Twists the space-time of the compressed air
The question of a dynamic of the dimensions of nature
This contribution aims to raise the question of a dynamic in the dimensions of nature. The focus is on the theory of relativity. Einstein had described how time and space can enlarge and shrink, twist and bend. However, he deliberately excluded any question of what could be the cause of such a dynamic. In his opinion, questions of this kind do not contribute to a better understanding. It is sufficient to collect all the results of natural research and use them to develop a theory that can explain everything without contradiction with the least amount of resources and effort. He allowed "Occam's razor" to be the only principle.
In the end, however, this has led to the opposite. Different methods of illustration were sought in order to condense as much as possible in complex images. In the Minkowski diagrams, coordinate crosses are twisted into one another. Space and time on the one hand and speed through space and time on the other are often mixed up. Principles of physics such as the principle of inertia are included in the construction of the coordinate system (inertial system) so that they can no longer be recognized as independent principles. Acceleration appears as the compression of geodesics in a curved gravitational field that curves in itself, since there is no more absolute space available to represent its curvature.
In order to be able to pose the question of the dynamics of the dimensions of nature, this has to be disentangled and disassembled step by step, at the risk of occasionally asking into the indefinite. The interpretations of Epstein and Hansen provided inspiration.
Lewis C. Epstein's interpretation of the special theory of relativity
Movements are usually represented by space-time diagrams in which the x-axis means space and the t-axis means time. A curve in the diagram basically runs from bottom to top and shows at what time a body is at which point. That is his world line.
Figure 1 world line in the space-time diagram
"Each point on the plane of the drawing corresponds to an event. (In this sense, the plane of the drawing represents space-time reduced to two dimensions). The line drawn in red becomes World line of the particle called. It provides information about Where the particle when is. The inverse of the slope of the world line in an event is the (instantaneous) speed. If the particle moves uniformly, its world line is a straight line (the slope of which is equal to the inverse of the speed). Such graphical representations are Spacetime diagrams "Excerpt from the highly recommended presentation of the special theory of relativity by Franz Embacher.
Author: own drawing
Diagrams of this kind are completely sufficient to describe all movements in the familiar environment. There are some striking phenomena: If, for example, something falls to the ground in a moving ship, its movement is not deflected to the side according to the speed of the ship, but it falls exactly as if the ship were standing still. Inside the ship it is not possible to tell whether the ship is moving as a whole. Aristotle was already familiar with this phenomenon (Phys. 212a). Galileo systematically analyzed it and found mathematical descriptions for its representation (Galileo transformations). - Something similar happens when the occupants of a stationary train see a neighboring train pulling away and have the impression that they themselves are being set in motion and the other train is stopping. - For all phenomena of this kind, rules can be found that describe the movement as a whole without contradiction.
A completely new situation only arose when, contrary to these rules, it was established that the speed of light is always constant. This broke through all of the transformations that had been in place until then and led to a completely new formalism: When an observer moves, the dimensions of space, time and mass change for him: At greater speed, space appears shorter (length contraction), time longer (time dilation ), the mass and energy are greater. This change in the frame of reference guarantees that the speed of light remains the limit speed.
It is most extreme when the world is viewed from the perspective of a photon flying at the speed of light. Nothing can approach the photon or move away from it, because these movements would reach superluminal speed. All distances to other objects have shrunk to 0 due to the length contraction, i.e. the photon encompasses the world in a single point. Mass and energy have condensed infinitely at this point, have become infinitely large. The photon looks like the cosmos at the moment of the Big Bang or like an infinitely large black hole. However, the world around the photon ages infinitely quickly due to time dilation, i.e. this state only lasts infinitely briefly. However, the photon itself does not age at all, it remains in this state forever. It encompasses complete time in eternal duration and at the same time complete space in a single, infinitely short moment of time. The photon thus realizes the laws established by Nikolaus von Kues of the collapse of opposites into a unity (coincidentia oppositorum), which are no longer the result of philosophical and religious considerations, but a strict consequence of the special theory of relativity. (Helmut Hansen adds in a comment: "This state only lasts infinitely briefly in order to make way for a new state, which in turn only lasts infinitely briefly. In short: Everything changes every moment. And that is exactly the conceptual meaning of the Eternal Now ".)
This is difficult for the human mind to understand, because how can the same space be larger for one observer and smaller for another? In 1984 Epstein published a new approach ("Relativity visualized"). "What makes time desynchronize? What makes space contract? What makes time run slow? What makes the speed of light unexceedable" (Epstein, p. 76).
The law of the constancy of the speed of light is a statement about speeds and not about space and time. Therefore it is more obvious to look for symmetries in the tangent space, in which the possible velocities are entered for each point of the space, instead of in the space itself. Epstein therefore replaces the space-time diagram with a velocity diagram, the axes of which are no longer Space and time, but rather the speeds through space and time, that is, mathematically speaking, it changes from space to its tangential space. However, this transition is often not clearly stated and statements about the tangential space of the velocities are presented as statements about the underlying space. The coordinate cross of the tangential space TxM is the point x where the tangential space is "attached" to the space M. On the whole there is a tangential bundle of all tangential spaces to spacetime.
