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A further look at cubical simplexes
by martin concoyle
Email: martinconcoyle (nospam) hotmail.com
11 May 2012
The stable order of material is observed, and the professional science and math communities try to use the ideas (really, dogmas) of indefinable randomness and non-linearity to describe this order, but they cannot, eg the stable spectra of the general nuclei.
The control (and the stability needed to achieve this control) that a living system has over itself is observed, and the professional science and math community, again, try to describe these stable properties with the assumptions, that such descriptions must be based on indefinable randomness, and again they cannot describe these properties, eg the origins of life, and life’s capability to intend and act etc.
Instead begin one’s descriptions with a descriptive math language whose structure is based on “stable math properties,” and see if the stable order of material systems can be described, and if the stable control that life has over itself can be described. [see first page, and see section, Getting serious about math and science, for the math and science part (model of molecular folding (which is highly controllable) is given).]
This is a paper mostly about how the properties of real and complex (fiber) isometry groups can be put in a relation to the stable (discrete) geometric shapes which fit naturally into metric-invariant (base) spaces, ie the stable shapes based on cubical simplexes.
The new context focuses on the math properties associated to stability, ie the shapes based on cubical simplexes, so that these cubical shapes are placed within a math structure of a dimensional layering of a set of containing spaces, so that these containing metric-spaces are, themselves, associated to the stable shapes of cubical-simplexes.
These metric-spaces (as well as Hermitian spaces) exists up to a space-time dimension of 12.
This (new) math structure is very simple, because the stable structures of mathematics are relatively simple, and it is consistent with the observed properties of material which appear (to us) to exist within a space-time dimension of 4 (but now there is a road-map into higher dimensions, so that these higher dimensions turn-out to be the (very real, and macroscopic) structure upon which life is built).
Currently a Hermitian-form invariant math structure fits into a 4-space-time structure, which nonetheless requires the structure of (non-linear) curvature.
The new context allows one to refine the structure of Hermitian-form invariance, where the refinements are geometric with the new dimensional structure much more rich in regard to (useable) geometric ideas, and it allows for more precise and accurate set of descriptions for the observed stable properties of physical systems, rather than an overly complicated description being centered on function spaces and subsequently focused on the useless ideas of indefinable randomness, non-linearity and involved in a process of carefully describing an inaccessible descriptive context, eg string-theory, ie materialism is assumed and the descriptions are outside of the bounds of materialism.
Thus, at best, these so called authorities, are trying to describe properties which only have an indefinable random relation to space-time, so that these properties exist down at the size-scale of mathematical point-particles. They have been trying to make this model useful for over 100 years, but, as yet, they have failed.
The best cultural position for society to consider, is that this attempt to describe the observed stable properties of material systems based on indefinable randomness and non-linearity has failed, and other math-science ideas should be considered. [But it is not the culture which is making such decisions within the US society, rather, it is the few owners of society who decide how language is used within society.]
This is the true failed state of the currently accepted (failed) descriptive structures of math-“science,” if the currently accepted description of matter is science at all? It seems more like arbitrary authority, sort of like a religion of an indefinable “but highly valued” intellect (where social value is being determined by the owners of society), which, in turn, seems more like arbitrary authority upheld by violence. The current authorities have nothing about reality by which to recommend their ideas, they express (authoritative) ideas which are neither accurate nor of any practical value, but the probabilities of random particle collisions are related to rates of nuclear explosions, (thus the military business community makes them authorities), but this particle-collision probability model of reactions is also the 19th century model of chemical reactions, and despite the hype, the chemical engineers are also quite inept, (genetics is being used to put together chemicals whose reactions and structures are not understood) where the inept modern physicists contribute nothing except a useless intellectual hierarchy to the society.
New math structures, cubical simplexes in a containing space of layered dimensions (where the dimensional layers are associated to stable shapes which have stable spectral properties), in a new math context which emphasizes the math properties of stability, accounts for all the observed properties of material where its fundamental descriptions of material interactions are very similar to the math structure of classical physics yet it is a model which implies the atomic hypothesis so that it is a description which has many points in common with both statistical physics and quantum physics and its derivation of the property of quantum randomness are classical interactions (essentially, classical micro-collisions) defined in a statistical setting of atomic interactions resulting in Brownian motions and in the atomic context it is equivalent to quantum randomness (of small components interacting, as waves, between macro (or nano) geometries). However, many of these interactions can be resonant with the over-all high dimensional containing space which can lead to new stable systems (emerging from the interactions), thus it explains stable structures in a direct manner and the math is geometry, ie it is a useful description. Material interactions, in the new descriptive structure, are spherically symmetric in 3-Euclidean-space but not in 4-Euclidean-space, but spherical geometry is non-linear when perturbed and thus it is not useful, it cannot be used to understand the observed stable structures of material systems.
The simple axiomatic context from which this description emanates is from that of: numbers (counting), set containment (finding a context wherein the containing set is finite), operators (grouping, number-type [properties], order of operations), measuring (stable uniform unit), measuring a property as a function of a containment set (of independent) coordinates.
These ideas, though, can be claimed as the readers own, and used without fear of copyright infringement, except that they are original ideas, and thus as an originator of these ideas it must be stated…
(as to what the author’s opinion of the public domain is, as opposed to a justice system wherein the supreme court negotiates the “letter of the law” with the owners of society, for whom the supreme court serves)
… that these ideas cannot be used for military purposes, or for purposes of domination, or destruction, or they cannot be used within a society whose law is “not” based on equality (theoretically US law is based on equality, ie as stated in the Declaration of Independence), where the power to govern is obtained from the people so that the structure of law serves the society (instead of society serving the owners of society), where, in the law, all people are equal (society does owe things to the people, not the people owing to the owners of society, nor should people be attacked by the selfish interests of others).
