons in an atom. This transfers heat from the 3rd material density and is creating heat in the 4th material density. Also any  3rd density material object moving within the gaseous 4th density matter (aether), creates a  drag-force and carries way partially and locally only  the  gaseous 4th density matter (aether). This may resemble the De Broglie’s law , but its mathematical details are different.

9. The Law of Relativity

The content of this law is that the space measurements and time measurements, if attached to a particular material density only are relative not absolute. So the space and time as measured in the familiar 3rd material density only, may not be the same when measured in e.g. the 5th material density only. For a more unified measurement of space and time, many material layers have to be coordinated. E.g. we might measure time with duration that our bodies change and replace all cells, that in earthly human beings for the current level of evolution is about 7 years. And this because in our human existence we also define a coordination between the various material densities (e.g. 3rd material density, the 4th material density , the 5 material density.) This law has also a further meaning. Our level of evolution of the consciousness defines what is external material reality  (which has many layers, e.g. coarse matter by protons, neutrons, electrons, and fine matter like the electromagnetic field or aether) and what is internal reality (which again has layers e.g. like  mind and spiritual awareness and intention). As we evolve some of the internal layers become external, and new deeper reality becomes internal. 



10. The Law of Polarity
10.0 The content of this law is that as in the 3rd material density  there is the emergence of the electric positive, negative and neutral, so is in the 4th and 5th material density. Therefore there are triads of micro-electrons micro-protons, micro-neutrons in the 4th material density and triads of nano-protons, nano-electrons and nano-neutrons in the 5th material density. These triads create these densities as the triad of proton neutron electron creates the familiar 3rd material density. If we are not satisfied with triads, and want a single particle creating the triad, and then all the reality (of a specific frequency) then this would be the electron, in  our known material reality.

10.1  In the 3rd material density, it is traditionally described with the system of linear equations of Maxwell’s electricity. What we know today as the Maxwell’s electromagnetic field was called originally by Maxwell himself as the Electro-magnetised  aether.  Aether is a gas formation of the 4th material density.  Nevertheless, if after this law we introduce besides the polarity of electrons, protons, neutrons, the corresponding polarity in the gaseous 4th density (aether) and utilise the non-linear Navier-Stokes equations of the gases (that are derived by the energy and momentum conservation, but now applied after law 3 of action to the 4th material density) we may correct the Maxwell equations of electromagnetism to non-linear equations. This is part of the process of discovering the Unified Field Theory, which is nothing else than the theory of the gaseous 4th material density with its intrinsic polarity. This Unified Field Theory is not what was much discussed as Unified Field theory at the quantum scale of strong, weak and electromagnetic interactions which was never attained by quantum physics.  It is a different laboratory-scale unified field theory that unifies Universal Attraction of Newton, and Electromagnetism of Maxwell. In addition a 3rd new laboratory-scale and macroscopic field is introduced that underground physics sometimes calls anti-gravity, but it is rather a the gravitodynamics (or aethrodynamics) as contrasted to Newtonian gravitostatics, that escape even Einstein’s gravitation. The more complete understanding of the physics of the gaseous 4th material density seems that it may lead to the discovery of the ability to significantly reduce the inertia mass of slow moving (non-relativistic speeds) bodies without reducing their quantity of matter (as number of atoms).

Remark: Once we have discovered the macroscopic Unified Field Theory, the free energy or overunity magnetic devices have full and simple explanation. The same for more overunity devices (e.g. based on Oxyhydrogen or water). Not only they are explained but also understood as forms of renewable energy.   They are of course explained and computed within the energy conservation and exchange of the  of 3rd and 4th material densities simultaneously. Exclusion of the 4th material density and restriction to the 3rd material density keeps them unexplained and inaccessible to Academic Science explanations.
There are persons for whom the aether vision has opened.  They see the aether glow around objects and bodies. They can see also the weak colored light of aether around the aether-centers (charkas) of the human body. But maybe a day will come that most of the people will have aether-vision. All that it takes is to have higher spin frequency of our electrons, protons and neutrons in our physical body so as to have the aether vision opened. In other words higher energy too, and an easier interplay of consciousness and matter in us.
The key to start really understanding the electromagnetism as the dynamics of the electromagnetised aether is to identify the classical scalar and vector potentials of Maxwell’s electromagnetism to fluid parameters of aether. The 1st key is : The scalar electromagnetic potential a0 is proportional to the pressure of the non-neutral aether (or non-neutral gaseous 4th material density)
The 2nd key is:The vector electromagnetic potential A is proportional to the vector of momentum of the non-neutral aether ( or non-neutral gaseous 4th material density).





11. The Law of subjectivity

Human beings have and are developing their bodies not only in the 3rd material density but also in the 4th and  5th.  The classical literature of acupuncture or aetheric energy centres (=the chakras) exists because we have a living and evolving body in the 4th material density. It seems that our cells and  DNA exists and functions (also as field vibrations transmitter and receiver) not only in the 3rd material density (or 1st order material reality) but also in the 4th and 5th ( or 1st and 2nd aether, or 2nd and 3rd order micro-atomic material reality) By changing our beliefs about  the true reality of our living bodies a new order of identity and self emerges for us. The frequency of the spin of the protons, neutrons, electrons in our body (3rd density) (and if we want to be more sophisticated we would say the Compton-De Broglie frequency) as well as the frequency of the spin of the permanent particles of aether or aetherons  (4th density) depends on our level of consciousness and mind, and does affect the way that  matter (3rd density matter) interacts with the classical electromagnetic and gravitational fields, or aether (4th density matter) (And of course with the 2nd aether (or 5th density matter) which for the time being our civilization is not aware as external physical reality, although as human beings we are aware of it as internal reality and experience .) The idea that it is not the world that exists (objectivity) , but your existence in which there is the perception of  the world (subjectivity), enhances subjectivity as equivalent to objectivity, and both are absorbed in to one only mode.  



12. The Law of Creation

The content of this law is that the physical leveled material reality (and e.g. its protons, neutrons, electrons etc) has been created, in the same way that we consider plants and animals as created. It is I think interesting to speculate if normally (with probably some exceptions) for a material layer to be created, a coarser or finer material   layer must exist in advance or not. Or maybe the material layers are somehow more independent in their creation. Thus for the known material layer made from protons, neutrons and electrons, to be created, we could speculate if a finer material layer (like aether or sub-aether) must exist in advance or not, and/or if the 2nd material layers of stars, and blackholes must exist in advance. Our times  seem to be rare times, when (decade of 1990's) a new material layer has been created, through a simultaneous emission from all black holes in all the galaxies of  a particular frequency. 
In other words the particles of the new material reality , were created simultaneously from a frequency of higher resolution realities. It seems that this is the way that protons, neutron, and electrons (and anti-matter too) were, initially, created: from a "SOUND" FREQUENCY (in other words compression waves) of higher resolution realities,  emanated from black-holes (which in higher realities are super bright starts, and which exist before and outside the layer of physical reality they create) , therefore a macro-resolution reality involved too.  (Εν αρχη η ο λογος=At first there was the Word). (Counting from the visible material layer made from protons, neutrons, electrons as 1st, then this new material layer  is the 10th)
This may also mean that particles like protons, neutrons, electrons may have a tangled sub-particles content of many higher frequencies realities particles! That is such triads that create for example the current visible material reality ( 3rd density reality, or 1st frequency micro-resolution reality) already contain structure , sub-particles and function of higher frequencies realities. This may explain the well known phenomenon in quantum mechanics of entanglement or correlation of particles. (see e.g. http://en.wikipedia.org/wiki/Quantum_entanglement   the reason is that waves in higher frequencies realities propagate much faster than light in the current 1st frequency physical reality) Which is as Einstein remarked a (spooky) non-local interaction that goes faster than light (in spite the fact that it is not practical  to transmit information). Still the quantum entanglement  is one more well known experimental proof that there exist higher frequencies material physical realities. 


Here we may remark that according to the philosophy of the ancient Greek atomic philosophers Democritus and Leukippus , the world started with a smooth flow of atoms in almost parallel motion, and without external intervention, there comes a time (blow-up time) that through mutual collisions it resulted in to turbulent flows and all sorts of material combinations appeared.
Here the initial atoms may be considered as created from the initial "sound" (compression wave of higher atomic resolution or higher frequency  reality).
We notice again that this approach of the origin of frequency or atomic resolution material reality is completely different from a big-bang theory. 

REMARK ABOUT THE USE OF THE TERM "AETHER" IN THE CURRENT BOOK BECAUSE THE TERM AETHER IS KNOWN TO BE DISCREDITED AT LEAST FOR A WHOLE CENTURY, WE NEED TO CLARIFY ITS RELEVANCY TO THE THEORIES OF FIELDS WHICH ARE WELL ACCEPTED.

1) SPECIAL AND GENERAL RELATIVITY ARE NOTHING MORE THAN MODELS OF AETHER FOR INERTIAL MASS, AND FOR THE UNIVERSAL ATTRACTION. ANY MODEL OF AETHER  AS CLASSICAL "FIELD" (GRAVITATIONAL OR ELECTROMAGNETIC ETC) THAT DOES NOT GET DEEPLY TO THE INDIVISIBLE FREE AND PERMANENT PARTICLES MATERIAL STRUCTURE OF THE FIELD ITSELF, IS INADEQUATE FOR A FULL MODEL OF AETHER AS WE SHALL SEE.  THE INDIVISIBLEs OR PARTICLES OF A FIELD ARE OBVIOUSLY NONE OF THE KNOWN PARTICLES, BUT STILL THEY CAN BE AS SIMPLE AS THE KNOWN TRIADS OF PROTON, NEUTRON ELECTRON ONLY AT RADICALLY SMALLER SIZE.

OTHER CLASSICAL MODELS OF AETHER ARE 
2) THE MAXWELL'S ELECTROMAGMETIC FIELD (OR ELECTROMAGNETIZED AETHER AS MAXWELL WAS CALLING IT) 
3) THE QUANTUM VACUUM THAT AS SUBSTRATUM GIVES RANDOMNESS TO MOTIONS OF PROTONS, NEUTRONS, ELECTRONS THROUGH THE SCHRODINGER WAVE EQUATION (DEBROGLIE-SCHROENDINGER-DIRAC'S AETHER)

A MODERN SUCCESSFUL UNIFYING THEORY OF "AETHER" AS A GASEOUS FLUID ,OR IN BETTER TERMS 2ND FREQUENCY MATERIAL PHYSICAL REALITY (SEE ALSO POST 6 ) SHOULD BE ABLE TO DERIVE ALL THE ABOVE 20TH CENTURY CONCEPTS OF ""AETHER" IN OTHER WORDS 
FROM THE CONCEPT OF 2ND FREQUENCY MATERIAL REALITY AND IN PARTICULAR FROM THE CONCEPT OF GASEOUS FLUID FROM 3 TYPES OF PERMANENT AND FREE PARTICLES (LIKE MICRO-ELECTRON, MICRO-PROTON, MICRO-NEUTRON) WE SHOULD BE ABLE TO DERIVE AS SPECIAL TYPE APPROXIMATIONS UNDER SIMPLIFYING ASSUMPTIONS.

1) THE LABORATORY MID-SCALE LINEAR CLASSICAL MAXWELL EQUATIONS OF THE ELECTROMAGNETIC FIELD (LABRATORY MID-SCALE INTERACTION OF CHARGED MATTER WITH "AETHER" )
2) THE MACROSCOPIC EINSTEINS AND NEWTONS EQUATIONS OF TH GRAVITATIONAL FIELD, AND EINSTEINS EQUATIONS OF SPECIAL RELATIVITY (MACROSCOPIC INTERACTION OF NEUTRAL  MATTER WITH "AETHER")

3) THE MICROSCOPIC SCHROENDINGERS EQUATIONS OF MOTION OF PARTICLES (MICROSCOPIC INTERACTION OF NEUTRAL  MATTER WITH "AETHER" )

THE USE OF THE TERM "AETHER" IN THE CURRENT RESEARCH IS AS A SUMMARY OF THE WELL ESTABLISHED PHYSICAL THEORIES OF 1) ELECTROMAGNETIC FIELD 2) SPECIAL RELATIVITY AND GRAVITATIONAL  FIELD 3) QUANTUM "VACUUM" SUBSTRATUM IN THE SCHROENDINGER WAVE EQUATION.
THE INTENDED UNIFICATION AND DERIVATION OF THE ABOVE FIELDS, IS WHEN WE CONCEIVE "AETHER" AS THE PLASMA STATE OF THE 2ND FREQUENCY MATERIAL REALITY (NOT THE PLASMA OF THE 1ST FREQUENCY MATERIAL REALITY).


THE NEXT TABLE IS NOT AN EXPERIMENTALLY  PROVEN FACT AS PHYSICAL REALITY. BUT IS IN THE REALM OF MOST PROBABLE SPECULATIVE HYPOTHESES, THAT MAY EXPLAIN THE MAXIMUM OF THE OBSERVABLE PHYSICAL PHENOMENA. IN THE SAME WAY THAT FOR DEMOCRITUS AND LEFKIPUS, THE ONTOLOGY OF THE PHYSICAL ATOMS AS MATERIAL INDIVISIBLE AND PARTICLES WAS A SPECULATIVE HYPOTHESES NOT AN EXPERIMENTALLY PROVEN FACT , WHICH NEVERTHELESS  WAS EXPLAINING THE MAXIMUM NUMBER OF OBSERVABLE PHENOMENA.  THE NEXT TABLE IS A HYPOTHESIS OF THE SAME NATURE. ONLY THAT NOW ATOMS ARE IN A LEVELED WAY, IN SUCCESSIVE RESOLUTIONS OF FREQUENCIES OF MATERIAL REALITIES. 