Figure 2 tangent space
The plane surface TxM is at the point x the tangent space of a differentiable manifold M .; Wikipedia
Author: From derivative work: McSush (talk) Tangentialvektor.png: TN The originally uploading user was TN in Wikipedia in German - Tangentialvektor.png, Gemeinfrei, Link
The Epstein diagram describes a tangent space. In order to avoid confusion, in the following the axes in the underlying manifold M will be denoted by x for space and t for time and the axes in the tangential space TxM by vx and vt. The vx-axis of the tangential space no longer shows the respective point in space, but the respective speed at the point x. Different velocities are possible at each point: "Further out" in the tangential space on the vx-axis is not a spatially further distant point of the underlying manifold M, but a greater velocity. The zero point of the tangential space marks the rest point at which the tangential space is attached in spacetime, i.e. the point at which the velocities are measured. For example, a pedestrian, an airplane, a rocket or a light can be located at the point x in the space M. Among other things, they differ in their respective speed, and this feature is entered in the tangential space. The vx-axis is therefore initially not infinitely long like the x-axis of space, but has a finite length from 0 to the speed of light c.
Correspondingly, in the tangential space the vt-axis no longer shows the time, but the aging, i.e. the "speed through time". This term can be misunderstood because it is a transfer (metaphor) of the velocity vx, which is initially only known for space. At the zero point is the light that does not age at all. The slower something moves through space, the faster it ages in its movement through time. (This is the reverse of the well-known sentence from the special theory of relativity: Faster clocks tick more slowly.) What ages fastest is what is spatially at rest, i.e. what is neither spatially moving nor in an environment that is spatially moved.
How can a measure of aging, i.e. the speed vt, be found? The spatial speed vx is the usual speed, the relationship between space and time, but the measure of aging can only be derived from the superordinate principle that Epstein formulates as a generalization of the constancy of the speed of light: The total speed through space and time is always constant! Only the proportions of the temporal and spatial speed can change. If the velocities are entered in the tangential space, they must lie on a quarter circle that connects the vx and vt axes and meets the vx axis at point c for the speed of light. Then the general formula for the unit circle can be used
(1) x2 + y2 = 1
by transferring to the vx and vy axes and for a diameter c converted into:
(2) vx2 + vt2 = c2
If a speed vx is given, the aging is calculated from this, i.e. the speed vt:
(3) vt = √ (c2 - vx2)
With this quarter circle the symbol of the mandala appears in physics, which Helmut Hansen systematically pursues in his books. The constancy of the speed of light only describes the borderline case of a movement with zero rest mass, which has no temporal component.
Figure 3 Epstein Circle
The drawing is taken from Helmut Hansen's book "Linien des Alten", p. 51. Deviating from this, the axes are designated with vx and vt and are rotated analogously to the usual space-time diagrams.
This diagram is to be understood as the representation of a tangential space with the axes vx and vt. Theoretically, any combinations of spatial velocities vx and ages vt could be possible in the tangential space. However, the principle of the constancy of the speed of light states that only those combinations are possible that lie on the Epstein circle shown here. If the speed of light c is reached in the limit, then vt = 0. Otherwise, each speed vx is uniquely assigned a speed vt. All velocities lie on the circle and thus fulfill a symmetry that is identical to the symmetry of electrodynamics, U (1), which will be explained later.
The degree of aging results as the "remaining portion" in order to be able to comply with this law. As a measure of aging, the vt-axis describes a movement in absolute spatial rest, which only has a temporal component, the rest energy. With a suitable choice of coordinates, the important conclusions such as E = m · c2 can be read directly from these diagrams, where E0 is on the vt axis and c as the speed is on the vx axis. Eckstein therefore chooses a representation in which the speed vx is multiplied by the constants m and c as the unit of the vx axis. m stands for a given mass, so m · vx is the momentum and c is the speed of light. The unit of the vt-axis is the rest mass for the same value m. Then the formula E = m · c2 can be read off at point c. Overall, this representation shows that the constancy of speed formulated by Epstein is a consequence of the conservation of energy. The space and time share of the speed must always add up to the same speed so that the total energy is retained.
Figure 4 total energy
E0 is the rest energy, Etot the total energy, p · c the pure kinetic energy. According to conservation of energy, Etot must be constant. (The idea for the drawing comes from David Eckstein's online script.) Source
With this representation, Epstein and Eckstein can very intuitively and clearly understand all of Einstein's conclusions. However, they do not seem to be aware of the scope of their knowledge. While Einstein wanted to make statements about spacetime, according to which, for example, lengths and time can change (time dilation, length contraction), they understand his principle as statements about tangential spaces. It is not the length that contracts, but the movement through space becomes larger or smaller, and not the time dilates, but the speed through time (aging) becomes larger or smaller.