A dominant unequal group of people within a society cannot acquire selfish value from these ideas, though a creative group of people can use them but not to gain dominance within society.
These ideas (this representation of knowledge) are meant for practical creativity, where the creativity is motivated by selflessness.
People should be opposed to science and math in its current (violently) authoritative state, because of “how it is used within the US society.”
It is used to justify the destruction of the earth in order to maintain the growth of unwise resource dependent big businesses.
It is used to divide the public into “the experts,” who are required to be within absolute institutions which serve big businesses, and “those who are inferior.”
However, because of propaganda, science and math have come to be neither accurate in its (their) descriptions nor is it useful for practical development. This remark is about the genius of the propaganda system, and the high-value identified (by the public, due to propaganda) with experts, ie the failure of the education system.
Where it should be noted that the education system should be based on equal and free-inquiry as well as the relation that equal and free-inquiry has to practical creativity and to new ways in which to build a (precise) language based on assumptions and new contexts.
The religious right uses science in the same manner as do the owners of society, they oppose its development because of their own agenda of domination within some (their own selfish) absolutist context, yet they use any part of it for their own petty needs, eg the fertilized human-egg is a human being.
Perhaps they should consider the equality of all life. But even science is in opposition to this idea, yet science has no valid description of life‘s origins.
In the west it is always the model of an overly domineering religious-owner-empire. (where religion is essentially absolutist dogmas, and this includes both material-based-science and religion)
The Declaration of Independence (of the US) was a split from the western culture of that time, a western culture dependent on domineering psychopathy and a (western) culture which is opposed to equal freedoms (western culture is always placed in a context of absolutists languages).
What is “the west?” Answer: The west is the Judeo-Christain-Islam religious based cultures whose theme, is a culture dominated by the few. In history, its defining point seems to be the Roman-Catholic-Empire begun by Constantine, and perhaps copied by Mohammed. Communism is also an oligarchic social structure, but it has the language of equality and perhaps it was vaguely an attempt to realize equality. Perhaps Lenin wanted to bring America to realize its historic intent of equality, but Lenin’s actions were those of an Emperor. The US Declaration of Independence was a conscious effort to break-apart from the oligarchic social structure of Europe, and to realize a society base on equality and creativity, but the residuals of European economic exploitation seemed to be too prevalent in the US society around 1776. Furthermore, at the time of Lenin, the Robber Barons of the US society saw a threat to their selfish rule due to the rhetoric of equality associated to communism, and its natural resonance with the US society. Thus people of the type as J E Hoover got their jobs within the US justice system which has always been compliant with (or served the interests of) the ruling oligarchy of the US society, despite this being in opposition to US law, as sated by the Declaration of Independence.
If the US society should suddenly flip, perhaps a way to poetic justice would be for many of those in prison (eg the drug offenders) to be given the administrative positions of the 1-million, or so, key managers which are today (2012) within the US institutions, and allow those 1-million administrators to become homeless, and for them to then be treated as the justice system (which they depend on and manage) now treats the poor and “minorities,” but do not allow the (hypothetical) new police-force to murder with impunity. [Perhaps one should seek this justice, and not the western oligarchic model of justice; “of slaughter,” as expressed in the French revolution and by Lenin, if the society’s social structure flips.]
Why is science and math not causing wide-ranging (new) technical development?
Why is the main form of technical development of our society based on 19th century classical physics? eg electronics, optics, thermal physics, and classically defined statistical physics.
Why cannot quantum physics describe, in a convincingly precise manner, the very stable definitive discrete spectra observed for general quantum systems? [Answer: The assumptions which are implicit in regard to containment sets, function spaces, domain spaces, and sets of operators are not sufficient to describe the observed patterns of the material world.]
It is also because indefinable randomness (and non-linearity) is not a valid descriptive basis for the descriptions of the ordered systems which are observed to exist, eg the stable definitive spectral properties of (1) atoms (2) nuclei (3) crystals (4) solar systems etc.
Is the set of overly authoritative assumptions which define the absolute institutions of western culture, such as western science, the basis for the endless failures (of the absolutist institutions of western culture)…,
(absolute institutions) such as the “peer reviewed (to ensure dogmatic purity)” and hierarchical authoritative sciences, whose research is associated to the businesses of the owners of society, eg particle-physics is about nuclear weapons engineering, [Is such hierarchical authority a search for truth? Answer: (no) hardly, it serves business interests, both production and the lack of other creative developments eliminates competition]], and fixed scientific authority is an expression of how the human behavioral trait of autism can be manipulated within the social structures to consolidate a fixed authoritarian basis for the endless failures of math and science, eg cheap clean fusion energy was supposed to be developed by 1955,
…, when it comes to both accurate descriptions…,
(of which very few exist for general quantum systems, neither the general nucleus nor the general atom have valid measurable descriptions, ie the spectra of these systems cannot be found from calculations (which are supposed to be derived from physical law) thus the discrete, stable spectral properties of these systems are [or form] the indefinable set of random events, upon which both quantum and particle physics are, supposedly, based)
…, and when it comes to providing wide-ranging (new) technical development (creativity) on new scientific frontiers?