Natural name
Other  name (in the web) 
Usual name in the web
Other names in books
2nd frequency macro-resolution reality
1st density physical reality
1st dimension  reality
Many-worlds reality
1st frequency macro-resolution reality
2nd density physical reality
2nd dimension  reality
Aether-cosmic-ball
1st frequency micro-resolution reality
3rd density physical reality
3rd dimension physical reality
Material reality
2nd  frequency micro-resolution reality
4th density physical reality
4th dimension  reality
1st Aetherial reality
3nd  frequency micro-resolution reality
5th density physical reality
5th dimension  reality
2nd Aetherial reality
4th  frequency micro-resolution reality
6th density physical reality
6th dimension  reality
3rd Aetherial reality
5th  frequency micro-resolution reality
7th density physical reality
7th dimension  reality
4th Aetherial reality
6th  frequency micro-resolution reality
8th density physical reality
8th dimension  reality
5th Aetherial reality
7th  frequency micro-resolution reality
9th density physical reality
9th dimension  reality
6th Aetherial reality
8th  frequency micro-resolution reality
10th density physical reality
10th dimension  reality
7th Aetherial reality
9th  frequency micro-resolution reality
11th density physical reality
11th dimension  reality
8th Aetherial reality
10th  frequency micro-resolution reality
12th density physical reality
12th dimension  reality
The new-born reality during the 90s. It was created by "sound"-like waves in higher physical realities  emanating simulateneously from all black holes, at a frequrency between 500-700 terra Hz (the visible physical reality has a frequency 1-2 Terra Hz , in the infrared range. 
To these realities we
must add also their  
  dual made from 
 their antimatter
Some reliable 21st century links of inventions not possible to explain with the classical academic 20th century physics is the next Blog

We present here an axiomatic system of the Euclidean geometry, without the dificlties of the infinite,  where lines and planes have material atoms and (1st) ethereal atoms, so as to link the above ideas with the traditional mathematical thought and to make them more transparent and with better clarity.
It is a new integrity between thinking, feeling and acting in the physical worlds. It is an educational and conceptual revolution. For more in the Blog       thedigitalmathematics.blogspot.gr


Here is in the form a table, the comparison of some of the beliefs of our scientific-educational system, that seem to require abandoning and replacement with some hypotheses closer to truth.


BELIEFS OF SCIENTIFIC- EDUCATIONAL SYSTEM THAT DO NOT SEEM TRUE

  HYPOTHESES WITH WHICH THEY SHOULD BE REPLACED THAT SEEM CLOSER TO TRUTH



A1. PHYSICAL REALITY



A1. PHYSICAL REALITY
1)All matter starts with the triad of permanent and free particles of protons , electrons neutrons. In smaller scale there is nothing but vacuum. All celestial bodies are those that we can observe with electromagnetic waves (light).
1)Physical reality, continuous deeper, than protons , neutrons, electrons, with finer such triads , of permanent, free particles.  There are many such layers of matter (Called densities or dimensions  or aethers or frequencies , the term frequency being relevant to the frequency of the spin of the triads) . We may call the reality of protons,   neutrons electrons, the 1st frequency reality. And there are many invisible to us planets, and celestial bodies, in the higher frequency material realities.

2)The universe was  created by a Big Bang 13-14 billion years ago.

2) This current (not higher)  frequency visible physical reality most probably was created 21 billion years ago, in other words all protons , neutrons electrons, were created from , waves emanating simultaneously from all from black holes for a limited time. This cosmo-sphere in a vortex motion, contracts and expands, giving the impression of big-bang creation. If we include higher frequency realities the universe might be at least 21 trillion years old.

3)All that exists, is the cosmological manifold (cosmo-sphere) that we can observe with the light in astronomy.

3)All that we can observe with light is only one cosmo-sphere., in this frequency reality. There are more such , in this (not higher) frequency reality that we cannot observe with light in astronomy.


4) Nothing goes faster than light













5) The interior of earth is full of magma


4) There is no such a limitation in nature. In the same way that there is no limitation that in the air that bodies cannot move faster than the sound. The formula E=mc^2, which is essentially the main deduction of special relativity which has been experimentally verified, can be deduced from assumptions that derive the Lorentz transformations too but do not include neither constancy of the speed of light neither that nothing goes faster than light.




5) All planets, satellites and stars, from their way that are shaped and gradually cooled, are hollow inside , because outer cooling is much faster than we thing compared to inner cooling, at gaseous stages that the dimensions are still large. Earth may have at its center an interior bluish small gaseous sun. 

6) Moon is a natural celestial body, that was there, billion years ago.

6) Moon is too large to be a natural satellite of earth. It might have been a natural satellite of a quite larger planet.  The equality of its spin and orbital period with the spin of the sun, seems to be artificial rather than natural outcome. It seems to be an intelligent intervention in the solar system, far in the remote past. 

7) The salt of the oceans is part of the natural evolution

7) The salt in the oceans is 3.5% of them, and is the huge amount of 50 thousand trillion tones! If it was to spread this salt on the surface of earth it would have the height of a 40 store building  There is no such abundance of salt in the ground, that is in contact with the sea. If there was so much NaCl , in the ground, then the lakes too would be salty. It seems to be an intelligent intervention in the earthly ecology, far in the remote past. 

A2. BIOLOGICAL LIFE

A2. BIOLOGICAL LIFE

8)All life  emerged from inorganic matter, and evolves in the Darwinian   way through randomness

8)Life seems more reasonable that it was created, by spiritual creators  in higher frequency physical realities, and only later, by lowering of the frequency, it appeared, in this frequency  visible physical reality, in a few planets of the galaxies. Then it went one evolving also in the Darwinian way.

9) All plant, animal and human life on earth were created in the above way, exactly on earth

9)I is by fat more probable and reasonable that most of the ecological life of living planets among the galaxies , was transplanted and transported by intelligent advanced civilizations from other living planets,. This should holds also for all plant , animal and human life on earth.



10)The dinosaurs was a natural evolution of earthly animal life



10)It seems by far more probable that the dinosaurs were artificial animals, and artificial evolution  on earth, after the Permian extinction of mammals (about 220 million years ago). After the KT extinction of the dinosaurs about 70 million years ago, it is highly unreasonable that so many mammals, appeared again suddenly! It is by far more probable and reasonable that they were  transported on earth by intelligent advanced civilizations from other living planets, of the galaxies. 

11)There is no other intelligent life in our solar system, but on earth

11)If we include planets and satellites at higher invisible frequencies we have not really proof, that one of them in our solar system, does not have (higher frequency) intelligent human life. 

12)There is no other sentient life on earth that speaks logical language than the human.

12) Modern experts tend to believe that Dolphins, and Whales , are sentient animals, and they have logical  language with their whistling system.  







A3.HISTORY OF THE CIVILIZATION







A3. HISTORY OF THE CIVILIZATION

13) Homo sapience emerged not earlier than 150-200 thousand years ago.

13)If the above hypotheses are more correct then intelligent humanoid life on earth in smaller numbers might have existed existed billions and millions years ago.

14)Most probably homo-sapience is an evolution of the primates, which is a 6 million years old animal

14) Again if the previous hypotheses are more true then some animals , including some primates may have been  artificial animal created by genetic engineering , by intelligent humanoid  civilizations of the past. 

A4. HUMAN EXISTENCE

A4. HUMAN EXISTENCE

15)Human existence, is mortal. After death there is nothing more

15)Human existences are immortals. Only the physical body dies. 

16)There is an immortal human soul, but we live only one mortal life.

16)The human soul is immortal, and we live many lives, with the same evolving soul but different physical bodies, in different civilizations of various solar system.

17)There are no miracles. Humans cannot make miracles. They can make only scientific technology.
Or 
Only God can make miracles. And the humans cannot. They can  only in the name of a God.

17)In higher frequencies realities, humans might have the natural ability of miracles. But also in this 1st frequency visible reality, there is a limited ability  of rare smaller  miracles, especially by a particular minority of individuals, when the full inherited DNA is functioning. The historic "miracles" might not have been mere fantasy and lies. The higher the frequency a human being exists the more "Godly" it is. 



Here is also a relevant interesting video 



And also








AXIOMATIC SYSTEM OF MATERIAL ATOMIC  EUCLIDEAN GEOMETRY

The axiomatic system adopted here, is that of Hilbert axiomatic system for the Euclidean Geometry, with modifications so that there is no need for the abstraction of the infinite, and therea re only finite many points visible and invisible for the geometric lines.  (See e.g. http://en.wikipedia.org/wiki/Hilbert's_axioms )

THIS PROJECT IS UNDER THE NEXT PHILOSOPHICAL PRINCIPLES

1) CONSCIOUSNESS IS INFINITE. CONVERSELY  THE INFINITE IS A FUNCTION AND PROPERTY OF THE CONSCIOUSNESSES.

2) BUT THE PHYSICAL MATERIAL  WORLD IS FINITE.

3) THEREFORE MATHEMATICAL MODELS IN THEIR ONTOLOGY SHOULD CONTAIN ONLY FINITE ENTITIES AND SHOULD NOT INVOLVE THE INFINITE. 



THIS PROJECT THEREFORE IS CREATING AGAIN THE BASIC OF MATHEMATICS AND ITS ONTOLOGY WITH NEW AXIOMS THAT DO NOT INVOLVE THE INFINITE AT ALL.




Our perception and experience of the reality, depends on the system of beliefs that we have. In mathematics, the system of spiritual beliefs is nothing else than the axioms of the axiomatic systems that we accept. The rest is the work of reasoning and acting. 

The abstraction of the infinite seems sweet at the beginning as it reduces some complexity, in the definitions, but later on it turns out to be bitter, as it traps the mathematical minds in to a vast complexity irrelevant to real life applications.

We may visualize euclidean space holographic representation simply confines all straight lines and planes within a spherical ball (like atmosphere of a planet), without applying the above transformation. 

1) All visible points (or low precision points ) are finite in number. And of non zero but minimum possible dimension in the single or Low precision, but not in the double or High precision. Between two visible points there is not always another visible point.

2) All invisible points or pixels or atoms (or high precision points ) are finite in number and of minimum dimension in the single and double precision.


The axiomatic system adopted here, is that of Hilbert axiomatic system for the Euclidean Geometry, with modifications. (See e.g. http://en.wikipedia.org/wiki/Hilbert's_axioms )

The idea so as to keep the tradition of "infinite"  extended straight lines , is to utilize when drawing and visualizing the geometric lines and planes, the above transformation. The intended functionality , is that the visible points close to the observer, must be detailed , have less invisible points and possible to manipulate by the "hand" while the very distant points are only visually observed and can have few only details (the visible points are large and have many invisible points). Therefore we do not describe and formulate here the  physical atomic space, itself, but an holographic optical representation of it with pixels (invisible points) and visible points.
Surfing among the euclidean figures of this geometry , is like turning pages in a e-book of a touch-screen mobile or i-pad. 

Quote: "It is not the world we experience but our perception of the world"

Nevertheless we give also an alternative formulation of the Euclids axiom of parallels, which corresponds to a non-optically distorted holographic representation of the Euclidean space, but non- distorted and assumed confined within a very large spherical ball. 


Once the resolution (double precision) of the geometry is fixed, the new axiomatic system is categorical (it has only one model, or all models are isomorphic), as is also the case with the classical infinite-points Euclidean Geometry.


We make here some small modifications of the Hilbert axioms of synthetic visual (non analytic or Cartesian with coordinates ) Euclidean  Geometry. Some of the axioms of Hilbert will not hold, (like that which claims that  between  two points here is always a third) , and some new initial concepts will be added, like that of two types of points, plus some relevant axioms. 

We may notice in the next axioms, that although we postulate, some geometric structure , for the visible part of the space, we do not postulate any geometric structure e.g. of the invisible points inside a visible point. We do not in particular postulate the invisible points being in a rectangular grid, or in hexagonal grid etc. They my be randomly placed, or even moving, exactly as the atoms in physical matter, or the molecules of air or the particles of aether. 
This in particular mean that when we represent the real numbers on the points of a geometric line, the high precision digital real numbers become invisible points of the line, and inherit structure and order, that the axiomatic geometry initially does not give to the invisible points. 


To enjoy in a preliminary way , what the realization of the geometry as digital geometry (e.g. with visual representations in a  computer screen), we may watch the next digital geometric animation , of the dynamics of the score, of a musical piece by Chopin (Nocturne) 

As well as some Beethoven, in digital lines and circle

Or some Chopin again with digital circles and digital geometric transformations 

Finally some Bach (Toccata and Fugue)


Without a digital ontology of lines, circles etc, the previous animation of the music, would not be possible. 

As I am a computer programmer too, besides being a mathematician, it became easier for me to , find out the necessary changes of the axioms of traditional mathematics, to derive axioms for the digital mathematics. 

All of the euclidean geometry, can be , defined axiomatically through the initial concepts of belonging εequality = (congruence) and order < 

We show also how this axiomatic formulation of Digital Euclidean Geometry links with the theory of cellular automatons. We define appropriate cellular automatons, for the pixels (invisible points) and visible points transformations of e.g. a Euclidean plane. 





AN AXIOMATIC SYSTEM FOR A PHYSICAL OR DIGITAL BUT CONTINUOUS  3-DIMENSIONAL EUCLIDEAN  GEOMETRY , WITHOUT  INFINITE MANY POINTS.

Konstantinos E. Kyritsis*
* Associate Prof. of TEI of Epirus Greece. ckiritsi@teiep.gr
C_kyrisis@yahoo.comTEI of Epirus, Dept Accounting-Finance Psathaki Preveza 48100


Abstract
This paper is concerned with finding an axiomatic system, so as to define the 3-dimensional Euclidean space, without utilizing the infinite ,that can imply all the known geometry for practical applied sciences and engineering applications through computers , and for more natural and perfect education of young people in  the Euclidean geometric thinking. In other words by utilizing only finite many visible and invisible points and only finite sets, and only real numbers with finite many digits, in the decimal representation. The inspiration comes from the physical matter , rigid, liquid and gaseous, which consists of only finite many  particles in the physical reality. Or from the way that continuity is produced in a computer screen from only finite many invisible pixels . We present such a system of axioms and explain why it is chosen in such a way. The result is obviously not equivalent, in all the details, with the classical Euclidean geometry.  Our main concern is consistency and adequacy but not independence of the axioms between them. It is obvious that within the space of a single paper, we do not attempt to produce all the main theorems of the Euclidean geometry, but present only the axioms.
Key words: Axiomatic systems of Euclidean geometry, Digital Mathematics, Digital space, Constructive mathematics, Non-standard mathematics.
Mathematical Subject Classification03F99, 03H99, 93C62, 51M05






§1. INTRODUCTION.
Changing our axiomatic system of the Euclidean geometry so as as to utilize only finite points, numbers and sets, means that we change also our perception our usual mental images and beliefs about the reality. This project is under the next philosophical principles
1) Consciousness is infinite. Conversely the infinite is a function and property of the consciousnesses.
2) But the physical material world is finite.
3) Therefore mathematical models in their ontology should contain only finite entities and should not involve the infinite. 
This paper is part of larger project which is creating again the basic of mathematics and its ontology with new axioms that do not involve the infinite at all.
Our perception and experience of the reality, depends on the system of beliefs that we have. In mathematics, the system of spiritual beliefs is nothing else than the axioms of the axiomatic systems that we accept. The rest is the work of reasoning and acting. 
Quote: "It is not the world we experience but our perception of the world"
The abstraction of the infinite seems sweet at the beginning as it reduces some complexity, in the definitions, but later on it turns out to be bitter, as it traps the mathematical minds in to a vast complexity irrelevant to real life applications. Or to put it a more easy way, we already know the advantages of using the infinite but let us learn more about the advantages of using only the finite, for our perception, modelling and reasoning about empty space. This is not only valuable for the applied sciences, through the computers but is also very valuable in creating a more perfect and realistic education of mathematics for the young people. The new axioms of the Euclidean geometry create a new integrity between what we see with our senses, what we think and write and what we act in scientific applications.
The Euclidean geometry with infinite many points creates an overwhelming complexity which is very often irrelevant to the complexity of physical matter. The emergence of the irrational numbers is an elementary example that all are familiar But there are less known difficult problems like the 3rd Hilbert problem (see [Boltianskii  V. (1978)“]). In the 3rd Hilbert problem it has been proved that two solid figures that are of equal volume are not always decomposable in to an in equal finite number of congruent sub-solids! Given that equal material solids consists essentially from the physical point of view from an equal number of sub-solids (atoms) that are congruent, this is highly non-intuitive! There are also more complications with the infinite like the Banach-Tarski paradox (see [Banach, StefanTarski, Alfred (1924)]) which is essentially pure magic or miracles making! In other words it has been proved that starting from a solid sphere S of radius r, we can decompose it to a finite number n of pieces, and then re-arrange some of them with isometric motions create an equal sphere S1 of radius again r and by rearranging the rest  with isometric motions create a second solid Sphere S2 again of radius r! In other words like magician and with seemingly elementary operations we may produce from a ball two equal balls without tricks or “cheating”. Thus no conservation of mass or energy!. Obviously such a model of the physical 3-dmensional space of physical matter like the classical Euclidean geometry is far away from the usual physical material reality! I have nothing against miracles, but it is challenging to define a space that behaves as we are used to know. In the model of the 3-dimesional space, as  new axiomatic system where such balls have only finite many points such “miracles” are not possible!

