Clarification of Dimensional Extrapolation, euclidean AND NON-EUCLIDEAN SPACE, and THE DERIVATION OF TRUE units
To understand Dimensional Extrapolation and the nature of TRUE units, it is necessary to unlearn some of the things mainstream physicists think they know. The idea that mass warps space, which is generally accepted as implied by the Theory of Relativity, is one of those things. Einstein himself raised doubt about this when he said in his Note to the Fifteenth Edition of “Relativity, the Special and General Theory” June 9th, 1852: “In this edition I have added, as a fifth appendix, a presentation of my views on the problem of space. … I wished to show that space-time is not necessarily something to which one can ascribe a separate existence. … Physical objects are not in space, but these objects are spatially extended. In this way, the concept of ‘empty space’ loses its meaning.” If space-time does not exist apart from distinct objects, the idea that the 3-D space of our physical observations is warped by mass is, at best, a conceptual analogy that may be useful in helping us to grasp the idea of gravitational gradient in a four-dimensional domain. Is space-time warped by mass? If so, how can we measure the warping if the point from which we observe and measure it is also within the same warped space-time domain?
We exist in a dynamic multi-dimensional reality; but our physical senses only convey the information of one-, two- and three- dimensional conceptual images to our physical brains. We deduce the existence of a fourth dimension based on comparison of current observation with the memory, actually pictures stored in the present brain, of previous observations. Our observations are conceptually limited to one-, two- and three- dimensional slices of a dynamic multi-dimensional reality. “Dimensional Extrapolation” is the term I use for the process of rotation and extension that is necessary to move from an n-dimensional domain into an (n+1)-dimensional domain.
The necessary angle of rotation in the dimensional extrapolation needed to expand our awareness from a one-dimensional Euclidean domain (a line) to a two-dimensional Euclidean domain (a plane) is 90 degrees. This is because this amount of rotation places the extension equidistant in angular degrees from the two directions of freedom of movement in the one-dimensional domain. Similarly, the angle of rotation needed to expand our awareness from a plane, which is a two-dimensional Euclidean domain to a volumetric three-dimensional Euclidean domain is also 90 degrees, and in general, moving from any n-dimensional Euclidean domain into an (n+1)-dimensional Euclidean domain requires rotating a unitary extension to a point that is equiangular in rotation away from all of the dimensions of the n-dimensional domain. This angle of rotation for Euclidean domains is always 90 degrees, and the magnitude of the unitary projection from any n-dimensional domain into the (n+1)-dimensional domain is determined by using the Pythagorean Theorem. With the unitary orthogonal projections in the n-dimensional domain as sides of the Pythagorean triangle, the magnitude of the projection into the (n+1)-dimensional domain is calculated as the hypotenuse.
The Pythagorean equations are sub-sets of the expression described mathematically by Σki=1 (Xk)n = Zn. where n = 1,2,3,…, the natural integers. The Pythagorean equations are obtained when k = n = 2, and two other important equations are obtained when k = 2 and n = 3, and when k = n = 3. Integer solutions to the first of these three equations, the Pythagorean equation, yield the Pythagorean triples. The lack of integer solutions to the second equation, proved by application of Fermat’s Last Theorem, and the integer solutions of the third equation prove to be of primary importance in determining how the stable structures of elementary particles are formed, as we shall see below. It is probable that other values of k and n will produce additional important equations useful in describing combinatorial relationships between particles in the quantized reality of the universe.
A second, related concept that requires unlearning is the idea that there is something called non-Euclidean space. The problem here arises from confusing dimensions of different types. While the dimensions of space and time are both measured in units of distinctions of extent, space domains and space-time domains are quantitatively and qualitatively different. First, let’s see how the dimensions of space and time are quantitatively different: When moving from a 1-dimensional Euclidean domain to a 2-dimensional Euclidean domain and from a 2-dimensional Euclidean domain to a 3-dimensional Euclidean domain, the units of extent are the same, i.e., they can be measured mathematically consistently in real numbers. But when moving from a 3-dimensional Euclidean domain to a 4-dimensional Euclidean domain, rotating 90 degrees away from all of the directions of freedom of the 3-dimensional domain, requires that the extension into the 4-dimensional domain is imaginary, i.e. a multiple of the square root of minus one, as calculated using the Pythagorean Theorem.
Beyond and even more important than these problems with definitions and the basic logic of mathematical physics, is the failure to include the functioning of consciousness in the equations. Consciousness is intimately involved in our observations. Clarifying the fact that a non-Euclidean domain is only definable relative to the Euclidean domain of conscious observation provides an important step toward understanding how to put consciousness into the equations of mathematical physics.