Why turn science and math into a jeopardy contest? A “jeopardy contest” is a set of dogmatic and fixed interpretations of (or contexts for) “facts.” Narrowly focused attentions, which can be overly obsessive, win the contest, ie the computer as Jeopardy Champion.
Science and math are about developing new languages outside of dogmas and meaningless measurable verifications, remember the model of Ptolemy was measurably verified, furthermore, today the observed patterns are outside of the capacity of the dogmatic laws of physics and math to describe, which is also similar to the manner in which Ptolemy’s model could not keep-up with the observed structures seen in regard to the properties of the planets (eg with the aid of a telescope, the phases of the planets, yet their might have been a way to also fit that data into the scheme of Ptolemy [everything fits into the indefinably random descriptive structure of quantum physics and particle-physics, eg function spaces and elementary particle are equivalent to epicycle structures within a descriptive context (which is based on indefinable randomness and non-linearity, and) which is practically useless]).
Education is a somewhat gentle process guided by inquiry and integrity, not absolutes and authoritative truths. Most of what is being said, in regard to our understanding of the world, is wrong, but it could also be interesting and educational.
Authoritative knowledge is a construct most useful to the owners of society who exist in an unequal and far too narrowly focused society, so that the narrowly defined experts will do what the owners of society want these obedient experts to do.
One uses languages in new ways to discuss “what seem to be interesting ideas.” One does not speak so as to only express authoritative ideas which are absolute truths in a “perfectly structured” language. Rather, language changes at its simplest levels of:
so as to remain practically useful, so that one can create within a context of what one believes “existence to be.”
The only aspects of the current western society which have great success are the social structures of wage-slavery and inequality, and its associated institutional violence, which lends itself to “overly authoritative” technical outlooks (as a [narrowly defined] culture) and a fixed (and development suppressing) means of expressing an authoritative belief structure, and the (old stand-by of western culture) extreme violence, which its dehumanized culture can generate, where this extreme violence is expressed by narrowly defined media and by mean narrowly fixed managers of communication institutions, where extreme violence (which) includes the capacity to uphold a narrow dogma for science through “peer review” so as to demand science authorities to publish within “peer reviewed” journals, a dogmatic version of science, through which scientific authority is defined within the society.
This scientific dogma is then used as an example of an authoritative truth which results in a demand that all publishing uphold a particular standard of accuracy (truth) so that the public will be protected from non-truths which the inferior people will try to instill in the un-witting public. It is, instead, the owners of society who mis-represent the world to the un-witting public.
This idea that inferior people want to interfere with science is usually construed as the expressions of the religious dogmatists.
As long as there is a split between two sets of descriptive language structures, which both claim to express truth….,
so that one language is based on materialism and the other language is based on an assumption of “another world,” from which, so called, non-material experiences emerge, eg emotions, thoughts, life, etc (which the materialists deny, based on both their faith in materialism and their arrogance),
…, then one can exploit the idea of “protecting the truth” of one (all inclusive) dogma from the truth of the other (all inclusive) dogma.
Furthermore, this provides the context in which the idea of superior and inferior people can be expressed, where the superior people are the experts and authorities (the learned ones).
The correct conclusion concerning Godel’s incompleteness theorem…,
(that precise languages have great limitations as to the types of patterns (which exist within the context of precise description) which they are capable of describing)
…, is not that the assumptions involved in the conditions of Godel’s theorem, itself, should be questioned at length (rather one needs to consider the content of the incompleteness statement) but rather that it is the set of assumptions, interpretations, contexts, containments, well defined-ness, meanings, etc* which should be questioned for the whole of the “subject matter” which has been built upon a set of assumptions and interpretations. This can be: science, law, government, media, education etc.
That is, the language of a society should be at the very simple level of assumptions and interpretations concerning all social institutions, that is the public debate needs to exist at the level of an 8-year old, so as to both be meaningful and to facilitate education based on equal and free-inquiry and the relation of a described (measurable) truth to practical creativity. The property of being measurable is not so much about verification, though that is included, but rather measuring is about using a description used to build something, where building depends on measuring and planning and organizing practically useful information when putting together the components of a new system.
The context of assumptions and interpretations and word meaning are the fundamental aspects of forming a precise descriptive language upon which new observed patterns of existence, ie the data, can be described, and new contexts of relations (between properties) can be realized, and new creative developments explored.
One must change, reconfigure, re-organize, and both add new as well as re-word axioms, so as to change the descriptive language and to be able to change the patterns which one can describe and use in a practical measurable context.
Getting serious about math and science
If one criticizes the structure of math, its assumptions, contexts, and interpretations, then one aspect of a math-science description which is “passed over” far too quickly are the: number, geometric, and physical attributes of the property of stability.
Stable, reliable measures of (physical or mathematical) properties eg being able to define and count (random) events, and the property of being able to have within the description a stable uniform unit of measuring, or to have a description which remains quantitatively consistent as well as a descriptive language where the meanings of words are stable and fixed (this is clearly the point where the ideas of absolutes enters the descriptive structure so that limitations of a “language’s descriptive range” enter the constructive process of a precise (measurable) descriptive language) thus a set cannot be “too big,” otherwise “what are considered to be independent properties” can become mixed together within the set which is “too big,” eg the plane filling curve blur’s the distinction between the two different directions of the plane.
For example, Thurston-Perelman suggests that the stable shapes for a dynamic context are the spaces of non-positive constant curvature, where general (and/or non-linear) shapes either disintegrate or evolve towards the (just mentioned) stable shapes (though there may be a few exceptions).
