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SO(4) and SO(3) and the geometric structure of space

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 3-space and 4-space, and the various forms of material interactions which exist (and are already used) in these spaces (in regard to force-fields and material interactions),
But on-the-other-hand, one wonders, if we are always a part of a circle-space then,
Why does our "material containing space" not (appear to) have any holes?
How does one re-organize and re-interpret both math constructs as well as the observed physical patterns of stability, respectively, to be able to describe; to a sufficient level of: generality, accurate-precision, and useful relation to practical creativity (or technical development); the stable, definitive, discrete spectral-orbital properties, which can be (or are) used to identify physical systems, and where these stable spectral-orbital properties exist at all size scales (from nuclei to solar systems etc)?

Use the stable patterns of the "discrete hyperbolic shapes" placed into a many-dimensional containment space, ie an 11-dimensional hyperbolic metric-space, where each dimensional level is a hyperbolic metric-space (of that dimension) modeled to be a "discrete hyperbolic shape." But where the entire range of the set of metric-spaces which possess non-positive constant curvature can also be used within the (this new) context, defined by the hyperbolic metric-spaces modeled as the very stable "discrete hyperbolic shapes" defined in a many-dimensional context.
The fact that "discrete hyperbolic shapes" are "circle-spaces" and are composed of toral components, where each toral component models each one of the holes-in-the-space, 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 point-like 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 3-space and 4-space, and the various forms of material interactions which exist (and are already used) in these spaces (in regard to force-fields and material interactions),
But on-the-other-hand, one wonders, if we are always a part of a circle-space then,
Why does our "material containing space" not (appear to) have any holes?

[We are within an open-closed and bounded metric-space, 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 metric-space, which, we believe, contains no holes, but the fundamental domain, within which our metric-space is (truly) contained, is related to a circle-space.]

How does one describe the patterns of "spectral-orbital stability" for "the so very many," very fundamental many-bodied systems, which exist at all size-scales?

These spectrally and orbitally stable material systems are systems which are identified, by the professional math-science community, as being associated to patterns which are (to be) based on non-linear 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 many-dimensional context {Note: R(3,1) is space-time}, ie R(n,1) is a hyperbolic metric-spaces 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 metric-space 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 hyperbolic-dimensions "two to five," inclusive, there exist open-closed and bounded "discrete hyperbolic shapes," which can be used to model metric-spaces. Furthermore, these (bounded) hyperbolic metric-spaces (or discrete hyperbolic shapes) can be
metric-spaces which contain lower-dimension discrete hyperbolic shapes
they can be models of material components which are contained in open-closed, hyperbolic metric-spaces.

D. Coxeter: "Discrete hyperbolic shapes" which are 6-dimensions and higher are all unbounded, and the last defined "discrete hyperbolic shape" has a hyperbolic-dimension of ten (so the proper containing space for stable geometric shapes is a hyperbolic metric-space of dimension-11).

The assumption of the existence of a maximal-size for subspaces of hyperbolic-dimensions five and less (inclusive), for such (possibly bounded) shapes, which are contained within an over-all containing space of 11-hyperbolic dimensions, implies that such an over-all high-dimension containing space has its spectral-orbital structure... ,
{associated to the (mostly bounded, low dimension) discrete hyperbolic shapes which identify its stable geometric-spectral structure}
... ., determined by a finite set of "discrete hyperbolic shapes" which, in turn, determine a finite spectral-orbital 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 metric-spaces 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 11-dimensional containing hyperbolic metric-space, 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 multiplicative-constants 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 metric-spaces and material-component 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 metric-spaces, 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 n-space.
These properties of spatial position and stable existence of a pattern, are used in the (new) model of "interacting (n-1)-discrete hyperbolic shapes" (ie interacting material components) which are contained in an n-dimensional hyperbolic metric-space, where the interaction is determined by the existence of an n-dimensional discrete Euclidean shape (torus) which spatially links the spatial positions (ie the distinguished vertex-points of the discrete hyperbolic shapes) of the interacting material components, and this interaction torus exists in a (an) Euclidean metric-space, where the linking Euclidean n-torus is contained in an (n+1)-dimensional Euclidean metric-space.
The math of the interaction is about a differential 2-form defined on the n-torus which is contained in (n+1)-Euclidean space, where this 2-form, defined in (n+1)-Euclidean space, is from a local (fiber) space of differential 2-forms 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 n-torus (where the interaction torus forms for each small time interval, defined by the period of the spin-rotation of metric-space states (see below), so as to form with the property of action-at-a-distance 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 n-space and locally transformed by SO(n+1) in relation to the n-torus geometry, upon which both the 2-form is defined, and which is contained in R(n+1,0).

