SO(4) and SO(3) and the geometric structure of space
author: m concoyle
email: martinconcoyle@hotmail.com
The goal of this paper (chapter) is to consider the SO(3) and SO(4) link between a geometric structure of material interactions in regard to both 3space and 4space, and the various forms of material interactions which exist (and are already used) in these spaces (in regard to forcefields and material interactions),
But ontheotherhand, one wonders, if we are always a part of a circlespace then,
Why does our "material containing space" not (appear to) have any holes?

How does one reorganize and reinterpret both math constructs as well as the observed physical patterns of stability, respectively, to be able to describe; to a sufficient level of: generality, accurateprecision, and useful relation to practical creativity (or technical development); the stable, definitive, discrete spectralorbital properties, which can be (or are) used to identify physical systems, and where these stable spectralorbital properties exist at all size scales (from nuclei to solar systems etc)?
Answer:
Use the stable patterns of the "discrete hyperbolic shapes" placed into a manydimensional containment space, ie an 11dimensional hyperbolic metricspace, where each dimensional level is a hyperbolic metricspace (of that dimension) modeled to be a "discrete hyperbolic shape." But where the entire range of the set of metricspaces which possess nonpositive constant curvature can also be used within the (this new) context, defined by the hyperbolic metricspaces modeled as the very stable "discrete hyperbolic shapes" defined in a manydimensional context.
The fact that "discrete hyperbolic shapes" are "circlespaces" and are composed of toral components, where each toral component models each one of the holesinthespace, where the number of these holes is counted so as to define the genus of the "discrete hyperbolic shapes," and this close association (between discrete hyperbolic shapes and tori) leads to the "discrete Euclidean shapes" (or tori) also being a part of the interaction process, and in doing this, it allows that the material interactions, in the new model of existence, be essentially similar to classical material interactions.
This new model for material interactions also allows for the new descriptive context to be such, that the apparent fundamental randomness of quantum description has its (the randomness's) source identified, these types of interactions between atomic and molecular systems so as to be determined between a continual sequence of very small, discrete time intervals (~10^18 sec), where such an identification of the cause of a random pattern (based on discrete, geometric interactions applied to small material components) also depends on the fact that "discrete hyperbolic shapes" have associated to themselves a highly distinguished "vertex" point on their shapes. This distinguished "vertex" point causes their interactions to be centered around this "vertex" point, and this causes the appearance of pointlike material interactions, which exist in a random context.
The goal of this paper (chapter) is to consider the SO(3) and SO(4) link between a geometric structure of material interactions in regard to both 3space and 4space, and the various forms of material interactions which exist (and are already used) in these spaces (in regard to forcefields and material interactions),
But ontheotherhand, one wonders, if we are always a part of a circlespace then,
Why does our "material containing space" not (appear to) have any holes?
[We are within an openclosed and bounded metricspace, which defines a lattice, but we are only aware of the blocks of the lattice {of the discrete hyperbolic shapes} which extend away from the lattice's fundamental domain, so we see an unbounded metricspace, which, we believe, contains no holes, but the fundamental domain, within which our metricspace is (truly) contained, is related to a circlespace.]
How does one describe the patterns of "spectralorbital stability" for "the so very many," very fundamental manybodied systems, which exist at all sizescales?
These spectrally and orbitally stable material systems are systems which are identified, by the professional mathscience community, as being associated to patterns which are (to be) based on nonlinear and/or random (indefinably random) properties, so as to form a set of chaotic and random systems, whose descriptions, according to these experts, are subsequently, irrelevant to practical development, since it is claimed that these simple, fundamental systems are "too complicated to describe."
How can math be a subject about "quantity and shape," if there is no "math property which focuses on stability," or no concern about, the math properties which deal with, or are concerned about, the math property of "the stability of a mathematical pattern?"
(Or concern about determining the math properties needed for the stability of the patterns which math describes?) This is most often related by the professionals to defining set containment, but the set which "does the containing" is modeled to be "too big of a set," namely, a set which is a continuum.
Answer: One should be concerned about this issue, and one should also try to base mathematical descriptions (of systems) on finite quantitative sets.