While the useful aspects of the classical descriptions, ie the ideas upon which new creative developments depend, are related to the solvable math structures which are stable, linear, “separable geometries,” which exist in a metric-invariant context for shapes which are often shapes of non-positive constant curvature, though in Euclidean 3-space the geometrically separable shape of spherical symmetry seems to be an important shape for relating spatial displacements of the objects to the material shapes (which determine the shapes of force-fields) which represent the equivalent forces (associated to the spatial displacements) etc.
Furthermore the stable shapes of the spaces of non-positive constant curvature (where spaces of positive constant curvature are the spaces of the non-linear spheres, which are non-linear since their metric-functions change as one changes one’s position along a spherical shape, or in a spherical space), are shapes which are related to the “cubical” simplexes, ie (1) the cubes and (2) the cubes attached to one another at their vertices.
It should be noted that a 2-square can have its opposite 1-faces (or edges) associated to themselves so as to form first a cylinder and then a torus (or doughnut shape). This idea can be continued in a similar manner into higher dimensions (of higher dimensional cubes).
These shapes naturally lead up into higher dimensions, where both material and the metric-spaces, themselves, can both be given a stable shape (associated to themselves), so that these shapes are stable and they carry on themselves stable spectra. Thus these shapes form the model of stable material systems, but since the metric-space can also have these stable shapes (though at a different dimension from the dimension of the material shapes which they can contain), the metric-space can also have a stable spectra associated to their (or its) over-all high-dimensional containment set structure.
Furthermore, an odd-dimensional cubical simplex, which has an odd-genus number associated to itself, could well represent, a relatively stable, but unbalanced (charge), and thus it would be an oscillating, energy generating shape (since based on charge, and thus contained in hyperbolic space), ie a high-dimensional but simple model of life.
This is a mathematical context which the professional mathematicians do not consider, yet it is a context which gives more insight into the issue of “how to organize the math structure which is needed to be able to describe from whence the observed physical order comes?”, ie the observed stable, discrete, definitive, spectral-orbital properties of: Nuclei, general atoms, molecules (and how they fold), crystals (now, often referred to as condensed matter), the solar system, where none of these general systems is capable of being described, to a valid level of precision, by using the currently accepted (and very dogmatic) laws of physics.
Yet modern physics claims these domains as if they possess valid descriptions, while modern physics acts as if the only questions open for careful consideration are about the structure of elementary particle (eg strings) and the relation that these elementary particles have to gravity (or to geometry) and about “what dark matter ands dark energy are?” ie what elementary particle are they to be related?
Everything that is being discussed in this article is concerned with the math patterns which are a part of the correct description of those math-material-containing-space structures which underlie the observed order of the world, and no discussion in Physical Review (or any other professional math-science journal) has much to say (which is of any value) in regard to this problem.
If one is going to describe a stable pattern, either mathematical or physical, then it must be a quantitative (measurable) language based on “cubical” simplexes in a linear descriptive context, in metric-invariant spaces which are also spaces of non-positive constant curvature.
This allows both quantitative consistency, and stability, for the descriptions of measurable patterns, upon which a creative structure can be planned, and parts measured so as to be put together.
Creative development depends on a stable context upon which descriptive patterns are constructed.
This assumes set containment of the pattern (or process) one is describing to be within a quantitative pattern (construction) of independent measuring directions in regard to the measurements of (for) some stable (substance) pattern, and that substance-pattern is the “cubical” simplex.
However, the “cubical” simplex can model both material and its stable containing metric-spaces, which can exist in many higher-dimensional contexts, so that the material simplexes in the different dimensional levels…, or the geometrically-independent geometric processes defined in a context of (on) a “cubical” simplex…, are constrained to their containing metric-spaces (which are also “cubical” simplexes) since higher dimensional shapes (or material) will interact with the stable shapes of the same dimension…,
(higher-dimensional shapes (then the material shapes))
…, which are contained in higher-dimensional [material-containing (metric-invariant)] metric-spaces.
If this rule is not upheld, in a measurable context, which ensures stability of substance as well as the stability of the quantitative sets (which represent measurements of a system’s property), which are used to describe the measurable patterns, then the description:
1. is not quantitatively consistent,
2. does not properly fit into a quantitative measurably distinguishable descriptive structure, so
3. the patterns described become meaningless and
4. are neither accurate
5. nor practically useful.
When considering the set of stable substances (stable math structures) which can be contained in a metric-space, then this set can be limited to a “finite set,” related to the number of separate sub-spaces contained in the various dimensions, up-to Euclidean dimension-11, where 5-dimensional hyperbolic “cubical” simplexes are the last closed and bounded hyperbolic “cubical” simplexes [upon which are based discrete hyperbolic shapes], and furthermore, there is also a set of conformal (constant) multiplicative factors which exist between the different metric-spaces of either the same dimension (but different subspaces) or between adjacent dimensional levels.