With the idea of the curvature of space, Einstein wanted to represent the velocities through space in space itself. Instead of looking at the tangential space at every point in space and showing the speeds at which particles move through this space, he understood the speed through space as the curvature of space. The degree of curvature is mathematically identical to the speed vx.
After it had taken many decades to "get used" to the theory of relativity, this renewed rethinking must surely be difficult. But it will allow for a much clearer presentation.
Each tangent space shows the speed at a particular point. If the question is also asked whether the speed can change (acceleration), then the question is how neighboring tangential spaces can be linked to one another. When something moves on a world line through space, its speeds can change at the same time. For this, the world line is no longer considered, but the line through the associated tangential spaces. Mathematics calls this line the connection in the tangential bundle.
Physics of the mandala (Helmut Hansen)
Helmut Hansen would like to show that based on this interpretation, an understanding of speeds greater than the speed of light is possible. This could open a way to understand quantum mechanical tunnel effects, as it also suits religious beliefs such as ubiquity and synchronicity of events.
Essentially, it seems to me to be about a new interpretation of the twin paradox. When a spaceman leaves the earth, it can be measured from the earth how far the movement leads and how long it lasts. In this measurement from Earth, the spaceman does not exceed the speed of light. However, when he returns, his slower aging will mean he will have remained significantly younger compared to his twin who was left on Earth. If his speed is now calculated as the ratio of the time he has measured (how much he has aged during the journey) and the path of his journey measured from earth, then he may have moved far faster than the speed of light. Eckstein comments according to Epstein's interpretation: "Since B takes up part of this space-time segment for space, he has less left for movement in time. It is as simple as that." (Eckstein, C7, Link). The higher speed results from the fact that the time dilation of the moving object is taken into account, but not the length contraction measured by it. This speed is therefore denoted by vra: It takes into account the relativistic effects of time dilation in the moving object and, at the same time, the absolute measurement of length without length contraction from the stationary observer. Are there reasons, and can a mathematical space be constructed in which these different calculation methods can be represented? That would take Epstein's step further.
Hansen shows the velocities vra on a further axis that runs parallel to the vx-axis. It assigns the associated value vra to each value on the vx axis by means of a vertical mapping. As a result, the quarter circle is enclosed by a square and the mandala is perfect. Hansen calls this square the "Einstein Square".
Figure 5 Mandala Code (Epstein Circle in Einstein Square)
Source: Helmut Hansen "Linien des Alten", p.59. The labels of the axes and their rotation have also been changed here.
The velocities entered on the vx axis are projected onto the upper side of the square. There, the associated speed vra is assigned to each speed vx. The conversion factor is the gamma value adopted by Einstein:
0 is mapped to 0, i.e. vra (0) = 0.
The limit speed c is mapped to ∞, i.e. vra (c) = ∞.
The speed 1 / √2 · c is mapped to c, i.e. vra (1 / √2 · c) = c.
This illustration intersects the Epstein quarter circle at the same point where the diagonal of the square intersects the quarter circle. I.e. at this point the time and space components of the speed are identical. That is the limit from which vra exceeds the speed of light. Godel had already found this point. Hansen therefore calls it the Gödel point.
The experimental evidence for the theory of relativity shows that this direct comparison of time measurement by the moving object and length measurement from the earth has very practical consequences. When cosmic radiation hits atoms in the earth's atmosphere 15 km above the earth, muons are formed. Because of their half-life, they would have to decay so quickly that they cannot reach Earth. Since they age more slowly, their half-life, measured from Earth, is longer and many of them are able to get to Earth.
Helmut Hansen's presentation contradicts all current approaches of theoretical physics. It is justified from the intuitive insight that the symbol of the mandala that appears here has its own, higher symbolic power. In a similar way, Kepler had begun his analyzes of the planetary motions when he had intuitively tried to project them onto the spheres of nested Platonic solids.
The construction of the Einstein square and the enclosure of the Epstein circle by this square indicate a fundamentally new mathematical and physical approach. Today all quantum field theories are mathematically represented as bundles of fibers. The special importance of the speed of light was recognized by Maxwell's establishment of the field equations of electrodynamics. As a symmetry group in quantum electrodynamics, the unitary group U (1) of all complex numbers on the unit circle, i.e. all complex numbers with magnitude 1 (cf. Wikipedia), is considered today. Hardly any theory is as well proven experimentally as this one. The symmetry group U (1) is similar to the Epstein circle, with the vx and vt axes being understood as the real and imaginary axes of the complex numbers. I do not know whether the internal connection between the Epstein interpretation and quantum electrodynamics has already been examined more closely.