The continuous 3-dimensional space, defined axiomatically, is closer to what we know from the continuity of matter and fluids in physical reality, and strictly logically different from the traditional Euclidean space of infinite many points. It is not only the Hilbert’s 3rd problem, and the Banach-Tarski paradox which do not hold anymore for the physical or digital 3-dimensional Euclidean space, but also elementary topics like the constructability with ruler and compass.
We know e.g. that the squaring of the circle is not constructible with ruler and compass in the classical Euclidean Geometry with infinite many points. Because it involves the solution of the equation πR^2=x^2,                                                                                      (eq. 1)
and the number π is a transcendental irrational number. But in the digital Euclidean space E3(n,m,q) the (eq. 1) becomes the equation of rational numbers

[π]mR^2=m[x]m^2,                                                                                                       (eq.2)
Where by []m  we denote the truncation of a real number of infinite decimal points to m only decimal points in the precision level P(m), and by =the equlity within the precision level P(m).
And so the constructability with ruler and compass of squaring of the circle must be put  together with the next two facts
1)      Equation (eq.2)  is an equation of rational numbers
2)      Rational numbers, that is of the form k/l (k, l positive integers) , are constructible with ruler and compass (as linear segments in a line with a unit length)
Still we should not jump in to conclusions. General rational numbers of the form k/l as above may not necessarily belong to the precision level P(m). So it might be necessary to resort to a  higher precision level digital geometric space  E3(n’,m’,q’), n’>>n, m’>>m, q’>>q , make the construction with ruler and compass, and then return back to the lower precision level space E3(n,m,q), to construct a square with equal area with the initial circle.

We shall not only describe a new axiomatic system of the Euclidean geometry but also new axiomatic system of the natural numbers and real numbers , where only finite many numbers with finite many decimal digits are involved. Actually we could start in the meta-mathematics with new axioms and definitions of 1st order and 2nd order formal Logic where only finite many symbols, finite many natural numbers and proofs with finite only steps are involved. But we have not sufficient space for this in this paper, so we shall start only from the natural numbers.
We present such a new system of axioms and explain why it is chosen in such a way. The result is obviously not equivalent, in all the details, with the classical Euclidean geometry.  Our main concern is consistency and adequacy but not independence of the axioms between them. It is obvious that within the space of a single paper, we do not attempt to produce all the main theorems of the Euclidean geometry, but present only the axioms. The next step is obviously to define a digital differential and Integral calculus over such a digital Euclidean geometry and digital real numbers without convergence of infinite sequences or limits. But again this is not for the space of the current paper but probably of a future such paper.
The next presentation of such an axiomatic system is a design of logically organized realistic thinking in the area of numbers and space. It is also a realistic ontology of an operating system for numbers and space, for all practical scientific and engineering applications.
§2. The new axioms
As I am a computer programmer too, besides being a mathematician, it became easier for me to, find out the necessary changes of the axioms of traditional mathematics, so as to derive axioms for the digital mathematics.  
The axiomatic system adopted here, is that of Hilbert axiomatic system for the Euclidean Geometry, with modifications. (See e.g. http://en.wikipedia.org/wiki/Hilbert's_axioms )
Surfing among the euclidean figures of this geometry , is like turning pages in a e-book of a touch-screen mobile or i-pad.
We make here some small modifications of the Hilbert axioms of synthetic visual  Euclidean  Geometry. Some of the axioms of Hilbert will not hold, (like that which claims that  between  two points here is always a third) , and some new initial concepts will be added, like that of two types of points visible and invisble, plus some relevant axioms. 

I do not claim here that the axioms of the Digital Euclidean Geometry, below, are independent, in other words none of them can be proved from the others. As the elements are finite, there may be such a case. But I am strongly interested a) at first that are non-contradictory, and b) second that are adequate many, so as to describe the intended structure. later simplified and improved in elegance versions of the axioms may be given. 
Before we proceed we remind the properties of the axiomatic digital natural numbers and axiomatic decimal digital real numbers, where again no infinite exists.


§2.1   SIMILAR TO PEANO, AXIOMS

We define the natural numbers in two scales (and later precision levels) that are two unequal initial segments of the natural numbers N(ω)< N(Ω) .The number ω  is called the  Ordinal size  ω of the local system of natural numbers Ν(ω) while the Ω is the cardinal size of the global system of natural numbers. ω<Ω. If we start with integers n1, n2, n3 from  N(ω), then their addition and multiplication, have the commutative semiring properties but without closure in N(ω), but with values in N(Ω). We call the N(ω), the local segment while the  N(Ω) the global segment. 
We have here an initial relation among the natural numbers which is called successor or next of a natural number x and it is denoted by S(x).

1) The number 1 is a natural number and belongs both to N(ω), and N(Ω)  .
2) There is no natural number whose successor is 1.
3) If x is a natural number of N(ω), its successor S(x) , is also a natural number belonging in N(Ω).
4) If two different numbers of N(Ω) , have the same successor, then they are equal,  Formally if S(x)=S(y) then x=y . 
5) (Peano axiom of induction)  If a property or formal proposition P()  holds for 1 (that is P(1)=true) and if when holding for x in N(ω) holds also for P(S(x)) with S(x) in N(Ω) , then it holds for all natural numbers of N(ω).

6) Axiom of sufficient large size. If we repeat the operations of the commutative semiring starting from elements of the local version N(ω), ω-times, the results are still inside the larger set N(Ω).

This last Peano axiom of induction is useful only if the natural numbers are formulated within a formal logic (the axiom itself as a formal proposition is in 2nd order formal logic) that its size Ω(l) is less than the size of the objective system of natural numbers Ω. Otherwise for sufficient large 
Ω(L)>>Ω, we may simply construct a lengthy proof of this axiom starting from P(1) then P(2) ...and finally P(Ω), which then it is a theorem.

Any two models Μ1 Μ2 of the digital natural numbers Ν(Ω),Ν(ω) of equal size ω, Ω are isomorphic. 

§ 2.2 THE AXIOMATIC MULTI-PRECISION DECIMAL DIGITAL REAL NUMBERS R(n,m,q).
 


a)      The rational numbers Q, as we known them, do involve the infinite, as they are infinite many, and are created with the goal in mind that proportions k/l of natural numbers k,l exist as numbers and are unique. The cost of course is that when we represent them with decimal representation they may have infinite many but with finite period of repetition decimal digits.
b)      The classical real numbers R, as we know them, do involve the infinite, as they are infinite many, and are created with the goal in mind that proportions of linear segments of Euclidean geometry, exist as numbers and are unique (Eudoxus theory of proportions). The cost of course finally is that when we represent them with decimal representation they may have infinite many arbitrary different decimal digits without any repetition.
c)      But in the physical or digital mathematical world, such costs are not acceptable. The infinite is not accepted in the ontology of mathematics (only in the subjective experience of the consciousness of the scientist). Therefore in the multi-precision digital real numbers, proportions are handled in different way, with priority in the Pythagorean idea of the creation of all numbers from an integral number of elementary units, almost exactly as in the physical world matter is made from atoms (here the precision level of numbers in decimal representation) and the definitions are different and more economic in the ontological complexity.

We will choose for all practical applications of the digital real numbers to the digital Euclidean geometry and digital differential and integral calculus, the concept of a system of digital decimal real numbers with three precision levels, lower, low and a high.

Definition 2.2
 The definition of a  PRECISION LEVEL P(n,m) where n, m are natural numbers , is  that it is  the set of all real numbers that in the decimal representation have not more than n decimal digits for the integer part and not more than m digits for the decimal part. Usually we take m=n. In other words as sets of real numbers it is a nested system of lattices each one based on units of power of 10, and as union a lattice of rational numbers with finite many decimal digits. We could utilize other bases than 10 e.g. 2 or 3 etc, but for the sake of familiarity with the base 10 and the 10 fingers of  our hands we leave it as it is. 

THE AXIOMS OF THE DIGITAL REAL NUMBERS R(n,m,q)

We assume at least three precision levels for an axiomatic decimal system of digital real numbers R(n,m,q) : THE LOCAL LOWER PRECISION LEVEL P(n) , THE LOW PRECISION LEVEL P(m) , AND THE HIGH PRECISION LEVEL P(q). Each precision level of order k has 10^(4k+2log2) points or numbers where by log we denote the logarithm with base 10.  It has 10^k positive decimal numbers , which are doubled for the negative ones, thus in total at most 2*(10^(2k)) <= 10^(2k+log2). And again so many for the integer numbers, thus in total 10^(4k+2log2)). Now if for 3-dimensional geometric applications as coordinates of points, this will give 10^(12k+6log2)) points inside a big cube. 

Whenever we refer to a real number x of  a (minimal in precision levels) system of real numbers R(n,m,q) , we will always mean that x belongs to the local lower precision level P(n) and that the system R(n,m,q) has at least three precision levels with the current axioms.
Whenever we write an equality relation  =m we must specify in what precision level it is considered. The default precision level that a equality of numbers is considered to hold, is the low or standard precision level P(n).

Some of the Linearly ordered Field operations
The field operations in a precision level are defined in the usual way, from the decimal representation of the numbers. This would be an independent definition, not involving the infinite.  Also equality of two numbers with finite decimal digits should be always specified to what precision level. E.g. if we are talking abut equality in P(m) we should symbolize it my =, while if talking about equality in P(q) we should symbolize it by = .If we want to define these operation from those of the real numbers with infinite many decimal digits, then we will need  the truncation function [a]x  of a real number a , in the Precision level P(x). 
Then the operations e.g. in P(n) with values in P(m) n<<m would be

[a]n+[b]n=m[a+b]m                                                                                                        (eq. 3)

[a]n* [b]n=m[a*b]m                                                                                                        (eq. 4)

([a]n)^(-1)=m[a^(-1)]m                                                                                 (eq. 5)

(Although, the latter definition of inverse seems to give a unique number in P(m), there may not be any number in P(m) or  not only one number in P(m), so that if multiplied with [a]it will give 1. E.g. for n=2 , and m=5 , the inverse of 3, as  ([3]n )^(-1)=[1/3]m =0.33333 is such that still 0.33333*3≠m1 ).

Such a system of double or triple precision digital real numbers, has closure of the linearly ordered field operations only in a specific local way. That is If a, b belong to the Local Lower precision, then a+b, a*b , -a, a^(-1) belong to the Low precision level, and the properties of the linearly ordered commutative field hold: (here the equality is always in P(m), this it is mean the =m).
1) if a, b, c belong to P(n) then (a+b), (b+c),  (a+b)+c, a+(b+c) belong in P(m) and
 (a+b)+c=a+(b+c) for all a, b and c in P(m).
2) There is a digital number 0 in P(m) such that
2.1) a+0=a, for all a  in P(m). 
2.2) For every a  in P(n)  there is some b in P(m) such that 
   a+b=0. Such a, b is symbolized also by -a , and it is unique in P(m).
3) if a, b, belong to P(n) then (a+b), (b+a),   belong in P(m) and
 a+b=b+a 
4) if a, b, c belong to P(n) then (a*b), (b*c),  (a*b)*c, a*(b*c) belong in P(m) and
 (a*b)*c=a*(b*c).
5) There is a digital number 1 in P(m) not equal to 0 in P(m), such that
5.1) a*1=a, for all a  in P(m).
5.2) For every a  in P(n) not equal to 0,  there may be one or none or not only one  b in P(m) such that   a*b=1 . Such  b is symbolized also by 1/a, and it may not exist or it may not be unique in P(m).
6)  if a, b, belong to P(n) then (a*b), (b*a),   belong in P(m) and
  a*b=b*a 
7)  if a, b, c belong to P(n) then (b+c) , (a*b), (a*c),  a*(b+c), a*b+a*c, belong in P(m) and
 a*(b+c)=a*b+a*c 

Which numbers are positive and which negative and the linear order of digital numbers is precision levels P(n), P(m), P(q) is something known from the definition of precision levels in the theory of classical real numbers in digital representation. 


If we denote by PP(n) the positive numbers of P(n) and PP(m) the positive numbers of P(m) then

8) For all a  in PP(n), one and only one of the following 3 is true

8.1) a=0
8.2) a is in PP(n)
8.3) -a is in PP(n)  (-a is the  element such that a+(-a)=0 )

9) If a, b are in PP(n), then a+b is in PP(m)
10)  If a, b are in PP(n), then a*b is in PP(m)

It holds for the  inequality a>b if and only if a-b is in PP(m)
a<b if b>a
a<=b if a<b or a=b 
a>= b if a> b or a=b 

and similar for PP(m).

Similar properties as the ones from P(n) to P(m) hold if we substitute n with m, and m with q. 

Also, the Archimedean property holds only recursively in respect e.g.  to the local lower precision level P(n).
In other words, if a, b, a<b belong to the Local lower precision level P(n) then there is n integer in the Low precision level P(m) such that a*n>b. And similarly for the precision levels P(m) and P(q).