This set (of subspaces and sets of conformal factors) would determine the allowed spectral set of the substances (closed and bounded discrete hyperbolic shapes) contained in the dimensionally-layered material containing metric-spaces, ie a mass-energy spectra where, mass = energy, is to be interpreted as the stable charged-system’s energy, contained in hyperbolic space, (which) is related by resonances to the inertial properties contained in Euclidean space, which identify the local linear as well as geometric structure which are associated to material interactions through the action-at-a-distance properties
[of Euclidean “cubical” simplexes]
which are subsequently, also, associated to a geometric-local-coordinate-transformations of the spatial translations of inertial material (interactions) in Euclidean space, in regard to a set of discrete translations of material positions, where the transformations are defined by (both the just mentioned geometric properties and) the discrete time intervals, in turn, defined by the (time) period of the rotation of metric-space states.
of inertial material in Euclidean space in a set of discrete
The material interaction, if within energy bounds, resonates to and “enters into” the set of allowed stable structures, in regard to the spectra of the over-all containing space, due to the discrete subgroup structure of both the fiber group and the containing (base) space, where the brief transition, from one stable system to another stable system, inter-relates both the opposite metric-space states within the material containing space and the opposite vector structure of the fiber (semi-simple Lie) group’s root systems, which is a part of the geometric structure (within the group of local coordinate transformations) which finds an extremum (usually to be thought of as a minimal energy [action]) of an expression for action (usually an integral expression, which involves local spatial displacement transformations of the object) in a closed bounded shape of an allowed shape for the “material end-point” of the material interaction. That is, the resonance and allowable energy values cause the opposite metric-space states (in the base space) and the opposite root system structures (in the fiber group) to become consistent with one another, so that this is related to the maximal torus of the fiber group.
One must consider the stable shapes which exist in a metric-invariant context. That is, the diffeomorphism group as a fiber group over a non-linear geometry is describing an unstable and quantitatively inconsistent context which cannot accurately describe stable structures and is unrelated to practical development.
Only the non-positive constant curvature spaces have metric-functions with constant coefficients and thus their local descriptions can be related to a linear, metric-invariant context wherein the stable geometries have the property of being parallelizable and orthogonal (or geometrically separable shapes, [as they have been called in these papers], ie the local coordinate directions are always perpendicular to one another). Being metric-invariant allows these shapes to be related to some of the classical Lie groups [compact Lie groups, and its desirable that they be connected lie groups, ie compact, connected Lie groups]
eg SO(s,t), where s + t = n, where s is the spatial subspace and t the temporal subspace
(Note: The maximal torus will be defined in regard to the, s, subspace), SU(s,t), Sp(2s), etc. where Euclidean space is SO(n,0).
The simplest spaces to consider are SO (n,0) and SU(n,0) while SO(n-1,1) and SU(n-1,1), and SO(s,t) and SU(s,t) can also easily be considered.
If n is odd then SO(n) is related to SU((n-1)/2), if n is even then there is an extra (real) dimension in SU(n/2) in regard to its dimensional relation to SO(n).
The geometric shape which is being inter-related [between shapes in the base space and a maximal torus in the fiber group] is concerning (about) a geometric-coordinate relation between R(n) and C(n), where, in turn, complex circles, in C(n), touch opposite faces of a cube, R(n)/Z(n), where Z(n) are the integer coordinates (or integer lattice), and the plane (which is defined by the complex circle) is to be normal to each n-direction in R(n). Furthermore, the cube, R(n)/Z(n), where Z(n) are the integer coordinates, can also be thought of as a torus, T(n), a toral shape which is contained in the base space (of a principle fiber bundle), but this torus (in the base space) can be related to an equal dimension torus in the maximal torus of the fiber group.
If one assumes that the rank (ie the dimension of a maximal torus, T(n) (in)) of the isometry fiber group (over the base space) is n…, then n-cubes in R(m) [one may assume m=n (to make the idea easier to grasp), though m>n] correspond to maximal tori, T(n), in the fiber group.
The math of the fiber group must preserve metric-invariance, while the tori within the metric-space (base-space of the principle fiber bundle) can have different sizes. Otherwise the cubical tori in the two spaces (ie the fiber group and the base-space) are geometrically similar, and thus they also possess “conformally similar” spectra (or equivalent, but perhaps the different independent circles, which are components of a torus, are related to different spectral values [ie different size circles]).
[Note: Closed curves on the tori can be defined as lines with rational slope defined on R(n) which must, at some lattice point, intersect with an element of the lattice Z(n). These closed curves define the spectral lengths on a torus. However, one can think of the spectra as being the different edges of the cube R(n)/Z(n).]
(For a Lie group’s associated Lie algebra) a maximal torus so that
How are a set of Lie algebra “vectors” (in the adjoint representation of the Lie algebra)…,
which exist in Weyl chambers of the root space decomposition of the “vector space,” and which are associated to an (n-dimensional) maximal torus subgroup (of the associated Lie group),
…., related to the properties of arc length of the maximal torus’s unit circle components, and in turn, these arc-lengths are related to (eg equal to) the arc lengths of a corresponding set of circle component of an n-dimensional torus in a (base-space) metric-space (of a principle fiber bundle)?
This could be related to either a metric-space toral shape or to a toral component of a “discrete hyperbolic metric-space geometric shape” which has toral components.
When one wants the arc-length of a unit circle (within a maximal torus) to equal to 10, then define the “angular argument,” Y, of exp(iY) (ie on the unit circle), to be, [2(pi)/10]t, for t a real number, where t is the arc-length along a unit circle component of a (given) maximal torus, then when t=10 the argument will be 2(pi) and a full circuit (the first full circuit) around the given unit circle (component of the maximal torus) has been traversed.
