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Discover Bumpy Space

Outline

Discussions about space often start with proclamations of continuity because it is mathematically convenient and traditional. But EthnoPhysics starts with the premise that we can understand space based on sensory experience. And experimental observations clearly show that space is quantized. Moreover, quantization is a logical consequence of categorical descriptions, even for a continuous sensorium. So here is an exploration of spatial concepts based on analyzing discrete sensory experiences: Bumpy space.

Cartesian axes compared to the electric and magnetic axes.
Cartesian axes compared to the electric and magnetic axes.

Spatial Axes

Let events in the history of some particle be described by their phase angle  \theta. And recall that the unit vectors \widehat{m} \equiv (1, 0, 0), \widehat{e} \equiv (0, 1, 0) and \widehat{z} \equiv (0, 0, 1) mark the magnetic, electric and polar axes. These algebraic entities can be used to construct another set of vectors. To start, the axis of the abscissa is defined from scalar multiples of

\widehat{x} \equiv \cos{\theta} \, \widehat{m} - \sin{\theta} \, \widehat{e}

And similarly the ordinate axis is composed from multiples of

\widehat{y} \equiv  \sin{\theta} \, \widehat{m} + \cos{\theta} \, \widehat{e}

These definitions can be rearranged to give \widehat{m} = \cos{\! \theta} \, \widehat{x} + \sin{\! \theta} \, \widehat{y} and \widehat{e} = \cos{\! \theta} \, \widehat{y} - \sin{\! \theta} \, \widehat{x} . The new axes can be visualized by rotating the electric and magnetic axes by \theta degrees around the polar axis. The new vectors together with \widehat{z} are called Cartesian basis vectors.

Spatial Extension

Let some particle P be described by its mechanical energy  E, and its angular momentum quantum number \textsl{\textsf{J}}. The orbital radius of P is defined by

R \equiv \dfrac{hc}{2\pi} \dfrac{ \sqrt{\textsl{\textsf{J}} \; }}{E}

ShapeDefinition
a pointR = \lambda = 0
a line\lambda > R = 0
a plane R > \lambda = 0
a discR \gg \lambda > 0
a rod\lambda \gg R > 0
a cylinder\lambda \sim R > 0

where h,  c and  \pi are all constants. Since  E and \textsl{\textsf{J}} have been defined from tallies of quarks, the orbital radius is thus established from quark counts too. We use it to make a rudimentary account of the expanse or extent of P.

A particle that is described by its wavelength  \lambda and orbital radius  R may be classified by its shape as shown in the accompanying table.

Any non-rotating particle has no orbital radius because \textsl{\textsf{J}}=0 when N^{\mathsf{U}}=N^{\mathsf{D}}. So particles shaped like points or lines are mathematically described by saying   R \! = \! 0 \, .

For photons, the angular momentum \textsl{\textsf{J}} is always one. And the wavelength  \lambda is related to the photon’s energy by  \lambda = h c \! / \! E. So the orbital radius of a photon can be written as R \left( \gamma \right) =  \lambda / 2\pi \, . Then a circular perimeter of 2 \pi R is the same as one wavelength.

Spatial axes are suggested by this French engraving of sea urchins.
Jean-Baptiste Lamarck, Echinus, Tableau Encyclopédique et Méthodique des Trois Règnes de la Nature, Paris 1791. Photograph by D Dunlop.

Displacement

Let some particle P be characterized by a repetitive chain of events written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \, \ldots  \, \right)

where each repeated cycle  \mathsf{\Omega} is the bundle of quarks

\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2}, \,  \mathsf{q}_{3} \, \ldots \, \mathsf{q}_{ N_{ \! \mathsf{q}} } \right)

The total number of quarks in  \mathsf{\Omega} is  N_{ \! \mathsf{q}} , a finite integer. The orbital radius of P is noted by  R ,  \lambda is its wavelength and  \theta is the phase angle. These characteristics are used to define the following quantities for describing events

d \! x \equiv  -4 \pi \! R \sin{\theta} / N_{\! \mathsf{q}}

d \! y \equiv  4 \pi \! R \cos{\theta} / N_{\! \mathsf{q}}

d \! z \equiv \lambda / N_{ \! \mathsf{q}}

Then using the Cartesian basis vectors  \hat{x},  \hat{y} and  \hat{z}, the displacement of P is defined by

d \! \bar{r} \equiv d \! x \, \hat{x} + d \! y \, \hat{y} + d \! z \, \hat{z}

From here on, we switch to implicitly using Cartesian basis vectors, and the displacement is written as d \! \bar{r} = ( d \! x, \, d \! y, \, d \! z ) \, . Note that  N_{ \! \mathsf{q}} appears in the denominators of all these expressions and can be eliminated using d \! \theta the phase angle increment to write

d \! \bar{r} = R \! \left(\rule{0px}{11px} - \! \sin{\theta}, \; \cos{\theta}, \; \lambda / 4\pi \! R \, \right) d \! \theta

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of quarks in a description may be large, but not infinite. In principle  N_{ \! \mathsf{q}} is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we may assume that  N_{ \! \mathsf{q}} is large enough to make an approximation to spatial continuity, only then allowing for the use of calculus.