The problem with the concept of non-Euclidean space as an existing reality is partly because of confusion due to poorly defined terminology, partly because of ignoring the role of consciousness, and partly because of the loss of detail in mathematical generalization. The Perception of a non-Euclidean dimensional domain is necessarily relative to the dimensional domain of the conscious observer. The qualitative differences between the observations of space and space-time domains exist because of the way we perceive reality through the limited physical senses and the way memory is stored in the neurological functions of the brain. Because the observation and measurement of a non-Euclidean domain is defined relative to the dimensional domain of the observer, it should be recognized as a relatively non-Euclidean domain. With these points in mind, we can no longer accept the assumption that space and time exist apart from, and without reference to consciousness.
Mathematicians have generalized the concept of space by considering Euclidean space as only one of many possible ‘spaces’ with different degrees of curvature. In that view, Euclidean space just happens to be the one with zero curvature. Three-dimensional Euclidean space, however, is the most important spatial domain in respect to all physical observations because it is the only space of observation available to our limited physical senses. It is, therefore, the only reference we have from which to define non-Euclidean dimensional domains. Three-dimensional Euclidean space is the natural arena of our physical observations and what we perceive as reality. And because the images of reality created by our physical brains are created within the limits of our physical observations, Euclidean space is the space of consciousness. The terminology can be improved by using the term ‘space’ for the three-dimensional domain only and Euclidean space as the space of reference for all observations.
When n ≠ 3, an n-dimensional domain, Euclidean or otherwise, can only be conceptualized in reference to the 3-dimensional Euclidean domain of our physical observations. For example, if a 2-dimensional plane is curved, i.e. non-Euclidean, its curvature can only be seen from the viewpoint of a 3-dimensional Euclidean domain. Similarly, the curvature of a non-Euclidean 4- or 5- dimensional domain can only be meaningfully measured and described from the viewpoint of our 3-dimensional Euclidean domain of observation. We know from Einstein’s relativity, validated many times by experimental evidence, that the measure of 1-dimensional time is distorted by relative velocity and mass. This means that the 4-dimensional space-time domain is observed as Euclidean or non-Euclidean, depending on the observer’s relative velocity and proximity to massive objects.
Thus ‘Space’ can only be defined unambiguously as the domain of the first three dimensions, within which, because of the limitations of our physical senses, all our physical observations are made. Through the physiological and the neurological processing of the data supplied by our senses, we perceive the 3-dimensional domain of the universe as Euclidean. In fact, if any part of the 3-dimensional domain of our observations is non-Euclidean, it can only be said to be non-Euclidean relative to the 3-dimensional domain of our conscious observations.
The very real differences, both perceptual and mathematical, between space and time domains are generally ignored by human beings, including scientists. They can be ignored without serious consequences in non-mathematical verbal descriptions when comparing one measurement of time with another, but ignoring them in the mathematical equations describing space-time-matter-energy relationships is problematic. This problem, in conjunction with the inappropriate application of Newtonian continuous calculus to discrete quantum phenomena, leads to virtually all of the so-called “weirdness” of quantum physics, as well as most of the apparent conflicts between quantum mechanics and relativity. So, while there may be many non-Euclidean domains relative to our Euclidean domain of observation, the three dimensional domain of space cannot be said to be existentially non-Euclidean.
Many of the problems of mathematical physics, unsolvable in the context of the current scientific paradigm, are relatively easily solved by the concepts presented here and by use of the Calculus of Distinctions, which is applicable where Newtonian calculus is not. Dr. Neppe and I have already published some of these solutions in the book “Reality Begins with Consciousness” (www.BrainVoyage.com) and in papers explaining the spin number of fermions, why quarks are always found combined in groups of three, and the Cabibbo mixing angle.
These problems with the current paradigm become paramount when dealing with the physics of elementary particles, starting with the quarks that make up the protons and neutrons of atomic structure. In a particle accelerator like the Large Hadron Collider (LHC), particles are accelerated to very high relative velocities and caused to collide, breaking compound particles apart into smaller particles. The mass and energy of the elementary particles flying away from a collision are deduced from three-dimensional snapshots of their apparent paths in a magnetic field. The fact that none of the measured amounts of the mass and energy of these elementary particles obtained in this manner are unitary, or even integer multiples of the unit commonly used, tells us that the unit being used is not the truly minimum sub-quantum unit.