Assume the containing space is primarily R(n,1), which is contained within a manydimensional context {Note: R(3,1) is spacetime}, ie R(n,1) is a hyperbolic metricspaces of dimension, n, but each dimensional level is a "discrete hyperbolic shape," and the construct possesses the further property, that for each subspace (of each dimensional level) there is a "discrete hyperbolic shape"... ,
(representing either a metricspace or a material component, depending on the dimension of the frame from which the "discrete hyperbolic shapes" are observed [see below])
... , which has a maximal size, where it should be noted that for hyperbolicdimensions "two to five," inclusive, there exist openclosed and bounded "discrete hyperbolic shapes," which can be used to model metricspaces. Furthermore, these (bounded) hyperbolic metricspaces (or discrete hyperbolic shapes) can be
either
metricspaces which contain lowerdimension discrete hyperbolic shapes
or
they can be models of material components which are contained in openclosed, hyperbolic metricspaces.
D. Coxeter: "Discrete hyperbolic shapes" which are 6dimensions and higher are all unbounded, and the last defined "discrete hyperbolic shape" has a hyperbolicdimension of ten (so the proper containing space for stable geometric shapes is a hyperbolic metricspace of dimension11).
The assumption of the existence of a maximalsize for subspaces of hyperbolicdimensions five and less (inclusive), for such (possibly bounded) shapes, which are contained within an overall containing space of 11hyperbolic dimensions, implies that such an overall highdimension containing space has its spectralorbital structure... ,
{associated to the (mostly bounded, low dimension) discrete hyperbolic shapes which identify its stable geometricspectral structure}
... ., determined by a finite set of "discrete hyperbolic shapes" which, in turn, determine a finite spectralorbital set. Thus, the "discrete hyperbolic" shapes which exist within such a containment set must be in resonance with this finite spectral set, ie both the hyperbolic metricspaces and the material components, where both of these constructs are modeled as very stable "discrete hyperbolic shapes."
It is this finite spectral set upon which the quantitative structure... , used for the descriptions of all the stable material components which compose (or are contained within) this set... , depend. {for their precise descriptions and math constructs}.
This construct is both the quantitative and geometric basis for the stable patterns which are observed physically, and which exist as (stable) math patterns within the set.
That is, all stable, bounded "discrete hyperbolic shapes," which are contained in this (or any such) particular 11dimensional containing hyperbolic metricspace, are in resonance with the finite spectral set, which is defined by the different dimensional levels (each dimensional level modeled as [with] discrete hyperbolic shapes).
Note: There can also be multiplicativeconstants which are defined between dimensional levels, as well as being defined between different subspaces of the same dimension. [ie subspaces which have the same dimensional value]
Math properties, or physical properties, associated to metricspaces and materialcomponent interactions
Both the properties of "position in space" and concerning material interactions of stable material components (or stable math patterns), where the interactions are determined by spatial relationships, are defined within Euclidean and by hyperbolic metricspaces, respectively, where spatial positions of the stable shapes {which are defined in the hyperbolic context as the stable "discrete hyperbolic shapes"} exist within a spatial set, which depends on R(n,0), or Euclidean nspace.
These properties of spatial position and stable existence of a pattern, are used in the (new) model of "interacting (n1)discrete hyperbolic shapes" (ie interacting material components) which are contained in an ndimensional hyperbolic metricspace, where the interaction is determined by the existence of an ndimensional discrete Euclidean shape (torus) which spatially links the spatial positions (ie the distinguished vertexpoints of the discrete hyperbolic shapes) of the interacting material components, and this interaction torus exists in a (an) Euclidean metricspace, where the linking Euclidean ntorus is contained in an (n+1)dimensional Euclidean metricspace.
The math of the interaction is about a differential 2form defined on the ntorus which is contained in (n+1)Euclidean space, where this 2form, defined in (n+1)Euclidean space, is from a local (fiber) space of differential 2forms which have the same dimension as the (tangent to the) SO(n+1) fiber group. Thus, the local transformation of the distinguished points, which identify spatial positions of the material components, is carried out in relation to the toral geometry of the Euclidean ntorus (where the interaction torus forms for each small time interval, defined by the period of the spinrotation of metricspace states (see below), so as to form with the property of actionatadistance for each time interval), thus the spatial displacements of the distinguished points [of the discrete hyperbolic shapes which model the stable material components which are interacting] can be placed in the Euclidean lattice defined in nspace and locally transformed by SO(n+1) in relation to the ntorus geometry, upon which both the 2form is defined, and which is contained in R(n+1,0).