Position

Let particle P be described by a chain of events written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \, \ldots \, \mathsf{\Omega}_{k} \, \ldots  \, \right)

where each cycle is characterized by  d \! \bar{r} its displacement vector. Then the position of the kth event is defined by the sum

\displaystyle \bar{r}_{k} \equiv \bar{r}_{\mathsf{o}} + \sum_{j=1}^{k} d \! \bar{r}_{j}

where  \bar{r}_{\mathsf{o}} is arbitrary. By this definition the displacement is an incremental change in the position of P. That is, the displacement can be viewed as the smallest possible change in  \bar{r} that arises from an interaction with a single quark.

If all events are assigned a position, then  \Psi can be expressed as a historically ordered set of position vectors written as

\Psi \! \left(\bar{r}\right)^{\mathsf{P}} = \left( \bar{r}_{1}, \, \bar{r}_{2}, \, \bar{r}_{3} \, \ldots \,  \bar{r}_{k} \, \ldots \, \right)

Consider a pair of events from  \Psi noted as  \mathsf{P}_{i} and  \mathsf{P}_{\! f} \, . The separation vector between these two occurrences is defined by

\Delta \bar{r} \equiv \bar{r}_{\! f} - \bar{r}_{i}

And the norm of their separation vector is defined as the distance between events

\Delta r \equiv \left\| \Delta \bar{r} \, \right\|

Spatial Quantization

The foregoing definitions imply that position, separation and distance are all quantized. Their variation is discontinuous because EthnoPhysics is based on a finite categorical scheme of binary distinctions.

Shapes are an important way of classifying experience as suggested by this French engraving of sea urchins.
Jean-Baptiste Lamarck, Echinus (detail) , Tableau Encyclopédique et Méthodique des Trois Règnes de la Nature, Paris 1791. Photograph by D Dunlop.

Quantization comes from the logical structure of the descriptive method, even for a continuous sensorium. In principle, motion is always some sort of quantum leaping or jumping from event to event. Phenomena like this have certainly been observed in twentieth-century physics and can, for example, be used to understand Zener diodes and the Stern–Gerlach experiment.

For EthnoPhysics, smoothly continuous motion is therefore presumed to be a macroscopic approximation. We are cautious about using calculus because the logical foundations of both differential and integral calculus are proven using assumptions about continuity. So EthnoPhysics does not require calculus. Instead calculations are designed to be implemented on digital computers, in a finite number of discrete steps.

Cartesian Coordinates

For EthnoPhysics, the displacements  d \!x ,  d \! y and  d \! z are finite increments of position. They are not infinitesimal. So we can just add them together in sums like other finite numbers. Thus the  x-coordinate or abscissa of event k is defined by

\displaystyle x_{k} \equiv x_{\mathsf{o}} + \sum_{i=1}^{k} d \! x_{i}

where x_{\mathsf{o}} is arbitrary and often set to zero. The  y-coordinate or ordinate is defined as

\displaystyle y_{k} \equiv y_{\mathsf{o}} + \sum_{i=1}^{k} d \! y_{i}

And the  z-coordinate or applicate of event k is

\displaystyle z_{k} \equiv z_{\mathsf{o}} + \sum_{i=1}^{k} d \! z_{i}

Recall that the  z-component of the displacement  d \! z is not a function of k or the phase angle. So if P is isolated then it moves in regular steps along the polar-axis, and the applicate of P can be written as

z_{k} = z_{\mathsf{o}} + k  d \! z

The three numbers  x,  y and  z are called the Cartesian coordinates of event k after the work of René Descartes . More exactly, they are the rectangular Cartesian coordinates. We use them to express the position of an event as \bar{r} = ( x, y, z ) where \bar{r}_{\mathsf{o}} = ( x_{\mathsf{o}}, y_{\mathsf{o}}, z_{\mathsf{o}} ) .

Spatial Isotropy

The hypothesis of spatial isotropy is a presumption that almost all of the particles in a description have phase symmetry, and also charge symmetry along both the magnetic and electric axes. This condition is easily satisfied for protons, electrons and hydrogen atoms.