The Einstein square with its two parallel speed axes cannot be represented mathematically in this way as a fiber bundle. Helmut Hansen justifies the entanglement of Epstein's circle and Einstein's square because the speed of light is independent of both the direction (isotropy) and the amount with which an observer moves through space. In his opinion, the two must be distinguished from each other and can only be proven through their own experiments (Michelson-Morley 1887 the direction, Kennedy-Thorndike 1932 the amount). For him, the following applies: direction = circle, amount = square, which together result in the mandala code.
It is important to me that his approach dissolves the wave-particle dualism into a smooth transition from a particle-like to a wave-like area. In an extended mandala code, the two sides of the Einstein square are lengthened to form two straight lines vx # ("vx increased / sharp") and vt #, which run parallel to the vx and vt axes (I added the names ). The projection of the spatial speed vx onto the speed vra can be read geometrically here:
Figure 6 Unfolded mandala code (Einstein's square unfolded into particle-like and wave-like areas)
Source: Helmut Hansen "About the Dual Parametrization of c", Figure 23, p. 12. Labeling and rotation of the axes have been changed. Speed arrows in the areas "time-like" and "space-like" have been added.
In the extended mandala code, the velocities vx are no longer mapped vertically on the upper side of the Einstein square, but here a straight line is drawn from the zero point that first intersects the Epstein circle and then meets the straight line vx #. If a perpendicular to the vx-axis is drawn there (shown here with blue dots), then the value can be read on the vx-axis, which is to be interpreted as speed vra. The illustration clearly shows how for all velocities whose spatial component vx is greater than 1 / √2 (read at the point where the Epstein circle is intersected), velocities greater than c are achieved for vra.
Hansen describes the area to the right of c as wave-like and space-like, the area above vx # as particle- and time-like. This leads me to the further assumption that in the Epstein circle there is an overlap of space and time as well as an overlap of particles and particles. These two overlaps cannot be distinguished from one another as long as they are superimposed on one another within a single representation. In order to make a distinction possible, a further axis is necessary, which shows the transition from particle to wave-like phenomena. This is to be achieved in the following by introducing the size axis and the idea of "speed through size", which is based on a discussion of the inertial systems.
In a further step I will interpret the mapping of the interval [0, c] to the interval [0, Ein] from the lower side of the Einstein square to its upper side as an example for two different speeds of counting. In this way, a further axis is to be derived from the dense representation of the mandala code in addition to the size, so that on the whole, the Epstein circle vx-vt is expanded to a sphere.
Before doing this, it should also be pointed out that Epstein comes up with an astonishingly similar representation. In a chapter on the big bang, he asks how old the universe is. Result: There is no definite age of the universe. In the Big Bang, all particles move apart at the speed of light, but the speed of light is again the total speed. The particles that move faster in space age more slowly. Measured from the Big Bang, they are therefore already very far away, but still very young. The paradoxical result is that the parts of the universe that are furthest away from us are also the youngest. If everything were to move with a speed vx = c, the maximum spatial size of the cosmos would be reached, but in time everything would have stopped at the Big Bang. This is another representation of the above-mentioned observation that from the perspective of the photon, the Big Bang has not yet left the Big Bang.
Figure 7 The Ultimate Horizon
Source: Epstein, "Relativity Visualized", Figure 6-5, p. 105 Epstein understands the zero point as the big bang. The diagram shows how far away from the Big Bang the particles are when they have all reached the same age. The faster a particle moves through space, the slower it ages. Hence the faster particles are much further away if they are the same age as the slower particles. There is no such thing as "the age" of space. As a measure, the age of those small parts can be chosen that do not move spatially, i.e. that remain spatially at the location of the Big Bang.
A closer look shows that this diagram mixes an absolute and a relativistic aspect, similar to the construction of the velocity vra. The term "age (proper time)" is misleading. This does not show the absolute time since the Big Bang, but the respective age that results for each moving particle according to its speed. Therefore this axis is basically a variant of the vt axis. Epstein has chosen a mixture of the space-time diagram and the vx-vt diagram and therefore comes to similar results as Helmut Hansen in the extended mandala code.
Are there other ways of representing synchronicity? I see five possible approaches:
(1) The light is initially only detected as a limit speed in space-time, in which effects spread through interaction particles that cannot move faster than light. It can be different in the tangential bundle. Effects can possibly propagate faster there via neighboring tangential spaces and would thus influence the environment in advance before the movement in space-time has arrived there.
Figure 8 images from one tangential bundle to another
A mapping from M to M and correspondingly from TM to TM can be selected here for the sake of simplicity. If symmetries spread in TM, this could be perceived as synchronicity in M. source
Author: By User: Fropuff ~ commonswiki -, Public Domain, Link
(2) Space-time and tangential bundles are themselves objects in a superordinate infinite-dimensional space (Hilbert space), which contains as axes both the space and time axes in M as well as all vx and vt axes of the tangential spaces TxM of the tangential bundle TM. Only in such a space can direct interactions between the spacetime axes and the velocity axes be considered, including symmetries that freely link relativistic and absolute aspects, as Hansen suggests. If, in a further step, this superordinate space itself is viewed as a manifold that changes and therefore has a tangential bundle for its part, the emergence and decay of symmetries can be studied there. (Considerations of this kind are used to prove the Poincaré conjecture).