We can determine the relative mass and energy of the minimal elementary particle unit by normalizing the mass and energy of electrons and quarks. Because of the relativistic light-speed limit to rotational velocity, using the mass of the minimal elementary particle, the electron, and its angular momentum, we are able to determine the minimum possible volume occupied by any elementary particle. Finally, setting the minimal mass-energy and minimal volume to unity (+1), we define the truly minimal unit, the proper unit with which to measure all physical phenomena. Defined in this way, all physical particles will be integer multiples of this sub-quantal unit. Because this sub-quantal basic unit is derived from the existence of a minimal-volume high-velocity rotating particle, and all particles are integer multiple of it, I call it the Triadic Rotational Unit of Equivalence, or TRUE Unit for short.
Given that the stable sub-atomic elementary particles, i.e. electrons and quarks, are high-velocity symmetrically spinning objects, we look at how they must combine to form compound symmetrically spinning objects. The reason symmetry is so important here is because asymmetric objects spinning at angular velocities approaching the speed of light would quickly fly apart because the centrifugal forces of angular momentum, unequal on opposing sides of the spinning object would pull it apart. Thus, the second law of thermodynamics, operating on all particles combining due to the attractions of opposite electrical charges and/or magnetic attraction, would cause any randomly-occurring asymmetric combination to decay almost immediately back to maximum entropy. This means that the complex physical universe as we know it cannot have evolved accidentally from flotsam from a giant explosion however many billions of years ago.
Normalized particle collider data tells us that up-quarks are made up of 4 of the minimum mass-energy-volume units described above, and down-quarks are made up of 9 minimal units. Applying Fermat’s Last Theorem to the equation resulting when k = 2 and n = 3, in the expression Σki=1 (Xk)n = Zn, tells us that two symmetric objects made up of any integer multiples of the minimum mass-energy-volume unit cannot combine to form a new symmetric object because there are no integer solutions to (X1)3 + (X2)3 = Z3. On the other hand, the equation (X1)3 + (X2)3 + (X3)3 = Z3, obtained when k = n = 3, has integer solutions.
Noting that Einstein’s E=mc2 , validated by empirical data, proved that mass and energy are two forms of the same thing, we deduce that there has to be a third form of reality, not measurable as mass or energy, producing stable, symmetric triadic particles. This is why under the normal conditions existing in the universe of our observational domain quarks are always seen in triadic combinations: two up-quarks and one down-quark form a proton, and one up-quark and two down quarks form a neutron.
The number of units of the third form needed to form symmetrically stable quarks, protons and neutrons are uniquely determined from the integer solutions of the equation obtained from the general expression Σki=1 (Xk)n = Zn, when k = n = 3.
Because of the continuing discovery of the relationship of the mathematical structure (rings, fields, etc.) of numbers studied in the discipline called number theory, to the structure of the observable universe, I believe that our discoveries strongly suggest that reality consists of the three forms of distinctions of content interacting in nine finite dimensions of extent: three dimensions of space, three dimensions of time, and three dimensions of consciousness, all contained and pervaded by a conscious transfinite substrate. The logical mathematical patterns of the conscious transfinite substrate are conveyed to the 3-dimensional subdomain of our observations by the expression Σki=1 (Xk)n = Zn. For this reason, I call this the Conveyance Expression and equations derived from it Conveyance Equations.
Experimental data tell us that the Hydrogen atom is unique, being the only element that consists simply of one electron and one proton; but Fermat’s Last Theorem tells us that this combination is asymmetric, and would therefore be extremely unstable, and would not exist long enough to form the universe we perceive without the addition of a particle composed entirely of units of the third form.
When we determine the number of units of the third form needed to stabilize the Hydrogen atom, and apply this analysis to all of the elements of the Periodic Table, we find that the most stable and most abundant elements in the universe are those that support life and sentient beings. Furthermore, gaps in symmetry that are found existing within the Periodic Table as we have known it, are filled by compounds prominent in amino acids and molecules of DNA and RNA, the building blocks of conscious life forms.
This fits nicely with our hypothesis that the third form of the substance of reality is the original primary form of consciousness itself, guiding the formation of a universe able to sustain life forms that are capable of experiencing consciousness. It also appears that the ratio of the third form to the mass/energy substance, based on abundance of life-sustaining elements in the universe and Hubble Telescope data, conforms with the conjecture that dark matter and dark energy are also composed of the third form.
Perhaps the most important, and consequently also the most controversial aspect of this analysis, is the unavoidable conclusion that consciousness in its primary form has always existed, and will always exist, in the quantized, relativistic universe that we experience, and that life, fully capable of supporting ever-existing consciousness, is the purpose of the physical universe.