Thus, the length around a unit circle (component of a maximal torus), when there is a factor of 1/10 in the argument of the circular component, is 10, where the factor, 1/10, is the value (or slope) of some Lie algebra vector, which is tangent to the maximal torus at the identity, and this “vector” is in some Weyl chamber. The slope of the Lie algebra vector can be determined as the tangent function of the angle which can be determined (using inner products) between a root vector of the set of Lie algebra “toral vectors” and another Lie algebra “toral vector.” Whether, the value of this slope (or angle between the set of nearest root vectors in the Lie algebra of a maximal torus), as in this case 1/10, is in one Weyl chamber or another depends on whether the angle between the nearest root vector (in regard to some given original [or first] root vector) is less than 90-degrees. If it is, then more than one Weyl chamber needs to be considered.
Does this angular relation between root vectors (in the Lie algebra of a maximal torus) provide a valid pattern to consider in a plane, since the real irreducible “diagonal” subspaces…, which are related to a particular circular component of a torus…, are 2-dimensional?
This factor (or value of a tangent function at a particular angle measured between a vector and a given original root vector), eg in the above case, 1/10, corresponds to a length of 10 around a unit-circle component of a maximal torus. However, it can also correspond to the measure of a circle’s circumference for a circle component of a torus contained in a metric-base-space, where in the metric-base-space the circumference of a circle component is 10 = 2(pi)r, and r = 10/2(pi), for the circle component of the torus contained in the metric-space (base-space).
Thus Lie algebra “toral vectors” can be related to tori contained in metric-spaces which have various sizes, where the “toral vectors” of the Lie algebra can be associated to particular conjugation classes of the Lie group.
This context seems to provide a description in regard to the range of arc-lengths…, for circles of a toral component (or a cubical component) defined in the base space…, which can stay within a given conjugation class of a maximal torus in the fiber group. Thus, if the spectral lengths (of circular components) stay within this range, then the corresponding stable geometric shape, in the base space, can have its shape related to a set of diagonal “local coordinate” transformations which “map out” the full geometric shape in the base space, ie the description of the stable shape in the base space stays within the simple context of diagonal local coordinate transformations. How can this be related to the math structures of opposites, where the structure of mathematical opposites (or involutions, also associated to a function [or operation] being its own inverse, eg multiplying by -1, or multiplying by the complex number, i, or multiplying by diagonal matrices which have 1’s and -1’s along the diagonal, defining reflections in a geometric context, etc) are so prevalent in the root space and in the cubical foundations of these stable shapes (in the base space), as well as to the structure of opposite metric-space states, through which the (physical) properties of metric-spaces, and spin rotations of metric-space states can both be described?
Is this the natural limit as to the types of spectra that a single stable shape can possess in the base space?
Can discrete conjugation (perhaps taking place at the distinguished point of the stable shape in the base space) between conjugation classes still be allowed for a stable shape in the base space to have a wider range of spectral values associated to itself?
If such a particular type of discrete conjugation take place so as to increase the spectral possibilities for stable cubical shapes in the base space then could these discrete conjugations (which change maximal tori in the fiber group) also be related to the folding structure of the stable geometric shape in the base space? This folding, based on discrete conjugation, at a specific geometric location within a stable shape, is a very good model of molecular folding, so that this folding could exist in a very controllable context.
torus in the arc-length
Consider a Euclidean 3-cube, there are three sets of 4-edges, ie 3 x 4 = 12 edges on a 3-cube, where the length of the edges (in one of these three sets) is equal (to, say, 10), then one wants to associate the length 10 to a related length of a circle’s circumference for a circle component on (of) a torus contained in a metric-base-space. In turn, one wants the torus contained in the base space to be related to the circle components of a maximal torus in the fiber group.
*The 3-cube is a 3-torus in a 4-dimensional Euclidean metric-space, where Euclidean 4-space is related to a fiber group SU(4) whose rank is three, ie the dimension of a (any) maximal torus in SU(4) is three. That is, the dimension of the 3-cube and the dimension of SU(3)’s maximal tori are consistent.
The spectra of Euclidean n-cube (or n-subcube) is exactly similar to the spectra of a maximal tori (of rank-n) in the fiber group of SO, while the geometry of the (very stable) discrete hyperbolic shapes in the base space of a SO(n-1,1) fiber group, is a shape which is (can be considered to be) equivalent to cubes (of the appropriate dimension [equal to the rank]) attached only at vertices, so as to form a diagonal of cubical shapes, where each cube in the diagonal would possess spectra similar to the spectra of a maximal tori in the fiber group, ie the edges of the cube. Note: The spectra of the discrete hyperbolic shapes are much more limited than are the spectra of the discrete Euclidean shapes, where the spectra of the discrete Euclidean shapes are all the rational sloped lines, whereas the spectra of the discrete hyperbolic shapes are (essentially) only the edges of the cubes.
This defines a context wherein the local transformations of a shape on the base space can be determined by a set of diagonal matrices in a particular maximal tori of a fiber group, where the circles (or higher-dimension faces) which compose the maximal tori correspond to the stable circular paths (or bounding shapes) around the tori on the base space.
The context of non-linearity cannot stay within a context of coordinates related to the maximal tori, ie non-linear shapes are transformed in a continuous path away from a diagonal context of local coordinate transformations associated to the non-linear shape.
The problem with toral-shapes in the Euclidean base-space is that either cubes or rectangles are equally “good” shapes upon which a torus can be defined (in a base space), and thus tori (in a base space) have the property of continuity but it is difficult to maintain the specific conditions for the stability of some specific rectangular shape (eg a rectangular prism) which exists in the base space, in relation to any other such rectangular shape. Whereas the discrete hyperbolic shapes, ie the shapes associated to a diagonal set of cubes attached at vertices, are very stable in hyperbolic space, ie associated to an SO(n-1,1) fiber group.