Quantum theory has been using this method for a long time. It does not consider the possible velocities that a particle can have at the point x and which are entered in the tangential space TxM, but the possible quantum states of the particle at the point x. The quantum states are entered in a fiber Fx in a fiber bundle FM. The fiber bundle FM is an infinitely dimensional space.
(3) So far it has been implicitly assumed that the earth is in absolute rest and that synchronicity could be established by an object that moves faster than the earth and then returns. Can't it be the other way around? The earth has its own movement, and it is conceivable, conversely, to switch to a moving object that is much slower. When it reports back, the earth has hardly grown older, has remained unchanged at the border crossing.
(4) The twin paradox - no matter in which direction - leads to the question of identity: which environment is valid upon return? How can the two different movements with their different dimensions in time, length and mass be brought together again at the moment of leaving and returning? This question leads to the general theory of relativity, since large accelerations or decelerations are necessary at the moment of start and return. Positive and negative acceleration re-pose the question of identity. Thought further, it is not only a question of different aging, but possibly also of different counting, which comes together in these moments and has to be integrated into the unity of one.
(5) This finally results in the following approach: In addition to space and time, further dimensions are necessary in order to consider questions of this kind. For systematic reasons, these are primarily the number and size that will be discussed below.
Transcendental interpretation in the sense of Kant
The clock that is permanently installed in one place, ie "unmoved" (room share = 0) applies simultaneously everywhere. It describes exactly the pure interaction that Kant spoke of in the "Critique of Pure Reason". Their signals are "infinitely fast" and their range is infinitely large. If, on the other hand, the clock is moved spatially, it is only valid in a local area that becomes smaller the faster the clock moves. In the other borderline case, the time component = 0, so that the clock coincides with the time axis. This is what Kant meant by pure causality. Here, in a one-dimensional sequence, one event causes the next. Everything has a clear reason and a clear effect. In summary, the vx-axis with the causality and the vt-axis with the interaction can be identified.
The theory of relativity describes the Gödel point, i.e. the speed 1 / √2 · c, a third, middle limit case, when the proportion of space and time are proportionally equal. At this point the forward cone takes on the shape as in the special theory of relativity. At this point, the physics of the mandala describes the borderline case of the special theory of relativity. Thus Einstein's physics is represented and "classified" as a limit value as is Newton's physics. However, it is no longer true that Newton's physics is a borderline case of Einstein's physics, but both are different borderline cases of higher physics.
The principle of relativity must be exactly reversed: the range and inclination of the forward cone are dependent on the respective speed of the watch, i.e. relative. They are not - as previously postulated in the special theory of relativity - everywhere inclined identically with the angle of the speed of light, i.e. not absolutely. The forward cones become more and more "steeper", more restrictive, the faster the clock is moved. Conversely, the slower the clock moves, the flatter they become. In the two extremes, the forward cones coincide with either pure interaction or pure causality. Einstein therefore not only provided a physics for which Newton is the limiting case, but he can also, mediated through the interpretation by Epstein, provide a new philosophical approach to the understanding of space and time, for which Kant is the limiting case. And with Kant, Euclidean space is also recognized as a borderline case: Euclidean geometry applies in the flat area of pure interaction on the vt axis. Euclidean geometry knows no time, it lies completely within time when the space fraction = 0. The non-Euclidean geometries lie in the inclined spaces with a space share> 0.
The theory of gravity in its current form describes "actions at a distance", so it applies simultaneously and, conversely, lies completely within space, if the time component = 0, on the vx-axis. It is not transmitted by any interaction particle that would have to move like any other particle. Therefore no gravitons can go (just as there are no tachyons on the pure time axis either). Quantum mechanics describes natural properties that are transmitted by interaction particles that each have certain speeds and ranges. Therefore it is structurally impossible to match gravitational and quantum theory. The Epstein Circle provides an approach to bring these theories into a larger context.
Helmut Hansen suspects that a bifurcation from Gödel point opens a hyperspace, which he calls Gödel trench, and from which extreme accelerations are possible. That needs to be looked at more intensively.
Size and number
Can the dynamic of space and time be expanded to a dynamic of all four dimensions of nature? To do this, the underlying manifold needs to be expanded: size and number complement space and time. In addition to the x- and t- two additional axes must be added, which are designated as the z-axis for the number and the g-axis for the size. In the tangent space the question is what the velocities through number and through magnitude are, corresponding to aging as velocity through time and spatial velocity. The circle in the area of aging and spatial speed is to be expanded by a second circle of dynamics of size and number, which is perpendicular to it, with the two additional vz and vg axes. When put together, this makes a ball.