The corresponding to the Eudoxus-Dedekind completeness in the digital real numbers also is relative to the three precision levels.
We define that two visible points A, B, are in contact or of zero distance distance(A,B)=0, if and only if in their Cartesian coordinates they are at a face , at an edge or at a  vertice successive. If this is so then there are invisible points A’ belonging to A (see axioms of incidence) and B’ belonging to B, so that distance(A’.B’)<=1/(10^2q). Two visible points in contact do not have in general the same Cartesian measures distance The distance of the invisible points is defined from the coordinates of the invisible points in the precision level P(q) of R(n,m,q) from the standard formula of Euclidean distance , that is a Cartesian measure as in  Definition 2.3.I.2   or with the Archimedean measures but the values are identical in the standard or low precision level P(n).

In other words for every visible point A  in the Low precision level , there are exactly two other points B1, B2 again in the Low precision level with B1<A<B2 , such that the distance between A and B1, and A, B2  is zero in the Low precision level, and there is no other visible point C strictly between A and B1 and a and B2. This can be derived also from the requirement that all possible combinations of decimal digits in the local lower, low and high precision levels are being used as numbers of the system of digital real numbers. 


Sufficient Mutual inequalities of the precision levels 
We impose also axioms for the sufficiently large size of the high precision level relative to the other two, and the sufficient large size of the low precision level relative to the local lower precision level. That is for the mutual relations of the integers m, n, q.
It may seem that these differences of the resolution or the precision levels are very severe and of large in between distance, and not really necessary. It may be so, as the future may show. But for the time being we fell safe to postulate such big differences.


1) If we repeat the operations of addition and multiplication of the linearly ordered commutative field starting from numbers of the local lower precision level P(n), so many times as the numbers of the local lower precision level P(n), then the results are still inside the low precision level P(m). (This in particular gives that (10^n)^(10^n)<=(10^m)). This may also be expressed by saying that the 10^(m) is seemingly infinite compared to the 10^(n)

2) The largest error in the high precision level P(q), which we may also identify as the smallest magnitude in the low precision level P(m) in other words the 10^(-m), will appear as zero error in the low precision level P(n),  even after additive repetitions that are as large as  the cardinal number of points of the lower precision level P(n). This is e.g. is guaranteed if 5n+2log2<m or rounded 6n<m (Where by log we denote the logarithm with base 10) . The points in 1-dimensional geometry are 10^(4n+2log2) and if an error of order 10^(-m) is repeated so many times and still be less than 10^(-n), then 10^(4n+2log2)*10^(-m)<10^(-n), thus 5n+2log2<m. For the Euclidean geometry cube, this requires that 10^(12n+6log2)*10^(-m)<10^(-n) thus 13n+6log2<m or rounded 14n<m. 
 This may also be expressed by saying that the 10^(-m) is seemingly infinitesimal compared to the 10^(-n)

3) The smallest magnitude in the high precision level P(q) in other words the 10^(-q), will appear as zero error in the low precision level P(m),  even after additive repetitions as large as the cardinal number of points of the low precision level P(m). This is e.g. guaranteed if 5m+2log2<q or rounded 6m<q, and for Euclidean geometry applications 13m+6log2<q or rounded 14m<q. 
This may also be expressed by saying that the 10^(-q) is seemingly infinitesimal compared to the 10^(-m)


If instead of three precision levels P(n), P(m), P(q), we would introduce four precision levels (still another (P(r)), with the same mechanism of recursive axioms, then we would denote it by R(n,m,q,r) and we would call it a 4-precisions levels system of digital real numbers.

Two digital systems of Real numbers R(n,m,q) , R(n’,m’,q’) with n=n’, m=m’, q=q’ and the above axioms are considered isomorphic.


§2.3 AN AXIOMATIC SYSTEM OF THE PHYSICAL OR DIGITAL BUT CONTINUOUS  3-DIMENSIONAL EUCLIDEAN GEOMETRY  E3(n,m,q).

We have as initial concepts of objects 

a) The High resolution or precision  points, or invisible points  or atoms 

b) The Low resolution or precision points, or visible points or pixels. 
 

c) The Lower or standard precision level of measurements.

Remark 2.3.1
We introduce in the digital Euclidean geometry the next two types of points:

1) All visible points (or low precision level  points ) are finite in number. And of non zero but minimum possible dimension in the single or Low precision, but not in the High precision level. Between two visible points there is not always another visible point. The case of non-existence on intermediate points will be used in the concept of completeness up to some density or resolution and continuity of the space. Visible points are called visible in our usual material realizations of geometric figures because if we put our eyes close enough to the paper surface or screen where a line or a circle is drawn, we can see the point, while at a normal distance we cannot see the points but only the linear segment or circle arc. E.g. pixels of lines on the computer screen. Nevertheless the smallest magnitude of the standard precision level is by far larger than the visible points.

2) All invisible points or pixels or atoms (or high precision level points ) are finite in number and of minimum dimension in the high  precision level. Invisible points are called invisible in our usual material realizations of geometric figures because no matter how close we may put our eyes to the  paper surface or screen where a line or a circle is drawn, we cannot see these points. E.g. atoms of a metallic material surface. The main reason of introducing here the invisible points is so as to have at least two alternative systems of measures (lengths, areas, volumes), that of Archimedes and that of Cartesius. The full significance of the invisible points will become apparent only when introducing digital curved space like digital Riemannian space or manifolds, which is not in the scope of the current paper. 


For the AXIOMATIC DIGITAL OR PHYSICAL EUCLIDEAN GEOMETRY we do not intent to use the system of Real numbers as it is defined as the minimal complete linearly ordered commutative field (in the order to topology), but instead all measurements of linear geometric segments lengths, areas, volumes etc will be done with a Low Precision level and a high precision level of real numbers. The definition of a  PRECISION LEVEL P(n,m) where n, m are natural numbers , is  that it is  the set of all real numbers that in the decimal representation have not more than n decimal digits for the integer part and not more than m digits for the decimal part. In other words as sets of real numbers it is a nested system of lattices each one based on units of power of 10, and as union a lattice of rational numbers with finite many decimal digits. We could utilize other bases than 10 e.g. 2 or 3 etc, but for the sake of familiarity with the base 10 and the 10 fingers of  our hands we leave it as it is. 

Whenever we refer to a real number x of  as (minimal in precision levels) system of real numbers r we will always mean that x belongs to the local lower precision level and that the system has at least three precision levels with the current axioms.

Whenever we write an equality relation we must specify in what precision level it is considered. The default precision level that a equality of numbers and geometric elements of geometric figures like length, area and volume, is considered to hold, is the standard or lower precision level


c) The visible lines

d) The visible planes

c) We may apply finite sets only on the points of the digital Euclidean geometry

d) And of course we may apply  digital formal logic to make arguments and proofs.

e) Besides the congruence  as equivalence relations we have the next   initial relations 
    among visible or invisible elements. 

An invisible point A belongs to a visible point B, denoted by A ε B

A visible point A    belongs to a line L  or  ,    denoted by A ε L

A visible point A   belongs to a plane P  or  ,    denoted by A ε P

 A line L belongs to a Plane P, denoted by  L ε P


A visible point A is between two visible points B, C. 

We design 7 groups of axioms

1) Of finite decimal coordinates
2) Of lengths, areas and volumes
3) Of  Incidence  
4) Of  Order 
5) Of Congruence 
6) Of Continuity
7) Of Resolution

In the next axioms the term point if we do not specify that it is invisible, refers to visible or low precision point. It has the minimum no-zero size (length) in the Low resolution real numbers that can be constructed on a geometric line, by the Cartesian coordinates as tiny cube, as we shall see.  We use the axioms of Hilbert, but we modify them and add more axioms.



I Axioms of finite decimal coordinates of points

1) Every  invisible point P has 3 numerical coordinates P(x1), P(x2) , P(x3) that are rational numbers that in decimal notation have finite many digits so many as the definition of the High measurement precision.
2)   Every  visible point P has 3 numerical coordinates P(x1), P(x2) , P(x3) that are rational numbers that in decimal notation have finite many digits so many as the definition of the Low measurement precision.
3) The density of the visible is uniform through-out  the spherical space. For every triad of decimal rational numbers of the low precision level P(m) , there is an invisible point with these coordinates.
4) The density of the invisible is uniform through-out  the spherical space. For every triad of decimal rational numbers of the high precision, there is an invisible point with these coordinates.


Remark 2.3.I.1
The visible and invisible points due to their orthogonal and  rectangular coordinates may be considered tiny little cubes. Then of course we may define their cross sectional length, area and volumes as 10^(-m), 10(-2m), 10^(-3m)  for the visible and
10^(-q), 10(-2q), 10^(-3q)  for the invisible points.

Definition 2.3.I.1  of the local lower LLS and low resolution LS finite sphere or  space. 
There is a central visible point O of the space with coordinates (0,0,0) such that all the visible and invisible points of the space that have distance at most ω of it, where  ω belongs in the P(n,n), and ω=10^n is called the  local lower resolution space or in short LLS. If we take the corresponding sphere from the center with coordinates (0,0,0) with all visible and visible points with radius Ω=10^m, is called the Low resolution space or LS.

Definition 2.3.I.2  Cartesian measures of length, areas and volumes
From the elementary Cartesian analytic geometry, we may define the distance of two points A(x1,y1,z1) B(x2,y2,z2) through the Pythagorean or Euclidean formula of distance (norm  with rule of parallelogram) . We may similarly define the area of three points not lying in a line, through the well known formula that is involving the determinant and their coordinates, and similarly for the 3-dimensional simplex or tetrahedron. Then we may define the area of finite sets of points that are in contact (see Definition xyz below) by triangulation with non-overlapping triangles. Similarly define the volume of finite sets of points that are in contact or connected (see Definition 2.3.VI.1 below) by simplicialization with non-overlapping tetrahedral (simplexes). Such measures of area, and volumes of finite sets of points that are in contact (connected) we call in the next the Cartesian measures of areas and volumes.

Remark 2.3.I.2
Notice that in the synthetic axioms that we introduce here we do not impose geometric structure to the invisible points, but only to the visible points. In other words we do not define invisible lines and invisible planes. But we could as well do so, from the coordinates of the invisible points and the standard equations of lines and planes in the analytic geometry.

 II Axioms of Archimedes measures of length, area, and volumes and compatibility with the coordinates


1) Every invisible point P, as belonging to a line L, has a non-zero length l(P) which is a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
2)  Every invisible point P, as belonging to a plane E, has a non-zero  area a(P) which is  a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
3)  Every invisible point P, has a non-zero  volume v(P) which is  a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
4) For every visible point P, there are a sets VIn(P) of invisible points of it, so that volume of the visible point is defined as the sum of the lengths, of the volumes  of the invisible points of the above sets correspondingly. These sets Vln(P) for the volume are not unique for the point P , but all the alternative such sets give the same values volume of the point, and the same for all visible points. 
5) For every visible point P, there are a sets LIn(P) , AIn(P) of invisible points of it, so that the length and are of the visible point is defined as the sum of the lengths, of the lengths and areas of the invisible points of the above sets correspondingly. These sets LIn(P) , AIn(P) and also their values for the lengths and areas are not unique for the point P, but depend and their values depend also, on the linear segment or plane correspondingly  that the point P is considered that it belongs.
6) The length of linear segment is defined as the sum of the lengths of its visible points that in their turn define a partition of the invisible points of the segment.  The length of the unit segment OA , with coordinates of O, (0,0,0) and (0,0,1) is equal to 1. (similarly by cyclic permutation of the coordinates and the other unit lengths from O).
7) The area of figure (set of visible point) is defined as the sum of the areas of all of its visible points that in their turn define a partition of the invisible points of the figure.  The length of the unit square OA-OB , with coordinates of O, (0,0,0) and (0,0,1), (1,1,1), (1,0,0)  is equal to 1 (similarly by cyclic permutation of the coordinates and the other unit squares)
8)  The volume of a figure is defined as the sum of the volumes  of its visible points that in their turn define a partition of the invisible points of the figure.  The volume of the unit cube  , with coordinates  (0,0,0) and (0,0,1), (1,1,1), (1,0,0) , (0,1,0) , (1,1,0), (0,1,1) ,(1,0,1)  is equal to 1
9) Congruent sets of points (of the LLS)  have length, area, and volumes either in the Cartesian measures or the Archimedean measures , correspondingly that differ only by errors that are zero in the standard or low precision level. Furthermore they remain zero error, even if are repeated additively as many times as the cardinal number of elements of the low precision level P(n).
10) For a finite connected set of visible points (of the LLS)  the difference of its measure  in the Cartesian measure and  the Archimedean measure , correspondingly  differ only by error that is zero in the standard or low precision level. Furthermore it remains zero error, even if it is repeated additively as many times as the cardinal number of elements of the low precision level P(n).




Remark 2.3.II.1

Both  types of measures Cartesian measures and Archimedes measures of, areas and volumes have the additive property of disjoint unions of finite sets of points in contact (connected sets of points see Definition 2.3.VI.1 ).

After the above axioms and definitions of such measures , it can be shown that the lengths, areas and volumes, are set functions l, a, v of sets of visible points, (but also of invisible points) ,with values in the positive Low precision level of decimal numbers, with the additive property of disjoint unions: (By ᴖ we denote the intersection and by ᴗ the union of sets)

l(A ᴗ B)=l(A)+l(B)-l(A ᴖ B)

a(A ᴗ B)=a(A)+a(B)-a(A ᴖ B)

v(A ᴗ B)=v(A)+v(B)-v(A ᴖ B)

AB congruent to A'B' then d(AB)=d(A'B')

Furthermore  more properties for linear segments AB, BC, AD hold like 

l(AB)=l(BA)
l(AC)<=l(AB)+l(BC)

Also angular measures ang() again with values in positive Low precision level are defined , through areas of circular sectors of unit circular discs or of the length of the corresponding circular segment of  unit circler discs. 

Axioms of Sufficient many points (visible and invisible) and mutual inequalities of the precision levels for length, areas and volumes. 
We impose also axioms for the sufficiently large size of the high precision level relative to the other two, and the sufficient large size of the low precision level relative to the local lower precision level. That is for the mutual relations of the integers m, n, q.
It may seem that these differences of the resolution or the precision levels are very severe and of large in between distance, and not really necessary. It may be so, as the future may show. But for the time being we fell safe to postulate such big differences.

The axioms are essential those of the digital real numbers R3(n,m,q) with numbers 1) 2) 3) as coordinates of the visible and invisible points of the digital Euclidean space E3(n,m,q).

In all the next axioms we start from visible points and geometric elements of the Local Lower resolution space LLS (which belongs in the coordinates cube P(n,m)) , and we result in to the Lower resolution space LS (which belongs in the coordinates cube P(m,m)), because of the recursive and not absolute closeness in the digital real numbers R(n,m,q). Angles in LLS are essentially circular sectors of length of radius equal to one unit.