Reiteration concerning quantum randomness
This model of material interaction defined on very small material components, which can be either neutral or charged (and changing very rapidly in regard to their charged properties), would define a Brownian motion, on the set of discrete time intervals [identified by the period of the spinrotation of opposite metricspace states]. Thus, this description is equivalent to the properties of quantum randomness applied to the distinguished points of the discrete hyperbolic shapes, ie the interaction would be both random and appear to be pointlike. [E Nelson showed such Brownian motion {defined for a set of distinguished points} is equivalent to quantum randomness, 1967, 1957 (?)]
Physical properties of metricspaces
The (math) properties of quantity (in regard to spatial measures of position) are contained in Euclidean space, while the properties of stable math patterns, whose positions of their distinguished points are measured in space, are the very stable "discrete hyperbolic shapes."
That is, position and inertia are contained in Euclidean space while charge and energy are contained in hyperbolic space.
This construct of properties associated to metricspaces (of nonpositive constant curvature) implies the existence of pairs of opposite metricspace states. For Euclidean space these are the positions, which are defined in regard to either the fixed distant stars or the rotating distant stars. For hyperbolic shapes this is (+t) or (t), where energy conservation is associated to invariance to time displacements (E Noether), so energy is also associated to time.
Containment within complex coordinate systems, ie finite dimensional Hermitian spaces, and thus related to SU(n), or SU(s,t)
The pairs of opposite metricspace states can be placed in the real and pure imaginary subsets of complexcoordinates. Thus, the new description naturally has a context associated to unitary fiber groups as well as a spin rotation fiber groups. The period of the spinrotation of a metricspace's oppositetime states, defines a time interval (~10^18 sec) within which distinguished points have their positions locally transformed, and the transformation can be in relation to the opposite metricspace states, which are defined between (spinrotation) the set of time intervals, which define a sequence of time intervals.
"Condensation" of material components, and resonance with the finite spectral set of the 11dimensional containment (hyperbolic) metricspace
The ndimensional discrete hyperbolic shapes model stable material components (which possess distinguished points), which are contained in both (n+1)dimension hyperbolic space, and their distinguished points are identifiable within an (n+1)dimensional Euclidean space.
However, it might be noted that, the ndimensional hyperbolic material component can be defined in any dimension metricspace which has higher dimension than (n+1). However, these stable material components would tend to "condense" on (or within, by penetration (within) by means of [relative] highenergy collisions) the stable, highdimensional "discrete hyperbolic shapes" which it encounters, due to either interactions or collisions.
Crystals are about such condensation, but it seems that atoms are also about such condensation, where both crystals and atoms are held together by either resonance with the finite spectral set (of the entire 11dimensional containing space), or (if) by the condensed material being statistically of lower energy (than its thermal environment) so a "discrete hyperbolic shape" can form, but it is not (may not be) in resonance with the (overall stability defining) finite spectral set.
Thus, the energy of a condensed set of material is defined to be as much about either its (unstable) "condensed" discrete hyperbolic shape, or about its inertialinteraction, and its relation to a discrete Euclidean shape.
[Note: The discrete hyperbolic shape exists in a sea of thermal energy, which apparently is relatively low thermalenergy, which allows the condensed state to be slightly stable and to define a "discrete hyperbolic shape" which may not be in resonance with the finite spectral set which defines stability. However, atoms clearly have the property of being in resonance with the finite spectral set, which defines stability within the containment set.
How come atoms are strongly resonant, and thus relatively stable, but many crystals are not "as strongly" stable?
Consider hyperbolic 3space (ie spacetime):
In an 11dimensional hyperbolic metricspace there are, [11C3]=165, such 3dimensional subspaces, where each such 3dimensional subspace has a discrete hyperbolic shape associated to itself (assume they are all bounded shapes, but this assumption is not necessary).
This defines a spectral set for "discrete hyperbolic 3shapes," but the 3flows, which are defined on the "discrete hyperbolic 4shapes," also define part of the hyperbolic 3dimensional spectra for the 11dimension hyperbolic containment set, etc.