"The speed vg through the size" is the speed of the transition between the different sizes such as solid, liquid, gaseous, chaotic. This is not to be confused with entropy. The entropy shows the degree of inner movement. Here, on the other hand, it is about the dynamics, how this measure can change, i.e. a state of higher entropy changes into one with lower and vice versa. This is the respective stability, how quickly something stabilizes or how quickly it threatens to disintegrate.
It is even more unusual to find the "speed vz by the number". That is the counting. So it is not the number line to be entered on a fourth axis, but the dynamics of counting, the "counting speed". Examples of different counting speeds: Slow, thoughtful philosophical counting or even the intuitive immersion in one. The simple counting follows. This is accelerated by multiplying and exponentiating. Real numbers summarize entire boundary processes, complex numbers run through circles and leaves, functions run through function values, spaces and functional spaces follow, and finally, as today's extreme extreme, virtual mathematics with its construction of problem and solution spaces.
Another approach to measure the speed vz by the number could be the complexity, corresponding to the different quantities. The different numbers and also not the number classes such as natural, whole, rational, irrational, real and complex numbers are to be entered on the vz-axis, but the complexity. Its measure could be the multiplicity of the one, starting from the simple one of the natural numbers via the plus-minus of the whole numbers, the fractions n / n, one as limit value up to one on the unit sphere. Here the complexity of the numbers turns into the symbol of the circle, which at the same time shows a certain level of resolution of the size and is the inner principle of the entire representation.
For the expansion to include the dynamics of size and number, there are fundamental considerations: In addition to causality and interaction, Kant introduced substance as a third equal term. That is what remains unchanged in time. There is interaction within the substance.
In the end, the Neoplatonists spoke - with Simplikios - of four measures of nature: In addition to time and place, these are number and size. I see substance stretched between vz and vg in a similar way as the total energy in the Epstein circle between vx and vt. Impulse and energy are functions of speed, mv and mv2. However, the pure laws of momentum and energy only apply to massPoints, i.e. for a plane in which the mass applies punctually. This contradicts nature and leads in physics to the insoluble problems that with integrals over the mass at the transition to the mass point infinitely large values arise. These can only be balanced by the purely hypothetical assumption of opposing virtual particles. The two-dimensional tangential space with the vx and vt axes shows a world in which mass is reduced to mass points. In order to get from the mass points to connected masses, the additional axes vg for the size and vz for the counting of times, spaces and sizes are necessary.
Conversely, there are similar complementarities in number and size as in space and time. A "pure number" paradoxically has no size, but is a pure statement of existence: something that occurs in one-number, two-number, ... Nothing can be said about its size from this. It is a fundamental law of arithmetic that its rules apply regardless of the size of the things that are counted, added, multiplied, and, more generally, that which is calculated. Physically speaking, arithmetic is completely scale-invariant, but this is not true anywhere in nature.
On the other hand, "pure greatness" doesn't have a number. A quantity is determined by the fact that it is internally coherent, that it cannot be torn into pieces. A quantity, as a quantity, is its own measure. This becomes clearest in the borderline questions, which Kant also considered: the size of the universe or the size of the individual. Pure quantities are therefore not countable, they are - as quantum theory has shown with full clarity - indistinguishable. Their number is indeterminate.
Between these two extremes, as in the case of space and time, there is a circle full of intermediate values: these are the probabilities. Today, however, probability theory only considers a single probability measure and the law of large numbers based on it. I therefore suspect that, corresponding to the Epstein circle, there is another vz-vg circle on which the probabilities lie with different number and size proportions. The larger the proportion of size, the more blurred the image and the more imprecisely the probabilities can be calculated. The larger the number component, the weaker the inner bond, which is given in terms of size by the inner relationship.
In order to bring relativity, gravitational and quantum theory together, such a further axis seems necessary to me. Einstein was by no means wrong with his concerns about the theory of probability. They have to be taken seriously and taken into account, in that probability theory becomes a borderline case in this way, with which Einstein might also "live".
The greatest difficulties still seem to me to be to convincingly clarify the concept of size and the speed vz of counting.
As a result, a completely different structure of mathematics could be established than is customary today and has remained unchanged since Bourbaki: (a) Through the Epstein circle, the special numbers of the circle such as π and thus e, (b) through the Einstein square the irrational numbers, here √2 * c as the length of the diagonal in relation to the side length of the unit square, (c) the imaginary numbers by including the group U (1), (d) orthogonal numbers (possibly symplectic form, energy concept), if this is how hyperspace arises.