Re-iteration 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 spin-rotation of opposite metric-space 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 point-like. [E Nelson showed such Brownian motion {defined for a set of distinguished points} is equivalent to quantum randomness, 1967, 1957 (?)]

Physical properties of metric-spaces

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 metric-spaces (of non-positive constant curvature) implies the existence of pairs of opposite metric-space 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 metric-space states can be placed in the real and pure imaginary subsets of complex-coordinates. 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 spin-rotation of a metric-space's opposite-time 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 metric-space states, which are defined between (spin-rotation) 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 11-dimensional containment (hyperbolic) metric-space

The n-dimensional 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 n-dimensional hyperbolic material component can be defined in any dimension metric-space 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] high-energy collisions) the stable, high-dimensional "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 11-dimensional 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 (over-all 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 inertial-interaction, 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 thermal-energy, 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 3-space (ie space-time):
In an 11-dimensional hyperbolic metric-space there are, [11C3]=165, such 3-dimensional subspaces, where each such 3-dimensional 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 3-shapes," but the 3-flows, which are defined on the "discrete hyperbolic 4-shapes," also define part of the hyperbolic 3-dimensional spectra for the 11-dimension hyperbolic containment set, etc.

However, because we (believe that we) exist within a 3-space; it seems that it is most natural that we exist in a spectral-orbital structure, wherein, when we transition from 3-space to 4-space, the discrete hyperbolic shapes get bigger so that
the "discrete hyperbolic 3-shapes" are contained within the "discrete hyperbolic 4-shapes" as free material components, eg atoms,
the 3-spectral-flows defined on the "discrete hyperbolic 4-shapes" determine the orbital structure of our planets.

Thus, one asks "What allows atoms to have a resonance with 3-shapes?"

In regard to the 3-space, within which we (believe we) are contained, there are, 11 - 3 = 8, separate dimensions, which are separate from the 3-space within which we are (or appear to be) contained. Thus, there can exist, [8C3]=56, 3-spaces which are separate from the 3-space within which we are contained.
Thus, each of these 56 separate 3-spaces (in the 8-dimensions which are separate from the "given 3-dimension subspace" of 11-space model of a containment set) can be a part of a "spectral-dimensional-size" sequence of discrete hyperbolic shapes which model the metric-space structure of these 3-spaces, 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
the spectral values descend in value (decrease in size) as the dimension increases
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 spectral-flows), or of condensed material components occupying orbits within a higher-dimensional (stable) geometry, as well as the existence of condensed material which determine "free" components (in the higher-dimensional hyperbolic metric-space).

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 6-dimensional (or higher-dimension) hyperbolic metric-space, with no orbital structures (unless unstable, non-linear models of material interactions in a many-body orbital system), and perhaps some condensation of components of the same dimension.