However, because we (believe that we) exist within a 3space; it seems that it is most natural that we exist in a spectralorbital structure, wherein, when we transition from 3space to 4space, the discrete hyperbolic shapes get bigger so that
both
the "discrete hyperbolic 3shapes" are contained within the "discrete hyperbolic 4shapes" as free material components, eg atoms,
and
the 3spectralflows defined on the "discrete hyperbolic 4shapes" determine the orbital structure of our planets.
Thus, one asks "What allows atoms to have a resonance with 3shapes?"
In regard to the 3space, within which we (believe we) are contained, there are, 11  3 = 8, separate dimensions, which are separate from the 3space within which we are (or appear to be) contained. Thus, there can exist, [8C3]=56, 3spaces which are separate from the 3space within which we are contained.
Thus, each of these 56 separate 3spaces (in the 8dimensions which are separate from the "given 3dimension subspace" of 11space model of a containment set) can be a part of a "spectraldimensionalsize" sequence of discrete hyperbolic shapes which model the metricspace structure of these 3spaces, and from which a sequence of spectral values is defined.
One can assume that for the set of the 56 different sequences of spectral values
either
the spectral values descend in value (decrease in size) as the dimension increases
or
the spectral values increase in value (increase in size) as the dimension increases.
In the increasing sequence, of spectral values, there can exist orbital properties of charges occupying orbits (or spectralflows), or of condensed material components occupying orbits within a higherdimensional (stable) geometry, as well as the existence of condensed material which determine "free" components (in the higherdimensional hyperbolic metricspace).
How stable spectra are organized in the current model (2012) of physical description
The decreasing sequence of the spectral values would require that the material components be contained in some 6dimensional (or higherdimension) hyperbolic metricspace, with no orbital structures (unless unstable, nonlinear models of material interactions in a manybody orbital system), and perhaps some condensation of components of the same dimension.
This is moreorless the model of materialism in 3space (or in spacetime), where material reduces to points, where, essentially, only nonlinear models of material interactions exist (in a context of indefinable randomness), and stable orbits are supposed to be caused by a 1/r potentialenergy term which is associated to a spherically symmetric forcefield for each pointparticle (ie for a manybody system the math of such a context will be nonlinear).
This seems to not work, yet the idea of higherdimensional spectral structures representing ever smaller geometric properties is the conclusion (or the underlying assumption) of particlephysics and its derived string theory viewpoint.
The existence of both increasing and decreasing sequences of spectral values is not necessary in regard to allowing the atomicsized spectra to exist. Though the existence of the decreasing sizes of spectral structures, as the dimension increases, makes it easier to identify smaller spectralvalues for "discrete hyperbolic 3shapes," and thus it is easier to imagine the spectra of atoms to be stable. However, of the 56 different possible sequences of spectral values, even for all the sequences characterized by increases of spectral values as dimension increases, many of these sequences could have the spectral values associated to the spectral values (sizes) of atoms.
However, if there were no hyperbolic 3spaces whose spectral values were not the size of atomic components, then the (relatively stable) atoms would have to be resonant with the lowerdimension 2spectral flows, associated to the "discrete hyperbolic 2shapes" which are associated to the interaction structures of interacting stable components of charge, namely, the "discrete hyperbolic 3shapes," of which the "discrete hyperbolic 2shapes" are faces (or spectral 2flows), where charges, nuclei, and electron clouds are all (stable) 2dimensional components, though the electron clouds may not be bounded "discrete hyperbolic shapes."
The [11C2]=55 set of "discrete hyperbolic 2shapes," which model the 2dimensional hyperbolic metricspaces and which define, part of, the 2spectra of the 11dimensional containment metricspace (set),... .. where the spectral set of the 11dimensional hyperbolic metricspace is a spectral set which is increased (added to) by the higherdimensional spectra of the discrete hyperbolic shapes, which model metricspaces, and this set of 2spectra (which are subspectra to higherdimensional spectral shapes) can be added to the 55, spectra of the 2dimensional separate subspaces of 11space,
... ., would be small (in spectral value) in the "three 2subspaces" of the particular 3space (defined by the particular "discrete hyperbolic 3shape"), due to the fact that constant factors defined between subspaces can (would) cause these particular subspace 2spectra to possess relatively small spectral values.
as opposed to a spectral sequence whose ****
New material can be defined, which is different from both inertial material and charged material
Consider that, higherdimensional material can exist in R(s,t), where s is the spatial subspace dimension and t is the temporal subspace dimension, and where s+t=n,
However, the interaction of any type of material would be spatially related to R(s,0), by the "new discrete shape's" distinguished point.