A different architecture of mathematics would also mean a new understanding of physical spaces. This could lead to a return to older considerations by Kepler and Proklos: Mathematics is systematically built up from principles such as the limit and the infinite, resulting in both the geometric objects such as point, straight line, circle, etc., as well as the number classes such as natural, rational, real, complex numbers. The number classes are developed apart. With the development, symmetries arise step by step. Forms such as point, straight line, circle and square develop in turn: At first it is about the elementary figures of geometry, later about symmetries in larger spaces. It was only with such a new approach to mathematics that an intuitive idea like the mandala that Hansen considered could be systematically understood.
Local and global view: The global view is based on the baseline. This is the line that is completely motionless and thus stands for earth, gravity or weight. From here everything can be observed that takes place in time. The faster than light speed only arises when the speed of the moving body is measured as the ratio of the local proper time (the moving body ages more slowly than the resting body) and the global distance traveled, as seen from the baseline. In order to be able to recognize and determine such a relationship, a location outside of this diagram is necessary. In order to get to this location, another dimension could be necessary, which could possibly be understood as the substance axis and which could open up hyperspace.
For a long time now, mathematics has not only spoken of numbers and points, but of functions and functional spaces. Functions can be added and derived in a similar way to numbers; there are limit functions of sequences of functions, just as there are limit values of sequences of numbers. One step further there are border areas of spatial sequences and qualitative properties of such sequences. And since the ingenious idea of Galois there have been limit constructions of construction sequences. How does this relate to the ideas developed here? The Fourier series could play a key role. Is it conceivable that a structure arises within the sphere defined above that has to do with consequences of this type?
And is it possible that the resonance phenomenon can be described within this sphere? Are there certain areas, possibly levels in this sphere, in which resonance effects occur pure or impure, and which therefore show different degrees of stability?
Can the helical lines drawn by Dürer be reconciled with these images? Is the helical line to be projected onto the sphere?
Finally, the principle of inertia should also apply as a borderline case to be defined in certain areas of this sphere. This could mean that the types of matter that Aristotle distinguished from the proto hyle, hyle topike and hyle noete possibly also find their place here and maybe even merge into one another ..
Mathematics of magnitude (magnitude and inertial system)
Size should not be understood here as a category, but as a measure of nature such as space and time. Recognizing this difference is initially the greatest difficulty. As a rule, the question is: what size is something, how big is something. This question is comparable to the determination of other categories: where is something, how old is it, what color is it, how does it taste, what other properties does it have. If the size is understood as a category, then it is basically identical to the volume of something in space: something is as big as how much space it takes up in space. Size as a category means: The question is what size a given substance is. Size as a measure, on the other hand, means: The question is no longer how big something is, but whether it is any size at all, to what extent it is size.
Vividly, something solid has more size than something perforated, as a group of individual, separate elements. How big is a flock of birds? For its size, does it also have to take into account all individually flying birds that may be in the process of separating completely from the flock? How big is a liquid? Do you include any drops in the area that have been splashed, and what about the part of the liquid that has just evaporated? How big is a flame? Size understood as a measure does not measure the respective size, but measures the accuracy with which the size can be measured.
It is difficult to break down the term size into two different terms and to distinguish it in order to avoid misunderstandings. However, the theory of relativity shows one approach. There the inertial system takes over the measure of size in a surprising way.
The principle of relativity assumes that all inertial systems are equal and move relative to each other. It is also tacitly assumed that the inertial systems can be clearly and unambiguously delimited from one another and do not merge into one another, do not dissolve or arise anew, i.e. are each absolute in itself.
This seems to be a matter of course based on the common examples: Movement on earth, in a ship, in a train or in a car. Everywhere it is confirmed that within the respective inertial system everything falls vertically downwards and is not deflected. Every inertial system appears as a space that exists for itself, in which inside and outside can be clearly distinguished. The only difficulties that have been discussed so far have concerned questions of whether the inertial system can accelerate or make a rotating motion. But it was always assumed that inertial systems would be preserved.
Without having to carry out large experiments, numerous counterexamples can be given:
(1) Inertial systems can be permeable. In a car with an open top, the inertial system of the car and the surrounding atmosphere will mix in the open area. If something falls to the floor of the car, it is partly carried along by the inertial system of the car and partly deflected by the inertial system of the surrounding air. - The asteroid belt can be viewed as an inertial system: if something falls from an asteroid, it is carried along in the direction of the overall movement of the belt. How big can the distances between the asteroids be until the inertial system breaks up into the individual smaller inertial systems of the individual asteroids and the surrounding space? - Epstein names a thought experiment by Galileo: A large stone falls as fast as a small stone, since the large stone can be thought of as a heap of small stones that form a common inertial system. How far apart do the individual stones have to be so that the outside wind or passing clouds with storms and hail can prevail between them? - Neutrinos are so small and fast that a stream of neutrinos penetrates everything. Are cases conceivable in which the neutrino current nevertheless causes changes in the respective inertial systems through which it penetrates?