This is more-or-less the model of materialism in 3-space (or in space-time), where material reduces to points, where, essentially, only non-linear models of material interactions exist (in a context of indefinable randomness), and stable orbits are supposed to be caused by a 1/r potential-energy term which is associated to a spherically symmetric force-field for each point-particle (ie for a many-body system the math of such a context will be non-linear).
This seems to not work, yet the idea of higher-dimensional spectral structures representing ever smaller geometric properties is the conclusion (or the underlying assumption) of particle-physics 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 atomic-sized spectra to exist. Though the existence of the decreasing sizes of spectral structures, as the dimension increases, makes it easier to identify smaller spectral-values for "discrete hyperbolic 3-shapes," 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 3-spaces whose spectral values were not the size of atomic components, then the (relatively stable) atoms would have to be resonant with the lower-dimension 2-spectral flows, associated to the "discrete hyperbolic 2-shapes" which are associated to the interaction structures of interacting stable components of charge, namely, the "discrete hyperbolic 3-shapes," of which the "discrete hyperbolic 2-shapes" are faces (or spectral 2-flows), where charges, nuclei, and electron clouds are all (stable) 2-dimensional components, though the electron clouds may not be bounded "discrete hyperbolic shapes."

The [11C2]=55 set of "discrete hyperbolic 2-shapes," which model the 2-dimensional hyperbolic metric-spaces and which define, part of, the 2-spectra of the 11-dimensional containment metric-space (set),... .. where the spectral set of the 11-dimensional hyperbolic metric-space is a spectral set which is increased (added to) by the higher-dimensional spectra of the discrete hyperbolic shapes, which model metric-spaces, and this set of 2-spectra (which are sub-spectra to higher-dimensional spectral shapes) can be added to the 55, spectra of the 2-dimensional separate subspaces of 11-space,
... ., would be small (in spectral value) in the "three 2-subspaces" of the particular 3-space (defined by the particular "discrete hyperbolic 3-shape"), due to the fact that constant factors defined between subspaces can (would) cause these particular subspace 2-spectra 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, higher-dimensional 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 "odd-dimension" and "odd-genus" 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 4-space, ie "the stable components" and "the oscillating components."
Thus, there can be defined a R(4,2) space which:
contains both material types,
it contains the oscillating type,
R(4,1) contains the stable type, etc.


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 1-dimensional inertial interaction structure within a 2-plane, 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 3-subspace 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 3-space 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) 2-plane, where this 1-hole constructed by material, can be used to define the magnetic field aspects of the electromagnetic 2-form defined in 4-space.
Electromagnetism has its "2-form, force-field" properties identified in space-time related to the geometry of charge and (usually) 1-currents... ..,
in relation to a solution to a wave-function (defined for both (+t) and (-t)) whose solution 1-form (supposedly related to potential energy of the electromagnetic system) is assumed to be single valued
... ., but this space-time structure could just as well be 4-space, R(4,0), which can be adjusted to accommodate the existence of the two 3-dimensional force-fields, ie E and B, of electromagnetism. The Lorentz force which relates the electromagnetic force-field (which is defined in a 4-dimensional space) to inertial properties, defined in regard to point-vertices of material components, naturally belongs to R(4,0), in a similar way in which a "discrete hyperbolic 3-shape" belongs to hyperbolic 4-space.
While, in regard to mass, in this (same) context, there seems to be a "true (x,y)-plane" associated to gravitational-inertial interactions, which are defined by (in relation to) an interaction 2-dimensional discrete Euclidean shape (a 2-torus) contained in 3-space, 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 2-form defined on the (Euclidean) interaction 2-torus, is consistent with the spherically symmetric geometry of gravitational interactions in 3-space.
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 2-plane related to SO(3)2 is what causes the inverse-square force field of free, and relatively motionless, charges, modeled as circles (or figure-eights) in a plane, to possess a spherically symmetric force field (associated to SO(3)2), whereas the geometry of magnetic force-fields would in this case be related to charged components possessing "discrete hyperbolic 2-shapes" 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) 11-dimensional space is of value, though there may be many such, different, finite spectral sets, for many different 11-dimension hyperbolic metric-spaces. 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, manipulate-able aspects of existence which could extend the properties of existence, so as to form new properties of an expanded existence.
For example,
"Can an 11-dimensional "discrete hyperbolic shape" be created?"
"Can the SO(3), SO(4) inter-related 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 free-inquiry, 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.


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 material-existence in either 3-space or within space-time.
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 "wage-slave 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, non-destructive 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 wage-slavery, 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 mis-used 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 free-inquiry, 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