In such a pattern one can assume that t increases with each new material type which can be defined.
For example:
The new material types are the "odddimension" and "oddgenus" discrete hyperbolic shapes, whose stable spectral flows (or orbits) are filled with charge, but the geometry of these types of discrete hyperbolic shapes (will) have a geometric charge imbalance, and thus these shapes would naturally begin to oscillate and subsequently they would generate their own energy.
So the two types of (relatively) stable discrete hyperbolic shapes of material components exist in hyperbolic 4space, ie "the stable components" and "the oscillating components."
Thus, there can be defined a R(4,2) space which:
either
contains both material types,
or
it contains the oscillating type,
while
R(4,1) contains the stable type, etc.
SO(4)
Mass can be defined as a circle in Euclidean space, where the mass of the circle is associated to k/r for the circle's radius, r, and some constant, k.
This defines a context for 1dimensional inertial interaction structure within a 2plane, but it can also be naturally related to an R(4,0) and SO(4), construct for changes of material positions. This is because of the geometry of SO(4).
Let R(4,0) have an (x,y,z,w)variable coordinate structure, where SO(4) = SO(3) x SO(3), then consider labeling SO(3)1 x SO(3)2, and a context in which the description is organized within a particular 3subspace structure, so that the SO(3)1 can be related to (x,y,z) subspace of R(4,0), and SO(3)2 is related to (x,y,w) subspace, and then, in regard to a description of material component interactions in the fixed (x,y,z)space, which is the 3space within which the interacting material is assumed to be contained, then [so that] SO(3)2 only affects (x,y) coordinates of the (x,y,z)space.
In this context, in regard to the descriptions of electromagnetism, a "circle" created by conducting material is defined in regard to a (deformed) 2plane, where this 1hole constructed by material, can be used to define the magnetic field aspects of the electromagnetic 2form defined in 4space.
Electromagnetism has its "2form, forcefield" properties identified in spacetime related to the geometry of charge and (usually) 1currents... ..,
in relation to a solution to a wavefunction (defined for both (+t) and (t)) whose solution 1form (supposedly related to potential energy of the electromagnetic system) is assumed to be single valued
... ., but this spacetime structure could just as well be 4space, R(4,0), which can be adjusted to accommodate the existence of the two 3dimensional forcefields, ie E and B, of electromagnetism. The Lorentz force which relates the electromagnetic forcefield (which is defined in a 4dimensional space) to inertial properties, defined in regard to pointvertices of material components, naturally belongs to R(4,0), in a similar way in which a "discrete hyperbolic 3shape" belongs to hyperbolic 4space.
While, in regard to mass, in this (same) context, there seems to be a "true (x,y)plane" associated to gravitationalinertial interactions, which are defined by (in relation to) an interaction 2dimensional discrete Euclidean shape (a 2torus) contained in 3space, so as to yield the math structures of Newton's gravitational laws, where the geometry of the local transformation structure of the SO(3) fiber group in relation to the geometry of the 2form defined on the (Euclidean) interaction 2torus, is consistent with the spherically symmetric geometry of gravitational interactions in 3space.
This would also account for the plane geometry of the solar system, as well as Saturn's rings.
That is, it is the geometric structure of SO(4) which allows us to perceive both gravity and electromagnetism as having particular geometric properties in R(3,0) and/or R(3,1).
Note: There is also the possibility that the 2plane related to SO(3)2 is what causes the inversesquare force field of free, and relatively motionless, charges, modeled as circles (or figureeights) in a plane, to possess a spherically symmetric force field (associated to SO(3)2), whereas the geometry of magnetic forcefields would in this case be related to charged components possessing "discrete hyperbolic 2shapes" being related to SO(3)1, and the (x,y,z) coordinates.
I do not claim to have identified any absolute, precise math structure within which all physical patterns can be described, but this structure "is the structure of stability," for both mathematical and physical description.
It is a simple context which leads to an even richer context for interpreting and modeling observed stable patterns.