III. Incidence

Terminology convention
1)      In all the axioms of incidence, order, and congruence below, when we say and write the term  “point” without specifying it to be an invisible point we will mean a visible point.
2)      when we say and write the term  “angle”, we will mean, a circular sector of unit radius.
3)      when we say and write the term  “line”, we will mean, a linear segment starting and ending at points of the Local Lower Sphere (LLS). The ending visible points of the line do not count as (interior) visible points of the line
4)      when we say and write the term  “plane”, we will mean, a circular disc , with boundary circle at surface points of the Local Lower Sphere (LLS). The boundary visible points of a plane do not count as (interior) points of the plane.

  1. For every two points A and B (in LLS) there exists a line a (in LLS) that contains them both. We write AB = a or BA = a. Instead of “contains,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common,” etc.
  2. For every two points (in LLS) there exists no more than one line (in LLS) that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
  3. There exist at least two points on a line (in LLS) . There exist at least three points that do not lie on a line.
  4. For every three points ABC (in LLS) not situated on the same line there exists a plane α (in LLS) that contains all of them. For every plane (in LLS) there exists a point which lies on it. We write ABC = α. We employ also the expressions: “ABC, lie in α”; “A, B, C are points of α”, etc.
  5. For every three points ABC (in LLS) which do not lie in the same line, there exists no more than one plane (in LLS) that contains them all.
  6. If two points AB of a line a (in LLS) lie in a plane α (in LLS) , then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc.
  7. If two planes α, β (in LLS) have an (interior) point A in common, then they have at least a second (interior) point B in common.
  8. There exist at least four points (in LLS) not lying in a plane.
9.      For every invisible point A, there is a visible point B, so that A belongs to B.
  1. Two invisible points A, B belong to the same visible point C is an equivalence relation among the invisible points.
IV. Order
  1. If a point B (of LLS) lies between points A and C (of LLS) B is also between C and A, and there exists a line containing the distinct points A,B,C.
  2. Of any three points situated on a line (of LLS) , there is no more than one which lies between the other two.
  3. Pasch's Axiom: Let ABC be three points (of LLS) not lying in the same line and let a be a line (of LLS) lying in the plane ABC and not passing through any of the points ABC. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.
V. Congruence
d)     If AB are two points on a line a, (of LLS), and if A′ is a point upon the same or another line a′  (of LLS), then, upon a given side of A′ on the straight line a′ , we can always find a point B′  (of LS) so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB  A′ B′. Every segment is congruent to itself; that is, we always have AB AB.
We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in at least one way (Always starting from the sphere LLS and resulting in the larger sphere LS).
  1. If a segment AB  (of LLS) is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB  A′B′ and AB  A″B″, then A′B′  A″B″.
Remark 2.3.V.1 Limited transitivity?
As we noticed in the axioms II.9 -II.10, congruent linear segments, and angles have equal measures with zero error in the standard precision level P(n), but non-zero in the low precision level P(m). Therefore in the transitivity of the congruence in the previous axiom, the error may be added and propagated. Still by the same axiom II.9 , the repletion may me s many times as the elements of the standard precision level P(n) and still be zero. Therefore we know that the transitivity of the congruence will still hold up to as many times as the number of the elements of the standard precision level P(n). Now if in the meta-mathematics of the formal logic we utilize the digital natural numbers N(ω) with ω=n, then certainly even the largest allowable number of formal propositions and therefore repetitions of the transitivity of congruence will not lead to an  non-zero error in the standard precision level. Therefore we may accept that the transitivity of the congruence is valid for all practical applications, although theoretically it is limited. Another way to keep the transitivity of the congruence is the next: We may define when modelling this axiomatic system to test the consistency, the congruence with a standard types transformation of the coordinates (e.g. isometric transformations). Then as the composition of two isometries are is an isometry, and the  error of an isometry  can be uniformly bounded for all isometries so as to be zero in the low precision level, the transitivity of the congruence is valid from the point of view of the standard precision level. 
  1. Let AB and BC be two segments of a line a (of LLS) which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ (of LS) having, likewise, no point other than B′ in common. Then, if AB  A′B′ and BC  B′C′, we have AC  A′C′.
  2. Let an angle ang (h,k) be given in the plane α (of LLS) and let a line a′ be given in a plane α′ (of LLS) . Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ (of LS) such that the angle ang (hk), or ang (kh), is congruent to the angle ang (h′k′) and at the same time all interior points of the angle ang (h′k′) lie upon the given side of a′. We express this relation by means of the notation ang (hk ang (h′k′).
  3. If the angle ang (hk) (of LLS) is congruent to the angle ang (h′k′) and to the angle ang (h″k″), then the angle ang (h′k′) is congruent to the angle ang (h″k″); that is to say, if ang (hk ang (h′k′) and ang (hk ang (h″k″), then ang (h′k′ ang (h″k″).
  4. If, in the two triangles ABC and A′B′C′ (of LLS) the congruencies
AB  A′B′, AC  A′C′, ang(BAC)  ang(B′A′C′) hold, then the congruence ang(ABC)  ang(A′B′C′) holds (and, by a change of notation, it follows that ang(ACB)  ang(A′C′B′) also holds).


VI. Continuity and Completeness up to some density or resolution, relative to the digital real numbers R(n,m,q).

The corresponding to the Eudoxus-Cartesius-Dedekind, completeness  also is relative to the three precision levels of R(n,m,q).
Definition 2.3.VI.1
We define that two visible points A, B, are in contact or of zero distance distance(A,B)=0 in P(n), if and only if in their Cartesian coordinates they are at a face , at an edge or at a  vertice successive. If this is so then there are invisible points A’ belonging to A (see axioms of incidence) and B’ belonging to B, so that distance(A’.B’)<=1/(10^2q). Two visible points in contact do not have in general the same Cartesian measures distance The distance of the invisible points is defined from the coordinates of the invisible points in the precision level P(q) of R(n,m,q) from the standard formula of Euclidean distance , that is a Cartesian measure as in  Definition 2.3.I.2   or with the Archimedean measures but the values are identical in the standard or low precision level P(n).


1.        Axiom of Digital Continuity and Completeness: For every non-ending visible point A of LLS , of a linear segment a, there are exactly two other divisible points B1, B2 on a  in LS, and with B1<A<B2 , such that the distance between A and B1, and A, B2  is zero, and there is no other visible point C strictly between A and B1 and a and B2. This can be derived also from the requirement that all possible combinations of decimal digits in the low and high precision levels are being used as numbers of the system of digital real numbers and correspond to visible and invisible points. 

Remark 2.3.VI.1 An alternative way that we could formulate the completeness of points is the next. An extension of a set of visible points on any  line , plane and the space , with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-VIII based on the given density of Coordinates in R(n,m,q), is impossible. In short we cannot add more visible points relative to the Low precision level of measurements and coordinates, and the same for the visible points and low precision level of coordinates. This comes also from the axiom of the density-completeness of all possible but finite many coordinates of points in R(n,m,q).





VII. Axioms of Resolution or of Density
These axioms are of the same nature  as the corresponding axioms  of the digital natural numbers, and multi-precision digital real numbers, and the axioms 1,2,3  of the digital real numbers.

1.      Axiom of sufficient high  resolution or density . Let a line a passing from the center of the space O and  the units of measurements  OA on it,  and let ω(a), Ω(a), demote the finite cardinal number which is the cardinal number of visible and invisible points that belong to a. Estimates of them are ω(a)=10^(4m+2log2) and Ω(a)=10^(4q+2log2) where by log we denote the logarithm with base 10. And let  ω(n) be the size of the model of the natural numbers constructed on the line a  through congruence and the above axioms. An estimate of it is ω(n)=10^(2n+log2) Then it holds that  
           ω(n) <= ω(a)  or 10^(2n+log2)<=10^(4m+2log2)  
                                       (Strong version of the axiom 2^ω(n) <= ω(a) ) 
           ω(a) <=Ω(a) or 10^(4m+2log2) <=10^(4q+2log2) 
                                        (Strong version of the axiom 2^ω(a) <=Ω(a) ) 
  1. .  Let ω(S), Ω(S), demote the finite natural numbers which are the cardinal numbers of visible and invisible points that can belong to the spherical S 3-dimensional space. And let also ω(P) be the cardinal number of invisible points that any  visible point may contain .  (From the group II of axioms about lengths, areas and volumes and axioms 8,9,10, we have an estimate that ω(S)<=10^(12m+6log2) and Ω(S)<= (12m+6log2) where by log we denote the logarithm with base 10, and m and q are the orders of the precision level of the visible and invisible points respectively and ω(P)<=10^(q-m))
  2. Then it holds that  
a) Any number of visible points of the total spherical space is less than any                      number of invisible points that a visible point may contain . In particular
            (ω(S)) <= ω(P) or 10^(12m+6log2)<=10^(q-m)  
            Strong version of the axiom 2^(ω(S)) <= ω(P)). 
       Remark 2.3.VII.1  The axiom must guarantees that lengths, areas and volumes that are defined by summing                   the corresponding values of the invisible points, will have in  general total errors zero in the low precision, and that the failure if the transitivity of relations congruence due to  their limited character can be avoided to occur, with sufficient high resolution of   invisible   points relative to the size of the total space and our repetitive construction in  it.
            b) The diameter of the total spherical 3--dimensional space in integer number of units of length denoted by ω(n) ,is less that any maximum number of   visible points that the  spherical space may contain. An estimate of ω(n) is ω(n)=10^(2n+log2).  In particular  
               ω(n) <=ω(S) or 10^(2n+log2)<=10^(12m+6log2)
              (Strong version of the axiom  2^ω(n) <=ω(S)).  
Remark 2.3.VII.2       The axiom  guarantees that a lattice of points with say integer coordinates will always be by far less dense than the lattice of visible points.

 Remark 2.3.VII.3        Notice that the inequalities of the current axioms of  resolution are stronger than those of axioms 1,2,3 of the digital real numbers.
Remark 2.3.VII.4.
Notice that we did not postulate anything similar to the axiom of parallel lines of Euclid! One reason is that the digital lines are eventually linear segments and do not extend to infinite, as this in the current setting would make them have infinite many points. Furthermore for this reason there are more than one linear segment passing from a point outside another linear segment which do not have any point in common. But this of course does not make the digital geometry a Lobachevskian or hyperbolic geometry. Still in the low precision level there would be only one linear segment that passing from  a point outside a linear segment so that the angles from a third crossing both linear segment have sum exactly equal to 180 degrees. The main reason that we did not postulate the axiom of parallels is that in the digital Euclidean geometry the propertis of such parallels in the low precision level  are  deduced from the axioms of the    coordinates. Furthermore they are already non-unique in the low precision level P(m), but only in the lower or standard precision level.  

Remark 2.3.VII.4        
Any digital space E3(n,m,q) is determined essentially from the integer parameters n,m,q of the corresponding digital system of real numbers R(n,m,q) which is used as coordinate system. To have that any two finite models of E3(n,m,q) are isomorphic, one has to define and model appropriately the incidence , order and congruence at first before defining the appropriate form of isomorphism of the models of E3(n,m,q).


§ 3 Conclusions

The axiomatic system of the digital Euclidean geometry E3(n,m,q) may seem complicated and elaborate compared to the Hilbert’s axioms of the classical Euclidean geometry. But as we remarked from the beginning the cost of the simple Hilbert’s axioms with infinite many points is paid later with overwhelming complexity, and many non-intuitive paradoxes like Hilbert’s 3rd problem, and Banach-Tarski paradox. In addition the theory of the measures of areas and volume that require integration and limits is  very complex. In contrast in the digital Euclidean geometry we may start with elaborate axioms, but later the theory of measures of areas and volumes is very simple and intuitive!
E.g. The areas of circular discs are calculated in the Cartesian measure of area, by triangulation with non-overlapping triangles of all the (finite many) points of the circular disc! And in the Archimedean measure of area the calculation is even simpler as the sum of the areas of the (finite many) points of the circular discs that are tiny rectangles. Integration is simple finite sums.
Similarly other elementary or non-elementary theorems can have easier proofs in the digital Euclidean geometry! We even have a new type of proofs  not possible in the classical Euclidean Geometry: Proof by induction on the number of (visible) points! It is quite interesting to re-formulate classical non-solved so far problems in the context of digital Euclidean geometry e.g. Riemann hypothesis of the roots of the zeta function , and try to prove it by induction on the number of visible points!


References
Banach-Tarski paradox:
Boltianskii  V. (1978)“Hilbert’s 3rd problem” J. Wesley & Sons 1978
Euclid The 13 books of the  Elements  Dover 1956
Hilbert  D. (1977)“Grundlangen der Geometrie” Taubner Studienbucher  1977
Hausdorf paradox:
Hausdorff Felix (1914). "Bemerkung über den Inhalt von Punktmengen"Mathematische Annalen75: 428–434. doi:10.1007/bf01563735
Moise E. E. (1963) “Elementary geometry from an advanced standpoint” Addison –Wesley 1963
Von Neumann paradox:
On p. 85 of: von Neumann, J. (1929), "Zur allgemeinen Theorie des Masses" (PDF), Fundamenta Mathematicae13: 73–116
Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the Banach–Tarski paradox" (PDF). Fundamenta Mathematicae138: 21–22.
Wilson, Trevor M. (September 2005). "A continuous movement version of the Banach–Tarski paradox: A solution to De Groot's problem". Journal of Symbolic Logic70 (3): 946–952. JSTOR 27588401doi:10.2178/jsl/1122038921.







APENDIX

A DIALOGUE OF THE IMMORTALS OF GEOMETRY

(
This is a fictional dialogue of the immortally famous mathematicians of the past that have significantly contribute to the mathematics of the Euclidean Geometry and comment on the new axiomatic system of the Axiomatic Digital Euclidean Geometry. The list is only indicative, not exhaustive. 

THE DIALOGUE OF THE IMMORTALS MATHEMATICIANS ON THE OCCASION OF THE NEW AXIOMS OF THE AXIOMATIC DIGITAL EUCLIDEAN GEOMETRY BY NEWCLID




NEWCLID after presenting the immortals the new axioms of the Axiomatic Digital Euclidean Geometry, invites them in a free discussion about it.
NEWCLID, is an individual representing the collective intelligence of the digital technology but also of mathematics of the 21st century.  

The participants of the discussion are the next 20.