So how are the spectra of the diagonal cubes (or cubes in Euclidean space) defined in the base space, related to the spectra of maximal tori on the fiber group?
There is either a set of discrete local maps [f(g,t(i),x) to Lg(x,t)(dx) at t(i)] from the fiber group onto (changes of material positions, x, on) the base space. [in this set of conditions the “opposite roots”
(dg and -dg) define local opposite directions (for the local maps) for a pair of separate (opposite moving) paths whose interaction structure both begins and ends.]
Due to the system (in the base space) being at the “right” energy and there existing resonances, so that the “opposite roots” define a pair of opposite (and consistent) paths (orbits).
There is an orbit of a point, x, of a (material) system in the base-space of the principle fiber bundle due to a toral subgroup (of the fiber group) forming the orbit by the subgroup acting on x in a continual manner,
(however, this orbit space could also be a result of locally discrete actions by the fiber group on the orbit of x in the base space).
One can think of the (either discrete or continuous) action of the toral subgroup on x is expressed as a circular subgroup, S (of the subgroup T(n)), acting on x, as Sx, so as to form a circular orbit (or a stable closed orbit) in the base space. Thus, there could be a further (periodic) scalar function, h(x,t), (based on geometric scalar values, and physical constants) so that the orbit [of a material system] in the base space is given by h(x,t)Sx (along with its opposite orbit).
That is, the dynamics of material contained in a metric-space can be described by a mechanism which most often leads to non-linear patterns since the descriptive structure is so very similar to the (non-linear) connection-form related to the derivative (but in a non-linear context), where the derivative is a local linear and thus allowing a consistent measuring relation to exist between a function’s values and the same function’s domain values, yet the interaction structure between stable shapes can at certain times during the interaction enter into resonance with an equivalent (dimensional) stable discrete shape which is part of (or contained within) the over-all high-dimension containing metric-space (base-space), and thus the relation between that stable shape (if the interaction has the “correct” energy) and the containing (hyperbolic, Euclidean, Hermitian (unitary), symplectic) metric-space’s fiber group is quite different (than is the interaction process, though they are consistent, but this consistency depends on the interaction suddenly being related to a new resonating structure, eg new dynamic pathways, which the containing space (itself) upholds or supports. That is, the material of the interaction gets related to a new material [or new metric-space] shape related to the fiber group’s orbital structures, where the cubes (tori) within the group get related to the cubes (toral components) of [within] a containing-space shape [cubes related to the shape of a containing space]), ie the stable under-pinnings of a (the) containment set are built from the properties of the fiber groups, namely, their discrete subgroups, which are subsequently related to the fiber group’s maximal tori.
Can it (a locally diagonal relation between a dynamical path and its representation as an action by a fiber group) stay within the diagonal context of a maximal torus, or does it need to continually depart from a diagonal relation?
Can it (a locally diagonal relation between a dynamical path and its representation as an action by a fiber group) stay within the diagonal context of a maximal torus, or can it discretely change to other maximal tori?
Sets of discretely distinguishable opposites (sets of opposite roots, or characters, or eigenvalues) defined on a maximal torus (or its tangent space, within the context of the adjoint representation), in turn, define a discrete partition of a Lie group’s orbit structure into conjugation classes associated to these toral eigenvalues. This is observed to be due to the properties of reflections which can be defined for vectors [defined on the local tangent space of a maximal torus (at the identity)] (equivalent to eigenfunctions) through the hyper-planes which are defined by the pairs of opposite characters (or roots). Does this observed pattern mean that sets of opposite eigenvalues are central to the stable orbital properties of dynamic or material interaction systems in the base spaces of “isometry fiber bundles?” Is a system of opposite values, opposite directions, opposite states, and geometrically opposite orbital structures the basis for stability of an orbital system, or for (continuously) staying within a conjugation class?
Would such discrete changes in a conjugation class be related to an associated stable shape, in the base space, being folded (associated by means of orbital properties of the shape)?
For the diagonal array of cubes, composing a discrete hyperbolic shape, can each cube be related to the spectra of one of the maximal torus?
Does each cube need to be related, by conjugation, to the spectra of some of (all) the other different maximal tori?
Consider acting (by means of the fiber group) on the base space with the toral transformations of the fiber group, so that the rank (the dimension of a maximal torus), n, is the same dimension as corresponding cubes in the base space, then the diagonal (of a matrix in a torus of a particular conjugation class) is filled with the same value exp (i 2(pi) t(j)z(j)), where the j’s are summed to n, and the t(j)’s are real numbers from the real number continuum of the diagonal Lie algebra elements corresponding to the “diagonal Lie group elements” (from the maximal torus) and the z(j)’s are from the integer lattice Z(n). For the value t(j)z(j))…, where the j’s are summed to n…, to represent a closed curve on a torus, then either the value 2(pi), in exp (i 2(pi) t(j)z(j)), needs to be multiplied by a constant 1/a(j), or simply letting the a(j) be arbitrary real numbers]. For if the a(j) are constant real numbers then the [z(j)/a(j)]x=y line on the (x,y)-plane will intersect an element of the Z(x,y), or integer lattice defined on the (x,y)-plane when x=a(j)/z(j), thus assuring that such a curve will be closed on the T(2), or 2-torus, (or 2-subtorus of T(n)). Note: x is equivalent to the values t(j).