These patterns give many possibilities and thus there are many choices needed to be made, in order to determine a more accurate, and sufficiently precise, description, but it is a geometric and stable descriptive structure, thus it is useful for practical creativity.
Thus, the project of finding the finite spectral set for an (or for our) 11dimensional space is of value, though there may be many such, different, finite spectral sets, for many different 11dimension hyperbolic metricspaces. However, coupling between such spaces is (would seem to be) a solvable problem.
This is about the description, the knowledge, of existence, so that we, and our knowledge, are related to the creative process focusing on the creative, manipulateable aspects of existence which could extend the properties of existence, so as to form new properties of an expanded existence.
For example,
"Can an 11dimensional "discrete hyperbolic shape" be created?"
"Can the SO(3), SO(4) interrelated patterns be used in a practical context?" etc.
One wants high academic discussions to be about assumptions, contexts, interpretations associated to simple models of quantity and shape, and dependent on the properties of stability, and related to practical creative development.
The complex development of instruments, and/or fixed narrow descriptive contexts, should be the province of small groups, but the whole enterprise needs to be equal, and based on equal freeinquiry, motivated by visions of creative possibility.
Language can be manipulated in a great variety of ways, and because of the obvious failure of today's technical descriptive languages, new contexts of description should be taken seriously and encouraged, rather than measuring a person's "intellectual value" by their dogmatic purity within a preposterous, and failed, fixed descriptive context, which is primarily associated to only a few complicated instruments (most notably engineering of nuclear weapons).
Education based on developing incomprehensible complexity of descriptive languages, is mostly about the claim that people are not equal.
For example, the discussion of law needs to be about the nature of being human and the types of society one wants to develop, where inequality is about selfishness and extreme violence; while equality is about creativity, knowledge, and selflessness, etc.
Everyone must be considered to be an equal creator.
Copyrights
These new ideas put existence into a new context, a context for both manipulating and adjusting material properties in new ways, but also a context in which life and creativity (practical creativity, ie intentionally adjusting the properties of existence) are not confined to the traditional context of "material existence," and material manipulations, where materialism has traditionally defined the containment of materialexistence in either 3space or within spacetime.
Thus, since copyrights are supposed to give the author of the ideas the rights over the relation of the new ideas to creativity [whereas copyrights have traditionally been about the relation that the owners of society have to the new ideas of others, and the culture itself, namely, the right of the owners to steal these ideas for themselves, often by payment to the "wageslave authors," so as to gain selfish advantages from the new ideas, for they themselves, the owners, in a society where the economics (flow of money, and the definition of social value) serves the power which the owners of society, unjustly, possess within society].
Thus the relation of these new ideas to creativity is (are) as follows:
These ideas cannot be used to make things (material or otherwise) which destroy or harm the earth or other lives.
These new ideas cannot be used to make things for a person's selfish advantage, ie only a 1% or 2% profit in relation to costs and sales (revenues).
These new ideas can only be used to create helpful, nondestructive things, for both the earth and society, eg resources cannot be exploited to make material things whose creation depends on the use of these new ideas, and the things which are made, based on these new ideas, must be done in a social context of selflessness, wherein people are equal creators, and the condition of either wageslavery, or oppressive intellectual authority, does not exist, but their creations cannot be used in destructive, or selfish, ways.
There is inequality, selfishness, fixed authority and intellectual hierarchy, fixed ways of both doing things and organizing society, so that markets are kept fixed, so as to support the social structure of inequality, and law is based on property rights and minority rule (republics), which is violently maintained, knowledge is narrowly defined, and placed into special overly complicated languages, which are mostly associated to some complicated instruments which have been developed, so that experts are the only people allowed to make judgments about knowledgeable matters, and this allows knowledge to be easily controlled, kept narrow, and it helps create the impression that people are not equal, and all of this structure is maintained through extreme violence. Their violent law and violent way of operating will cause the world to be destroyed simply by demanding that resources be misused so as to support their social power.
Equality is about selfless creativity, simplifying language to be placed at the level of assumption and contexts and interpretations where language can be more directly be related to practical creativity, willingness to change both knowledge and social organization, allowing equal freeinquiry, law can be based on equality, and it can be the basis by which to oppose inequality, violence, and selfishness, where the use of resources will have to be balanced and easily changed within society

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