1.     Pythagoras
2.    Aristarchus from Samos
3.    Eudoxus
4.    Euclid
5.    Democritus
6.    Archimedes
7.    Apollonius
8.    Copernicus
9.    Galileo
10. Newton
11. Leibnitz
12. Cartesius
13. Cauchy
14. Dedekind
15. Weierstrass
16. Hilbert
17. Riemann
18. Cantor
19. von Neumann
And a mortal:
20. Newclid

NEWCLID:
Welcome honourable friends that you have become immortals with your fame and contribution in the creation of the science and discipline of Mathematics among the centuries on the planet earth!
Now that you have watched my presentation of the axiomatic system of the new Axiomatic Digital Euclidean Geometry, I would like to initiate a discussion that will involve your remarks and opinions about it. Who would like to start the conversation?
PYTHAGORAS:
Thank you Newclid for the honour in gathering us together. I must express that I like the new approach of the Axiomatic Digital Euclidean Geometry, that as you say is a resume of what already the beginning of the 21st century in the earthly Computer Science has realized through software in the computer operating systems and computer screens and monitors.
I must say that I like the approach! In fact, I was always teaching my students that the integer natural numbers are adequate for creating a mathematical theory of the geometric space! One only has e.g. to take as unit of  measurement of lengths, the length of a visible points and all metric relations in the low precision level of the figures , including the Pythagorean theorem, become relations of positive integer numbers, or solutions of Diophantine equations! But at that time no such detailed and elaborate axiomatic system, neither a well accepted concept that matters consists from atoms, was available in the mathematicians of the ancient Greece, Egypt or Babylon.

EUCLID: I am impressed Newclid for your elaborate axiomatic system. The axioms that I had gathered in my books with title “Elements” for the Euclidean geometry in my time were much less! I would like to ask you a question that puzzles me since I watched your presentation: How do we know that the more than 20 axioms of Hilbert about my Euclidean geometry, or your axioms of the Digital Euclidean Geometry are enough to prove all that we want to prove?
NEUCLID: This is a very good question, Euclid! Maybe our friend here Hilbert might like to answer it!
HILBERT: Well my friends, this is a question that I posed also to myself when writing my more than 20 axioms of the classical Euclidean Geometry! I have not read any such proof! It is by the rule of the thump as they say! I collected them , through my experience and according to the theorems of Euclidean geometry till my time but also according to the standards of proofs in my time!
NEWCLID: What do you mean Hilbert? That maybe in the future we might discover that we need more axioms?
HILBERT: Exactly! That is what the History teaches us!
CARTESIUS: If I may enter the discussion here, I propose that a proof that the axioms of Hilbert are enough could be proving from the Hilbert axioms, the basic numerical axioms of my Analytic Geometry with coordinates! This, in my opinion, would be a proof!
NEWCLID: Very good idea Cartesius! This in my opinion suggests also that my axioms of the Digital Euclidean Geometry, that involve coordinates too, most probably are enough. But I am almost sure that they are not independent and some of them can be proved from the rest. Still I cannot claim that I have any proof, more than just experience and a rule of the thump, that my axioms are adequate! Maybe in the future I may discover that I need a couple more!
ARISTARCHUS: May I ask Newclid if your concept of digital Euclidean space which is in the shape of a spherical ball is intended to be large enough so as to allow e.g. astronomical calculations like my calculations of the size of earth, moon , sun and their mutual distances?
GALILEO: I have the same question Newclid! Good that ARISTARCHUS asked it!
COPERNICUS: Me too Euclid!
NEWCLID: Certainly ARISTARCHUS! The spherical digital Euclidean space can be so large so as to include all the observable galaxies of the astronomical world as we know it! But it can be also small as a planet to accommodate for planetary calculations only too! The axioms do not specify how large or small it should be!
ARCHIMEDES: I like your axiomatic system and concept of space Newclid! It is as my perceptions! Actually my experimental work with solids that I was filling with sand or water to make volume comparisons is just an experimental realisation of your axioms of volumes through those of the points and finite many points!
DEMOCRITUS: Bravo Newclid! Exactly my ideas of atoms! Actually as in my theory of atoms, the water is made from finite many atoms, the volume experiments of Archimedes with water is rather the exact realisation of your axioms of volume through that of the invisible points! Here the atoms of the water are invisible, while the granulation of the sand may resemble your axioms of the visible points!
NEWCLID: Thank you, my friends! I agree!
EUDOXOS: Well in your digital Geometry Newclid, my definition of the ratio of two linear segments which is the base of the complete continuity of the line is not that critical in your axiomatic system, although I thing that it still holds!
DEDEKIND: As I reformulated the idea and definition of equality of ratio of linear segments of Eudoxus, as my concept of Dedekind cuts about the completeness of continuity of the real numbers, I must say the same thing as Eudoxus!
WEIESSTRASSE: The same with my definitions of convergent sequences though the epsilon-and-delta formulation! They still hold in your approach!
APOLLONIUS: I would like to know Newclid, if my theory of circles in mutual contact would be provable as I know it in the classical Geometry of Euclid. E.g. if tow circles are in contact externally, are they in contact in one only point, as I know it, or in more than one point in your geometry?
NEWCLID: I think APOLLONIUS that in my geometry what you observe in the real world is also more or less what is provable with the visible points. For sure two circles in contact even if they have only one common visible point they will have many common invisible points, all those inside the common visible point! But I am afraid that they may even have more than one common visible point , depending on their size and the definition of circle intersecting circle or line.   The reason is that it may happen that two different visible points have an error of distance from the centers of the circles which is zero in the Low precision although not zero in the High precision. Still one may gibe an appropriate definition where one of them has a maximal property thus a unique point of contact.
GALILEO: I would like to ask Newclid if your concept of invisible and visible points could be large enough and both of them visible, so as to account for the real planet earth (which is not a perfect sphere) as if a perfect sphere!
NEWCLID: Well GALILEO, the initial intention is the invisible points are indeed small enough to be invisible. But as you understand what is visible and invisible is not absolute and depends at least on the closeness of our eye. Theoretically one could conceive a model of my axioms where both visible and invisible points are visible and even large!
LEIBNITZ: I want to congratulate you Newclid for your approach! In fact my symbols of infinitesimal dx in my differential calculus suggest what I had in mind: A difference dx=x2-x1 so that it is small enough to be zero in the Low precision but still non-zero in the High precision! Certainly a finite number!
NEWTON: I must say here that the Leibnitz idea of infinitesimal as a finite number based on the concepts of Low and High precision is not what I had in my mind when I was writing about infinitesimals. That is why I was calling them fluxes and symbolized them differently. The theory of null sequences of numbers (converging to zero) of Cauchy and Weierstrassse is I think the correct formulation of my fluxes. So that such fluxes fit to a Geometry as Euclid and Hilbert was thinking it and not as Newclid formulated here. Still for physical applications I thing that Newclid's concept of space with finite many points only is better and closer to the physical reality! I was believing in my time that matter consist from finite many atoms , but I never dared to make a public scientific claim of it, as no easy proof would convince the scientist of my time!
I want to ask an important question to Newclid: Is your differential and integral calculus based on three levels of precision more difficult or simpler that the classical differential and integral calculus based on limits and infinite many real numbers?
NEWCLID: Well Newton thank you for the good words! Actually I have not yet developed all of a differential and integral calculus based on digital real numbers and digital Euclidean geometry, therefore the question runs ahead of our presentation. But I have thought myself about it, and I can remark the next: A differential and integral calculus based on three levels of precision is certainly less complicated than ( and also not equivalent to ) the classical calculus with infinite sequences or limits. But a differential and Integral calculus of 3 ,4 or more precision levels is by far more complicated than the classical differential and Integral calculus. Only that this further complication is a complexity that does correspond to the complexity of the physical material reality, while the complexities of the infinite differential and integral calculus ( in say Lebesgue integration theory or bounded variation functions etc) is a complexity rather irrelevant to the physical material complexity.  
CARTESIUS: I want to congratulate you Newclid for your practical , finite but axiomatic too approach for the physical space, and the introduction of my idea of rectangular coordinates right from the beginning of the axioms! I have a question though! You correspond points to coordinates, but they also have volume. If we think of a cubic lattice with its points and coordinates, which of the 8 cubes that surround the point you assume as voluminous point in your geometry?
NEWCLID: If I understand your question well CARTESIUS, it is the cube that its left upper corner is the point. Thanks for your praise!
CAUCHY: I wish I had thought of such an axiomatic system of space with finite many only points, and the concept of infinitesimals as Leibnitz mentioned with your Local, Low and High precision levels! But there is a reason for this! Your axioms are much more elaborate and complicated that the Hilbert axioms of Euclidean Geometry!
NEWCLID: Indeed CAUCHY! But later the proofs of many other theorems, on areas, volumes and even derivatives, will become much simpler!
HILBERT: I like your brave and perfect approach Newclid! No infinite in your axioms so as to have easy physical applications, as nothing in the physical material reality is infinite. Congratulations! I am glad that my axioms of the classical Euclidean Geometry were of a good use to your work.
Von NEUMANN: I like tooyou axiomatic system Newclid! I believe that I could easily make it myself, except at that time I was busy in designing a whole generation of computers! I believe your works is a direct descendant of my work on computers. As you said your ideas came from software developers in the operating system of a computer!
NEWCLID: Indeed von Neumann! Thank you!
CANTOR: Pretty interesting your axiomatic system Newclid! But what is wrong with the infinite? Why you do not allow it in your axiomatic system? I believe that the infinite is a legitimate creation of the human mind! Your Digital Euclidean Geometry lacks the magic of the infinite!
PYTHAGORAS: Let me, Newclid, answer this question of CANTOR! Indeed CANTOR the human mind may formulate with a consistent axiomatic way what it wants! E.g. an axiomatic theory of the sets where infinite sets exist! And no doubt that the infinite is a valuable and sweet experience of the human consciousness! But as in the physical material reality there is nowhere infinite many atoms, mathematical models that in their ontology do not involve the infinite, will be more successful for physical applications! In addition there will not be any irrelevant to the physical reality complexity as in the mathematical models of e.g. of physical fluids that use infinite many points with zero dimensions in the place of the finite many only physical atoms with finite dimensions. The infinite may have its magic, but the axioms of the Digital Euclidean Geometry have their own and different magic!
RIEMANN: Very impressive Newclid your logical approach to the Euclidean space! But what about my Riemannian geometric spaces? Could they be formulated also with Local, Low and High precision levels and finite many visible and invisible points?
NEWCLID: Thank you Riemann! Well my friend any axiomatic system of your Riemannian Geometric spaces, with finite many points would require at least 3 or 4 precision levels! The reason is that at any A point of a Riemannian Space, the tangent or infinitesimal space at A is Euclidean! And here the interior of the point A will be a whole spherical Euclidean space which already requires two precision levels and both the visible and invisible points of the tangent Euclidean space will have to be invisible, while the point A visible point! But let us have patience! When I will be able to develop fully the digital differential calculus on a digital Euclidean Geometry we will reach and answer your question with clarity!
NEWCLID: If there no more questions or remarks, let us end here our discussion, and let us take a nice and energizing walk under the trees in the park close to our building.



AT THIS POINT THE DISCUSSION ENDS.



AXIOMATIC SYSTEM OF MATERIAL ATOMIC  EUCLIDEAN GEOMETRY

We have as initial concept of objects 

a) The High resolution or precision  points  , or invisible points or material atoms 

b) The Low resolution or precision points, or visible points. They have  the minimum no-zero size (length and width or diameter) in the Low resolution real numbers that can be created on a geometric line as we shall see.

The Low precision or visible congruence of visible linear segments AB of two visible points points A, B, and the Low precision or visible congruence  of angles (to be defined below) 

The High precision or invisible congruence of invisible linear segments AB of two invisible points points A, B, and the High precision or visible congruence  of angles (to be defined below) 
Remark. MEASUREMENT PRECISION LEVEL, AXIOMS OF LENGTH, AREAS, VOLUMES, LIMITED TRANSITIVITY OF CONGRUENCE. If we are to model this axiomatic digital Euclidean geometry, within the classical euclidean geometry, with infinite many points, and within classical logic, infinite natural numbers and classical real numbers then we should model the visible points with small balls or cubes  at the verge of seeing them in a densest lattice. Similarly for the invisible points. We assume rational approximations of the diameter, area of cross 2-dimensional cut , and volume of these visible and invisible points  according a two levels of High and Low precision in real numbers  , which means up to a finite number of digits L and H L<<H in the decimal representation of numbers  respectively  for the Low and high precision. In addition in that case we must postulate and fix a real numbers measurement resolution or precision level which is the same with the low precision level, and in which we use for lengths , areas and volumes. We even model the invisible and visible points with Cartesian coordinates by rational numbers with finite number of digits in the decimal representation, so that Low and High precision corresponds to tow such precision levels defined by the number of decimal digits in the Cartesian coordinates.  We will use the Archimedean approach of lengths , areas and volumes (Archimedes was measuring volumes of 3D figures by filling them with as the sand) of the corresponding  rational approximation of the values of length, area and volume the invisible points. So here we will need an extra set of axioms compared to the Hilberts axioms, those of  the diameter, maximal cross area , and volumes of the invisible points. We will call them AXIOMS OF LENGTH, AREAS AND VOLUMES. The axiom of density or resolutions below must guarantee that the "error" (in the classical Euclidean)  of the measurement of the lengths , areas and volumes this way after  the rational approximation of the corresponding values of the invisible points plus the intermediate area between the spherical or cubic points if at all ,  for any figure inside the overall spherical or space, is below the threshold of the Low  measuring precision level.
And also we must have axioms about congruence of invisible and visible points, of visible lines, and planes , which mean equality with error of order non-zero  in the high precision only, and zero in the low precision. In addition all the such congruence relations have limited transitivity, in other words the transitivity of congruence holds only up to maximum number of repetitions (obviously of the order of maximum number of invisible points  that  a visible point contains).  





c) The visible lines and the invisible  atomic lines

d) The visible planes  and the invisible  atomic planes

c) We may apply the digital set theory on the points of the digital Euclidean geometry

d) And of course we may apply the digital formal logic to make arguments and proofs.

e) Besides the congruence equivalence relations we have the next   initial relations 

Among visible or invisible  elements. 

An invisible point A belongs to a visible point B , denoted by A ε B

A visible (or invisible)  point A    belongs to a visible (or invisible respectively) line L   ,    denoted by A ε L

A visible  (or invisible) point A   belongs to a visible (or invisible respectively) plane P   ,    denoted by A ε P

 A visible (or invisible) line L belongs to a visible (or invisible respectively) Plane P, denoted by  L ε P


A visible (or invisible) point A is between two visible (or invisible respectively) points B, C. 

We design 6 groups of axioms
·     
·     
·    1) Of  Incidence  
·    2) Of  Order 
·    3) Of Congruence 
·    4) Of  Parallels   
·    5) Of Continuity
·    6) Of Resolution
·     
In the next axioms the term point refers to visible or low precision point and the line an plane to visible line and plane or refers respectively to the invisible material atomic points , lines an planes. 