If one considers the extra (constant) factor of 1/a(j) then the different cubes (in the base space) associated to different a(j) values, in exp (i 2(pi) [t(j)/a(j)]z(j)), can have different relative sizes in the (base) metric-space (so as to be easily related to the maximal tori), in turn, associated to the eigenvalues,
exp (i 2(pi) [t(j)/a(j)]z(j)) (or conjugation class functions). This would allow for more variability in regard to the set of eigenvalues which can be defined on the base space.
On a maximal torus (or on a conjugation class within a fiber group) the function value,
exp (i 2(pi) [t(j)/a(j)]z(j)), (which identify “values” on a circle) remains a constant (stays [or is well defined eigenfunction] on the circle).
Conjecture: This corresponds to the lines in the Lie algebra associated to the Weyl chambers. This means that the range of values associated to (or allowed by) a particular conjugation class is determined by the values of slope, z(j)/a(j), (of the vectors in the Lie algebra) which stay within the “bounding walls” of the Weyl chambers, which, in turn, are determined by the root system structures, ie vector relations characterized by sets of opposite roots (or opposite vectors), related to the adjoint representation.
Assume that there is a set of “values” of the form, exp (i 2(pi) [t(j)/a(j)]z(j)), which define the set of values which are allowed on a conjugation class. The different functions of the form exp (i 2(pi) [t(j)/a(j)]z(j)), identify the different conjugation classes [on the adjoint representation of a Lie group] of the group (as well as identifying the different set of conjugate tori which cover the Lie group). Are these different conjugation classes defined by ranges of a set of values (or set of functions from the torus into the circle)? [If so] What is the range of allowed values (of these constant values which is associated to one conjugation class)?
That is, is there a set of such values (functions onto a circle) which define the same conjugation class? Answer: Yes, this would be the set of values which range over (z(j)/a(j)) as this slope stays within the Weyl chambers.
This is important, since each such different value (on one conjugation class) can be associated to one of the different cubes in the (base space) structure of the diagonal of cubes (which touch at their vertices). What range of constant values can be associated to a single conjugation class of a classical Lie group?
If the (fixed) value associated to a cube (in the diagonal of cubes in the base space) is not the value (of the eigenvalue of the maximal torus, which one wants) of a particular conjugation class (if this is truly the case for conjugation classes), then one would have to (discretely) conjugate between the different maximal tori in the fiber group, by means of the Weyl group. Thus, the eigenvalues of the set of diagonal cubes in the base space can have a wide range of values associated to any of each of the separate cubes, if the eigenvalue of each cube is discretely related to the (various) eigenvalues of maximal tori in the fiber group.
Is it “one eigenvalue for each conjugation class?”
Is it “A combination of a both a continuous range of eigenvalues defined by a continuous group action within each Weyl chamber, ie within a conjugation class, and (also) discretely related eigenvalues, defined by a discrete group action between the Weyl chambers” ?
If only one value (only one function [whose values are defined on a circle]) is allowed for one conjugation class to be associated to a cube, then the set of maximal tori (which the full set of conjugation classes identify) would then have to also be “the same set of values from whence the different cubes (in the base space, of the diagonal of cubes) are allowed to have the different values for their eigenvalues,” (so that the eigenvalues of the other cubes in the diagonal of cubes in the base space are different). That is, for a different cube in the diagonal of cubes on the base space to have a different value for an eigenvalue (from the eigenvalues of the other cubes) one would have to (discretely) conjugate between the different maximal tori in the fiber group, by means of the Weyl group, W = [The normalizer of T(n)]/ The centralizer of T(n) = (N(T(n))/C(T(n))=(N(T(n))/T(n))). where the order of W (the number of elements in W) is finite.
For example, in SU(n) the Weyl group, W = [The permutations of n-things, the symmetry group of n-things, ie n! permutations, on the n-elements which are defined along the diagonal of T(n)].
Note: The normalizer of T is the biggest subgroup in G within which T is a normal subgroup. The centralizer of T are the elements of G which commute with all the elements (or with each element) of T (one might believe that for (a) T this is T itself, ie the centralizer of (a) T is T itself, (yes, this is correct)).
But (if it is true that each conjugation class can only have one eigenvalue) and if each different cube has a different eigenvalue, then this would mean that there can only be as many cubes along the diagonal of cubes (in the base space), each with different eigenvalues, as there are conjugation classes in the Lie group.
Does a description based on non-linear geometry have any meaning, other than it being a quantitatively inconsistent description?
Does indefinable randomness have any meaning?
Do descriptions which are based on sets which are too big have any meaning? Do distinguishable patterns blend into other, different, distinguishable patterns so that the original distinguished property has no meaning, because the set is “too big” ? Are opposites truly distinguishable from opposites in sets which are “too big” ?
Can these math properties of: non-linearity, indefinable randomness, and defining math patterns on sets which are “too big,” be used to identify (describe) stable properties (stable patterns) in a accurate manner (or are the descriptions unstable), and can these properties be used to build something (based on inter-relating measurable properties of the new system’s regions and/or components) which is practically useful?
The above description based on cubical-related stable geometric shapes is ultimately based on a finite set, namely, the finite set of stable spectra which can be fit into an over-all high-dimension containing space where the containing spaces have stable shapes. This finite set determines the set of material or metric-space types which can compose the containing space. The highest stable dimension is an 11-dimensional hyperbolic metric-space, but there are no 11-dimensional hyperbolic shapes.
This work is in the public domain