(2) Inertial systems can dissolve. In a thought experiment, imagine an airplane made of wax that dissolves in the sun like the wings of Icarus. How long does the character of the inertial system remain, when do the individual parts of the dissolving aircraft become independent and each form their own inertial systems? - Other examples are smaller rivers that flow at their own speed into a larger river and dissolve there. During the inflow, the two independent inertial systems overlap until the larger current finally dominates. Transitionally, there can be turbulence that has its own internal stability for a while, so it can be viewed as an inertial system.
From these examples it follows that the measure of size is not "its size", i.e. the volume measured in a surrounding absolute space (the limits of the respective absolutely imaginary inertial system), but rather the stability and permeability, ie the uniqueness and continuity of the Border, which differentiates inside from outside. Before being asked which size is larger than another, i.e. which exceeds it when it is put on directly, it must be asked how consistent the respective size is. This does not mean the specific density either. If, for example, grains are pressed very closely together, they will still fall apart immediately as soon as the external pressure is loosened. A piece of brick, on the other hand, stands for itself.
These examples should have shown sufficiently that to this day, measurements are generally sought in order to measure size, whereas here, conversely, it is a question of understanding size as a measure by which something is measured. Today, as a rule, the solid and the liquid, or more generally the various states of aggregation, are rigidly demarcated from one another. A science of dimensions, on the other hand, should research the criteria according to which size is to be applied as a measure. There are different transitions: from particle-like to wave-like. From a regular internal structure to an irregular one. From a greater density of internal connections to a lower density. How Schelling described the heavy can be taken up as a borderline concept: The heavy is that which is obscured, opaque, which even swallows light and no longer releases it.
Mutual permeability and varying durability are intuitively the simplest parameters for determining what constitutes different degrees of size.
Entropy or heat can also only serve to describe the size as a measure. They help to determine the inner mobility of the individual components, which gives a unit the cohesion. It seems more obvious to me to choose the electromagnetic field as a paradigm. Each size has a flow (divergence) that shows its permeability and a rotation that creates and secures its cohesion. Different sizes can be recognized by measuring divergence and rotation. In the case of a solid material, the cohesive rotation is, so to speak, infinitely fast, which constantly secures the surface. In the electromagnetic field, the speed of light mediates the cohesion of magnetism and electricity.
The "speed through size" is therefore how quickly something big dissolves or builds up. This speed must not be confused with the internal speeds by which the limit of the great is maintained.
If the concept of size is to be defined in this way and to be distinguished as clearly as possible from the traditional understanding of size as volume, no other concept is as suitable as the inertial system. The inertial system is the great, that is, the one that has greatness. The "speed through magnitude" is the "speed through the inertial systems", i.e. the speed with which inertial systems can arise and pass away.
As a result, the principle of relativity is expanded again. The inertial systems are not only relative to each other, but they also only apply relatively in the process of their creation and disappearance.
Just as the concept of aging as the "speed through time" is misleading and is therefore spoken of as "proper time", so is the decay / build-up of a quantity. It is the greatest, or de-enlarging, size.
In Helmut Hansen's unfolded mandala code, the relationship between space and time and the relationship between particles and waves are superimposed. Here they should be separated and decoupled from one another. When asked about the total speed, it has not only a space and a time component, but also a size component, which says how much the size changes in the sense described here.
The mathematics of size (and therefore not the theory of mathematical size) will probably ask, on the one hand, how the great is structured inside and how it is wrapped around the outside and thus held together. In this sense, I expect mathematics of magnitude that will be on an equal footing with geometry (including topology and differential geometry, i.e. the mathematics of space) and arithmetic (mathematics of number). It will have its own elements (structural and structural elements, twisting and knotting elements). It looks as if different preparatory work was carried out throughout the mathematics of the 20th century, which can systematically find its place in a mathematics of size: differential topology, tiling, graph theory, knot theory, fractal geometry.
The mathematics of size will also move into areas of application (such as material strength, mathematical theory of macro- and / or nano-molecules, etc.), as will analysis into mechanics or function theory into electrodynamics.
These are the first ideas. The further question to be asked is whether there is also a mathematics of time that differs from the mathematics of space (geometry, topology). Today in mathematics not even such a question is asked. The work of Aristotle and the Neoplatonists on the concept of time should be read from this point of view. - As a mathematics of number, arithmetic will also completely change its character. On the whole, a new architecture of mathematics will result.
To be continued.
[Version November 21, 2010]
David Eckstein: Epstein explains Einstein, online
Franz Embacher: Special Theory of Relativity, Online
Lewis C. Epstein: Relativity Visualized: The Gold Nugget of Relativity Books, San Francisco 1991 
Lewis C. Epstein: What is the Principle of Relativity ?, San Francisco 1982
Helmut Hansen: The Physics of the Mandala, Aitrang 2007
Helmut Hansen: The Lines of the Old, Norderstedt 2009 (Online)
Helmut Hansen: About the Dual Parametrization of c, in: Proceedings of the Natural Philosophy Alliance Vol 7, Long Beach 2010
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