The incidence of invisible to visible elements is intended to define a homomorphism of the geometric structure of invisible material atoms to visible, relative to the relations, belonging ε, order < and congruence =

I. Incidence
1.    For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of “contains,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common,” etc.
2.    For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
3.    There exist at least two points on a line. There exist at least three points that do not lie on a line.
4.    For every three points ABC not situated on the same line there exists a plane α that contains all of them. For every plane there exists a point which lies on it. We write ABC = α. We employ also the expressions: “ABC, lie in α”; “A, B, C are points of α”, etc.
5.    For every three points ABC which do not lie in the same line, there exists no more than one plane that contains them all.
6.    If two points AB of a line a lie in a plane α, then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc.
7.    If two planes α, β have a point A in common, then they have at least a second point B in common.
8.    There exist at least four points not lying in a plane.
9.    There is a special point denoted by O, which is called the center of the space, and a special linear segment OA, which is called the unit of measurement of lengths in the geometry.  
10. For every invisible point A, there is a visible point B, so that A belongs to B.
11. For every invisible line a, there is a visible line b , so that a belongs to b.
12. For every invisible plane a, there is a visible plane b , so that a belongs to b.
13. The relation of belonging from invisible elements is transferred as holding to a belonging of corresponding visible elements that the invisible elements belong to.
14. Two invisible points A, B belong to the same visible point C is an equivalence relation among the invisible points.
II. Order
1.    If a point B lies between points A and CB is also between C and A, and there exists a line containing the distinct points A,B,C.
2.    Of any three points situated on a line, there is no more than one which lies between the other two.
3.    Pasch's Axiom: Let ABC be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points ABC. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.
4.    Every line a, has two points ω1 and ω2 so that every other point of the line , lies between ω1 and ω2 . We call them the end points of the line. All end points of lines define a spherical surface with center the point O (center of the space). All end points of lines of a plane define a circle, with center the center 0 of the space. 
5.    The relation of order from invisible elements is transferred as holding to an order of corresponding visible elements that the invisible elements belong to.
III. Congruence
1.    If AB are two points on a line a, and if A′ is a point upon the same or another line a′ , and as long the A'ω is larger than the AB, then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB ≅ A′ B′. Every segment is congruent to itself; that is, we always have AB ≅AB.
We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in at least one way.
2.    If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″.
3.    Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
4.    Let an angle ∠ (h,k) be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ such that the angle ∠ (hk), or ∠ (kh), is congruent to the angle ∠ (h′k′) and at the same time all interior points of the angle ∠ (h′k′) lie upon the given side of a′. We express this relation by means of the notation ∠ (hk) ≅ ∠ (h′k′).
5.    If the angle ∠ (hk) is congruent to the angle ∠ (h′k′) and to the angle ∠ (h″k″), then the angle ∠ (h′k′) is congruent to the angle ∠ (h″k″); that is to say, if ∠ (hk) ≅ ∠ (h′k′) and ∠ (hk) ≅ ∠ (h″k″), then ∠ (h′k′) ≅ ∠ (h″k″).
6.    If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′, AC ≅ A′C′, ∠BAC ≅ ∠B′A′C′ hold, then the congruence ∠ABC ≅ ∠A′B′C′ holds (and, by a change of notation, it follows that ∠ACB ≅ ∠A′C′B′ also holds).
7.    The relation of congruence from invisible elements is transferred as holding to a congruence of corresponding visible elements that the invisible elements belong to.

Remark. LIMITED TRANSITIVITY If we are to model this axiomatic digital Euclidean geometry, within the classical euclidean geometry, with infinite many points, and within classical logic, infinite natural numbers and classical real numbers then we should model the visible points with small balls at the verge of seeing them. Similarly for the invisible points. We assume rational approximations of the diameter, area of maximal 2-dimensional areal cut , and volume of these visible and invisible points  according a two levels of High and Low precision in real numbers  , which means up to a finite number of digits L and H L<<H in the decimal representation of numbers  respectively  for the Low and high precision. Then also we must have axioms about congruence of invisible and visible points, of visible lines, and planes , which mean equality with error of order non-zero  in the high precision only, and zero in the low precision. In addition all the such congruence relations have limited transitivity, in other words the transitivity of congruence holds only up to maximum number of repetitions (obviously of the order of maximum number of invisible points  that  a visible point contains).  

IV. Parallels 
1.    Euclid's Axiom ):Let a be any line and A a point not on it. Then there is at least one line b in the plane, determined by a and A, that passes through A and does not intersect a. 
V. Continuity
1.    Axiom of Archimedes. If AB and CD are any segments of two lines , then there exists a number n such that n segments CD constructed contiguously from A, along the ray from through B, will pass beyond the point B, as long as Dω is at least larger than twice the AB.
2.    Axiom of line completeness. An extension of a set of visible points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible. (This axiom states that there is always a fixed and constant number of visible points on any  line.) 

VI. Resolution (or Density)
1.    Axiom of sufficient high  resolution or density . Let a line a passing from the center of the space O and  the units of measurements  OA on it,  and let ω(a), Ω(a), demote the finite cardinal number of visible and invisible points that belong to a. And let  ω(n) be the size of the model of the natural numbers constructed on the line a  through congruence and the above axioms. Then it holds that  
            2^ω(n) <= ω(a)
            2^ω(a) <=Ω(a)  


Remark. If we are to model this axiomatic digital Euclidean geometry, within the classical euclidean geometry, with infinite many points, and within classical logic, infinite natural numbers and classical real numbers so not using the corresponding finitary axiomatic systems of logic natural numbers and real numbers then we should state this axiom in a different way:


1.    Axiom of sufficient high  resolution or density .  Let ω(S), Ω(S), demote the finite natural numbers which are an upper bound of visible and invisible points that can belong to the spherical S 3-dimensional space. And let also ω(P) be a common upper bound to the number of invisible points that any  visible point may contain .  Then it holds that  
           a) Any number of visible points of the total spherical space is less than any                      number of invisible points that a visible point may contain . In particular
            2^(ω(S)) <= ω(P). 
         (The axiom must guarantees that lengths, areas and volumes that are defined by summing                   the corresponding values of the invisible points, will have in  general total errors zero in the low precision, and that the failure if the transitivity of relations congruence due to  their limited character can be avoided to occur, with sufficient high resolution of   invisible   points relative to the size of the total space and our repetitive construction in  it)
            b) The diameter of the total spherical 3--dimensional space in integer number of   units of length denoted by ω(n) ,is less that any maximum number of   visible points that the  spherical space may contain. In particular if 
               2^ω(n) <=ω(S).  
             (The axiom  guarantees that a lattice of points with say integer coordinates will always be by far less dense than the lattice of visible points, and  that the "error" (in the classical Euclidean)  of the measurement of the lengths , areas and volumes this way after  the rational approximation of the corresponding values of the invisible points plus the intermediate area between the spherical or cubic invisible points, if at all,  for any figure inside the overall spherical space, is below the threshold of the   low measuring precision level.)


VII Axioms of finite decimal coordinates of points

1) Every  invisible point P has 3 numerical coordinates P(x1), P(x2) , P(x3) that are rational numbers that in decimal notation have finite many digits so many as the definition of the High measurement precision.
2)   Every  visible point P has 3 numerical coordinates P(x1), P(x2) , P(x3) that are rational numbers that in decimal notation have finite many digits so many as the definition of the Low measurement precision.
3) The density of the invisible and visible points is uniform through-out  the spherical space.
(An alternative although not equivalent way is to postulate that inversely for every  triad of decimal rational numbers of the high precision  , there is an invisible point with these coordinates and for every  triad of decimal rational numbers of the low (measurement)  precision  , there is a  visible point with these coordinates). 


 VIII Axioms of length, area, and volumes and compatibility with the coordinates

1) Every invisible point P, as belonging to a line L, has a non-zero length l(P) which is a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
2)  Every invisible point P, as belonging to a plane E, has a non-zero  area a(P) which is  a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
3)  Every invisible point P, has a non-zero  volume v(P) which is  a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
4) The length of linear segment is defined as the sum of the lengths of all of its invisible points.  The length of the unit segment OA , with coordinates of O, (0,0,0) and (0,0,1) is equal to 1. (similarly by cyclic permutation of the coordinates and the other unit lengths from O).
5) The area of figure is defined as the sum of the areas of all of its invisible points.  The length of the unit square OA-OB , with coordinates of O, (0,0,0) and (0,0,1), (1,1,1), (1,0,0)  is equal to 1 (similarly by cyclic permutation of the coordinates and the other unit squares)
6)  The volume of a figure is defined as the sum of the volumes of all of its invisible points.  The volume of the unit cube  , with coordinates  (0,0,0) and (0,0,1), (1,1,1), (1,0,0) , (0,1,0) , (1,1,0), (0,1,1) ,(1,0,1)  is equal to 1


Let is denote by ω(E) the finite cardinal number of all visible points, and by Ω(E) the finite cardinal number of all invisible points. If a line a passes through the center of the space O, and let us denote by ω(n) is   the size of the model of the natural numbers constructed on the line a and the unit of measurements OA,  through congruence and the above axioms.



If two set-theoretical models M1, M2 of the above Holographic Euclidean geometry have the same , ω(n), ω(E),Ω(E) , then they are isomporphic. 

In other words , the above axiomatic system is in some particular sense (not absolute) categorical up to the units of measurements and visible and invisible density.To achieve absolute categorical axiomatic system, some assumptions about how many invisible points has each visible point , if all visible points have equal number of invisible points etc must be made. 

Finally we give here still an alternative axiomatic system, in which we have two levels of precision and invisible points  , and we endowed the invisible points , with a similar geometric structure ,  as the visible points, lines and planes.
The first high precision invisible points are also called material atoms, while the second hither precision invisible points are called ethereal atoms. The second higher precision invisible points , lines and planes  are called ethereal.  The geometric structure of the invisible ethereal   points, lines and planes is again, that of incidence, order, congruence ,parallelism, and resolution.

Remark. In this axiomatic digital geometry the lengths of linear segments, the areas of plane figures and volumes of 3-dimensional figures are defined by summing the corresponding values of the visible points (as Archimedes was measuring volumes with sand in his book Psamitis), are always rational numbers of low precision!
Similarly all metric relations on figures like the Pythagorean theorem etc hold only within the Low precision of the measurements. So there is no need of irrational numbers for this geometry! (An old problem of the Pythagoreans). In addition as it is know in classical Euclidean geometry that any rational number (as linear segment ) is constructible with ruler and compass, then the classical problems of squaring the circle , trisecting the angle, are in the axiomatic digital Euclidean geometry constructible with ruler and compass !



AXIOMATIC SYSTEM OF ETHEREAL ATOMIC EUCLIDEAN GEOMETRY

The distinction of matter from aether is similar to the distinction of matter , and the electromagnetic and gravitational field, which are properties of a second atomic material layer called in history aether. 

We have as initial concept of objects 
a) The Highest resolution or precision  points  , or invisible points or ethereal  atoms
b) The High resolution or precision  points  , or invisible points or material atoms 

c) The Low resolution or precision points, or visible points. They have  the minimum no-zero size (length and width or diameter) in the Low resolution real numbers that can be created on a geometric line as we shall see.

The Low precision or visible congruence of visible linear segments AB of two visible points points A, B, and the Low precision or visible congruence  of angles (to be defined below) 

The High precision or invisible congruence of invisible atomic linear segments AB of two invisible points points A, B, and the High precision or visible congruence  of angles (to be defined below) 

The Highest precision or invisible congruence of invisible ethereal linear segments AB of two ethereal points points A, B, and the Highest  precision or visible congruence  of ethereal angles (to be defined below) 

Remark. MEASUREMENT PRECISION LEVEL, AXIOMS OF LENGTH, AREAS, VOLUMES, LIMITED TRANSITIVITY OF CONGRUENCE. If we are to model this axiomatic digital Euclidean geometry, within the classical euclidean geometry, with infinite many points, and within classical logic, infinite natural numbers and classical real numbers then we should model the visible points with small balls or cubes  at the verge of seeing them in a densest lattice. Similarly for the invisible points. We assume rational approximations of the diameter, area of cross 2-dimensional cut , and volume of these visible and invisible points  according a two levels of High and Low precision in real numbers  , which means up to a finite number of digits L and H L<<H in the decimal representation of numbers  respectively  for the Low and high precision. In addition in that case we must postulate and fix a real numbers measurement resolution or precision level which is the same with the low precision level, and in which we use for lengths , areas and volumes. We even model the invisible and visible points with Cartesian coordinates by rational numbers with finite number of digits in the decimal representation, so that Low and High precision corresponds to tow such precision levels defined by the number of decimal digits in the Cartesian coordinates.  We will use the Archimedean approach of lengths , areas and volumes (Archimedes was measuring volumes of 3D figures by filling them with as the sand) of the corresponding  rational approximation of the values of length, area and volume the invisible points. So here we will need an extra set of axioms compared to the Hilberts axioms, those of  the diameter, maximal cross area , and volumes of the invisible points. We will call them AXIOMS OF LENGTH, AREAS AND VOLUMES. The axiom of density or resolutions below must guarantee that the "error" (in the classical Euclidean)  of the measurement of the lengths , areas and volumes this way after  the rational approximation of the corresponding values of the invisible points plus the intermediate area between the spherical or cubic points if at all ,  for any figure inside the overall spherical or space, is below the threshold of the Low  measuring precision level.
And also we must have axioms about congruence of invisible and visible points, of visible lines, and planes , which mean equality with error of order non-zero  in the high precision only, and zero in the low precision. In addition all the such congruence relations have limited transitivity, in other words the transitivity of congruence holds only up to maximum number of repetitions (obviously of the order of maximum number of invisible points  that  a visible point contains).  


c) The visible lines , the invisible  material atomic lines, the invisible  ethereal atomic lines, 

d) The visible planes  ,  the invisible  material atomic planes ,  the invisible  ethereal atomic planes


c) We may apply the digital set theory on the points of the digital Euclidean geometry

d) And of course we may apply the digital formal logic to make arguments and proofs.

e) Besides the congruence equivalence relations we have the next   initial relations 

Among visible or invisible  elements. 

An material invisible point A belongs to a visible point B , denoted by A ε B

An ethereal invisible  point A belongs to a invisible material point B , denoted by A ε B

A visible (or invisible material or ethereal )  point A    belongs to a visible (or invisible material or ethereal respectively) line L   ,    denoted by A ε L

A visible  (or invisible  material or ethereal) point A   belongs to a visible (or invisible material or ethereal respectively) plane P   ,    denoted by A ε P

 A visible (or invisible material or ethereal ) line L belongs to a visible (or invisible  material or ethereal respectively) Plane P, denoted by  L ε P


A visible (or invisible  material or ethereal) point A is between two visible (or invisible  material or ethereal respectively) points B, C. 

We design 6 groups of axioms
·     
·     
·    1) Of  Incidence  
·    2) Of  Order 
·    3) Of Congruence 
·    4) Of  Parallels   
·    5) Of Continuity
·    6) Of Resolution
·     
In the next axioms the term point refers to visible or low precision point and the line an plane to visible line and plane or refers respectively to the invisible material atomic points , lines an planes or  refers respectively to the invisible ethereal atomic points , lines an planes.


The incidence of invisible material  to visible elements and invisible ethereal to invisible material , is intended to define a homomorphism of the geometric structure of invisible material atoms to visible, and of ethereal to material relative to the relations, belonging ε, order < and congruence =

I. Incidence
1.    For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of “contains,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common,” etc.
2.    For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
3.    There exist at least two points on a line. There exist at least three points that do not lie on a line.
4.    For every three points ABC not situated on the same line there exists a plane α that contains all of them. For every plane there exists a point which lies on it. We write ABC = α. We employ also the expressions: “ABC, lie in α”; “A, B, C are points of α”, etc.
5.    For every three points ABC which do not lie in the same line, there exists no more than one plane that contains them all.
6.    If two points AB of a line a lie in a plane α, then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc.
7.    If two planes α, β have a point A in common, then they have at least a second point B in common.
8.    There exist at least four points not lying in a plane.
9.    There is a special point denoted by O, which is called the center of the space, and a special linear segment OA, which is called the unit of measurement of lengths in the geometry.  
10. For every invisible material point A, there is a visible point B, so that A belongs to B.
11. For every invisible ethereal point A, there is a invisible material point B, so that A belongs to B.
12. For every invisible ethereal line a, there is an invisible material line b , so that a belongs to b.
13. For every invisible material line a, there is an visible  line b , so that a belongs to b.
14. For every invisible ethereal plane a, there is an invisible material plane b , so that a belongs to b.
15. For every invisible material plane a, there is a visible  plane b , so that a belongs to b.
16. The relation of belonging from invisible ethereal elements is transferred as holding to a belonging of corresponding invisible material elements that the ethereal elements belong to.
17. The relation of belonging from invisible material elements is transferred as holding to a belonging of corresponding visible  elements that the invisible elements belong to.
18. Two invisible material points A, B belong to the same visible point C is an equivalence relation among the invisible points.
19. Two invisible ethereal points A, B belong to the same invisible material point C is an equivalence relation among the invisible ethereal points.
II. Order
1.    If a point B lies between points A and CB is also between C and A, and there exists a line containing the distinct points A,B,C.
2.    Of any three points situated on a line, there is no more than one which lies between the other two.
3.    Pasch's Axiom: Let ABC be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points ABC. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.
4.    Every line a, has two points ω1 and ω2 so that every other point of the line , lies between ω1 and ω2 . We call them the end points of the line. All end points of lines define a spherical surface with center the point O (center of the space). All end points of lines of a plane define a circle, with center the center O of the space. 
5.    The relation of order from invisible ethereal elements is transferred as holding to an order of corresponding invisible material elements that the ethereal elements belong to.
6.    The relation of order from invisible material elements is transferred as holding to an order of corresponding visible  elements that the invisible elements belong to.
III. Congruence
1.    If AB are two points on a line a, and if A′ is a point upon the same or another line a′ , and as long the A'ω is larger than the AB, then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB ≅ A′ B′. Every segment is congruent to itself; that is, we always have AB ≅AB.
We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in at least one way.
2.    If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″.
3.    Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
4.    Let an angle ∠ (h,k) be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ such that the angle ∠ (hk), or ∠ (kh), is congruent to the angle ∠ (h′k′) and at the same time all interior points of the angle ∠ (h′k′) lie upon the given side of a′. We express this relation by means of the notation ∠ (hk) ≅ ∠ (h′k′).
5.    If the angle ∠ (hk) is congruent to the angle ∠ (h′k′) and to the angle ∠ (h″k″), then the angle ∠ (h′k′) is congruent to the angle ∠ (h″k″); that is to say, if ∠ (hk) ≅ ∠ (h′k′) and ∠ (hk) ≅ ∠ (h″k″), then ∠ (h′k′) ≅ ∠ (h″k″).
6.    If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′, AC ≅ A′C′, ∠BAC ≅ ∠B′A′C′ hold, then the congruence ∠ABC ≅ ∠A′B′C′ holds (and, by a change of notation, it follows that ∠ACB ≅ ∠A′C′B′ also holds).
7.    The relation of congruence from invisible elements is transferred as holding to a congruence of corresponding visible elements that the invisible elements belong to.
8.    The relation of congruence from invisible ethereal elements is transferred as holding to a congruence of corresponding invisible material elements that the ethereal elements belong to.
9.    The relation of congruence from invisible material elements is transferred as holding to a congruence of corresponding visible elements that the invisible elements belong to.


Remark. LIMITED TRANSITIVITY If we are to model this axiomatic digital Euclidean geometry, within the classical euclidean geometry, with infinite many points, and within classical logic, infinite natural numbers and classical real numbers then we should model the visible points with small balls at the verge of seeing them. Similarly for the invisible points. We assume rational approximations of the diameter, area of maximal 2-dimensional areal cut , and volume of these visible and invisible points  according a two levels of High and Low precision in real numbers  , which means up to a finite number of digits L and H L<<H in the decimal representation of numbers  respectively  for the Low and high precision. Then also we must have axioms about congruence of invisible and visible points, of visible lines, and planes , which mean equality with error of order non-zero  in the high precision only, and zero in the low precision. In addition all the such congruence relations have limited transitivity, in other words the transitivity of congruence holds only up to maximum number of repetitions (obviously of the order of maximum number of invisible points  that  a visible point contains).  
IV. Parallels 
1.    ( Euclid's Axiom):Let a be any line and A a point not on it. Then there is at least one line b in the plane, determined by a and A, that passes through A and does not intersect a. 
V. Continuity
1.    Axiom of Archimedes. If AB and CD are any segments of two lines , then there exists a number n such that n segments CD constructed contiguously from A, along the ray from through B, will pass beyond the point B, as long as Dω is at least larger than twice the AB.
2.    Axiom of line completeness. An extension of a set of visible points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible. (This axiom states that there is always a fixed and constant number of visible points on any  line.) 

VI. Resolution (or Density)
1.    Axiom of sufficient high  resolution or density . Let a line a passing from the center of the space O and  the units of measurements  OA on it,  and let ω(a), Ω1(a), Ω2(a) demote the finite cardinal number of visible , invisible material points and invisible ethereal points respectively that belong to a. And let  ω(n) be the size of the model of the natural numbers constructed on the line a  through congruence and the above axioms. Then it holds that  
            2^ω(n) <= ω(a)
            2^ω(a) <=Ω1(a)  
            2^Ω1(a) <=Ω2(a)  



Remark. If we are to model this axiomatic digital Euclidean geometry, within the classical euclidean geometry, with infinite many points, and within classical logic, infinite natural numbers and classical real numbers so not using the corresponding finitary axiomatic systems of logic natural numbers and real numbers then we should state this axiom in a different way:



1.    Axiom of sufficient high  resolution or density .  Let ω(S), Ω1(S), Ω2(S) demote the finite natural numbers which are an upper bound of visible and invisible material points and ethereal points that can belong to the spherical S 3-dimensional space. And let also ω(P) be a common upper bound to the number of invisible points that any  visible point may contain and ω2(P)   be a common upper bound to the number of ethereal points that any  invisible point may contain .  Then it holds that  
           a) Any number of visible points of the total spherical space is less than any                      number of invisible points that a visible point may contain . In particular
            2^(ω(S)) <= ω(P). 
         (this guarantees that lengths, areas and volumes that are defined by summing                   the corresponding values of the invisible points, will have in  general total errors zero in the low precision, and that the failure if the transitivity of relations congruence due to  their limited character can be avoided to occur, with sufficient high resolution of   invisible and visble  points relative to the size of the total space and our repetitive construction in  it)
            b) The diameter of the total spherical 3--dimensional space in integer number of   units of length denoted by ω(n) ,is less that any maximum number of   visible points that the  spherical space may contain. In particular if 
               2^ω(n) <=ω(S).  
             (The axiom  must guarantee that a lattice of points with say integer coordinates will always be by far less dense than the lattice of visible points, and  that the "error" (in the classical Euclidean)  of the measurement of the lengths , areas and volumes this way after  the rational approximation of the corresponding values of the invisible points plus the intermediate area between the spherical points,  for any figure inside the overall spherical space, is below the threshold of the low measuring precision level.)
c)    2^(Ω1(S)) <= ω2(P). 
    



VII Axioms of finite decimal coordinates of points

1) Every  invisible point P has 3 numerical coordinates P(x1), P(x2) , P(x3) that are rational numbers that in decimal notation have finite many digits so many as the definition of the High measurement precision.
2)   Every  visible point P has 3 numerical coordinates P(x1), P(x2) , P(x3) that are rational numbers that in decimal notation have finite many digits so many as the definition of the Low measurement precision.
3) The density of the invisible and visible points is uniform through-out  the spherical space.
(An alternative although not equivalent way is to postulate that inversely for every  triad of decimal rational numbers of the high precision  , there is an invisible point with these coordinates and for every  triad of decimal rational numbers of the low (measurement)  precision  , there is a  visible point with these coordinates). 


 VIII Axioms of length, area, and volumes and compatibility with the coordinates

1) Every invisible point P, as belonging to a line L, has a non-zero length l(P) which is a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
2)  Every invisible point P, as belonging to a plane E, has a non-zero  area a(P) which is  a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
3)  Every invisible point P, has a non-zero  volume v(P) which is  a rational number that in decimal notation has finite many digits so many as the definition of the High measurement precision while it is zero in the low measurement precision.
4) The length of linear segment is defined as the sum of the lengths of all of its invisible points.  The length of the unit segment OA , with coordinates of O, (0,0,0) and (0,0,1) is equal to 1. (similarly by cyclic permutation of the coordinates and the other unit lengths from O).
5) The area of figure is defined as the sum of the areas of all of its invisible points.  The length of the unit square OA-OB , with coordinates of O, (0,0,0) and (0,0,1), (1,1,1), (1,0,0)  is equal to 1 (similarly by cyclic permutation of the coordinates and the other unit squares)
6)  The volume of a figure is defined as the sum of the volumes of all of its invisible points.  The volume of the unit cube  , with coordinates  (0,0,0) and (0,0,1), (1,1,1), (1,0,0) , (0,1,0) , (1,1,0), (0,1,1) ,(1,0,1)  is equal to 1

Similarly for the highest precision invisible points
Let is denote by ω(E) the finite cardinal number of all visible points, and by Ω1(E) the finite cardinal number of all invisible material points and by Ω2(E) the finite cardinal number of all invisible ethereal  points . If a line a passes through the center of the space O, and let us denote by ω(n) is   the size of the model of the natural numbers constructed on the line a and the unit of measurements OA,  through congruence and the above axioms.



If two set-theoretical models M1, M2 of the above Ethereal Euclidean geometry have the same , ω(n), ω(E),Ω1(E), Ω2(E) , then they are isomorphic. 

In other words , the above axiomatic system is in some particular sense (not absolute) categorical up to the units of measurements and visible and invisible density.To achieve absolute categorical axiomatic system, some assumptions about how many invisible points has each visible point , if all visible points have equal number of invisible points etc must be made. 

Remark. In this axiomatic digital geometry the lengths of linear segments, the areas of plane figures and volumes of 3-dimensional figures are defined by summing the corresponding values of the visible points (as Archimedes was measuring volumes with sand in his book Psamitis), are always rational numbers of low precision!
Similarly all metric relations on figures like the Pythagorean theorem etc hold only within the Low precision of the measurements. So there is no need of irrational numbers for this geometry! (An old problem of the Pythagoreans). In addition as it is know in classical Euclidean geometry that any rational number (as linear segment ) is constructible with ruler and compass, then the classical problems of squaring the circle , trisecting the angle, are in the axiomatic digital Euclidean geometry constructible with ruler and compass !



References
1) D. Hilbert “Grundlangen der Geometrie” Taubner Studienbucher  1977

2) V. Boltianskii “Hilbert’s 3rd problem” J. Wesley & Sons 1978
3) E. E. Moise “Elementary geometry from an advanced standpoint” Addison –Wesley 1963
4) Euclid The 13 books of the  Elements  Dover 1956






And here we present a corresponding axiomatic system of the real numbers without the difficulties of the infinite , with 3-levels of precision. It is a new integrity between thinking, feeling and acting in the physical worlds. It is an educational and conceptual revolution. For more in the Blog       thedigitalmathematics.blogspot.gr



References
 0) H. A. Thurston “The number system” Dover 1956
1) Rozsa Peter “Playing with Infinity” Dover Publications 1961
2) R. L. Wilder “Evolution of mathematical Concepts”  Transworld  Publishers LTD 1968
3) Howard Eves “An Introduction to the History of Mathematics”,4th edition 1953 Holt Rinehart and Winston publications
4) Howard Eves  “Great Moments in Mathematics”  The Mathematical Association of America  1980
5) Hans Rademacher-Otto Toeplitz “The  Enjoyment of Mathematics”Princeton University Press 1957.
6) R. Courant and Herbert Robbins “What is Mathematics”  Oxford 1969
7) A.D. Aleksndrov, A.N. Kolmogorov, M.A. Lavrentev editos
“Mathematics, its content, methods, and meaning” Vol 1,2,3  MIT press 1963
8) Felix Kaufmann “The Infinite in Mathematics”  D. Reidel Publishing Company 1978
9) Edna E. Kramer “The Nature and Growth of Modern Mathematics”
Princeton University Press 1981
10) G. Polya “Mathematics and plausible reasoning” Vol 1, 2 1954  Princeton University press
11) Maurice Kraitchik “Mathematique des Jeux”  1953  Gauthier-Villars
12) Heinrich Dorrie “100 Great Problems of Elementary Mathematics”
Dover 1965
13) Imre Lakatos “Proofs and Refutations”   Cambridge University Press 1976
“Dtv-Atlas zur Mtahematik” Band 1,2,1974
14) Struik D. J. A Concise History of Mathematics  Dover 1987

15) S. Bochner The Role of Mathematics in the Rise of Science Princeton 1981

16) D.E. Littlewood “Le Passé-Partout Mathematique” Masson et c, Editeurs Paris 1964
17) A concise history of mathematics , by D. J. Struik, Dover 1987









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