The hypothesis is useful because it implies that even if the phase  \delta_{\theta} \, , the magnetic polarity  \delta_{\hat{m}} \, , or the electric polarity  \delta_{\hat{e}} \, , get mixed-up and change sign, the overall description of a particle remains unaffected. And if almost all particles share these symmetries, then we can greatly simplify analysis by usually ignoring  \delta_{\theta} \, ,  \delta_{\hat{m}} and  \delta_{\hat{e}} \, . These quantities determine the spatial orientation. Disregarding them implies that any specific direction is just about the same as another. That is why it is called a hypothesis of spatial isotropy.

XXX

Sensory interpretation: The phase can be explained as a representation of black and white sensations, the magnetic polarity depends on red and green perceptions, and the electric polarity is defined from blue and yellow impressions. So exercising this hypothesis, and setting aside further consideration of these explanations, is a way of objectifying a description. We stop paying attention to if an event looks black, or white, or red, or yellow, or any other color. Moreover interpreting the phase as some time-of-day becomes irrelevant. The assumption of spatial isotropy is an important way for descriptions to transcend these sensory details.

Under some circumstances the hypothesis may replace our earlier use of reference sensations. Indeed, here is a quick look at a three-step plan for systematically glossing over visual perceptions to obtain the necessary conditions for spatial isotropy.

First, Take a Different Viewpoint

As a first step to isotropy, the descriptive framework is changed from quark space to an ordinary space with a Cartesian coordinate system. Then by definition the electric and magnetic axes will automatically revolve around the polar axis of any particle as it moves. In the Cartesian view, particles are rotating. The four quark coefficients that describe the distribution of rotating quarks  n^{\mathsf{u}},  n^{\mathsf{\overline{u}}},  n^{\mathsf{d}} and  n^{\mathsf{\overline{d}}} are then used to define four atomic quantum numbers  \mathrm{n},  \ell,  s and  j that describe angular momentum. Explicit descriptions of colors like black, white or grey, are then dropped.

Second, Objectify Visual Details

The second step toward isotropy is to disentangle all the visual sensations involved in experimental research, and reconstruct them as factual reports.

A vast quantity of complex visual information is described in the scientific literature of experimental physics. Laboratory reports all have some dependence on visual sensations like seeing dials, meter-needles, printer output, computer screens, etc. As part of their daily activities, experimental physicists analyze all these visual sensations, and use them to produce a measurement. Subsequent discussion can then gloss-over the profusion of visual detail, and simply refer to the measurement.

Ethnophysics articulates this generic scientific task by first objectifying visual sensations as dynamic quarks. Then dynamic quarks are used to define the motion of larger particles like atoms and photons. Visual sensations are thus masked, and quickly forgotten. Talk of dynamic quarks is replaced with discussion about the dynamic characteristics of atoms and photons. Analysis shifts from quarks and colors, to lengths and wavelengths. Details about this shift depend on particle type, as discussed next.

Describe Atoms using Lengths

Atoms are important for understanding spatial isotropy because they are the smallest particles that are big enough to logically define length. And lengths are required to establish Cartesian coordinate systems.

But when considering complete atoms, subatomic variations in the distribution of leptonic quarks often cancel each other out. Leptonic quarks represent chromatic sensations. So for the most part, colors are already smeared-out1Red and green sensations cancel each other, and likewise for yellow and blue. of atomic descriptions. It is true that chlorine looks greenish, and sulphur is yellow. But most gases are colorless, and most metals are greyish. In any case, chromatic details are just not necessary for describing the mechanics of atoms.

Thus atoms are relevant, but their colors are not. So to make isotropic descriptions involving atoms, we stop talking about dynamic quarks. Instead, atoms are described using dynamic characteristics like velocity and acceleration. These characteristics are in-turn defined, and experimentally determined, from lengths. Atomic colors are just ignored.

Describe Photons using Wavelengths

The color of a photon is one of its most important characteristics, it cannot be simply ignored. And so EthnoPhysics has employed a crude categorical system for describing colors. This method actually does ignore a huge amount of chromatic detail, but not 100%. Meanwhile wavelengths can be experimentally determined using prisms and gratings. Color vision is not required to measure a wavelength. And wavelengths are correlated with color.

So to do physics in an isotropic space, quantitative measurements of wavelength supersede the categorical description of a photon’s color.

For example, seeing a Balmer-alpha photon is a great example of seeing a red photon. But for isotropy, we stop associating this sensation with any leptonic quarks. Instead, the color of Balmer-alpha is explained by the common human experience that any photon with a wavelength of about 650 (nm) looks red.

Third, Adopt Some New Conventions

The final step to obtaining isotropic space is to make some new conventions. The direction of the  x and  y-axes cease to be defined by specific colors. The axes are still relevant, but their directions are now established by complex visual experiences like seeing mountains, obelisks and benchmarks. Likewise, electric and magnetic polarities are no longer set by color. They are conceptually retained, but their signs are established by new conventions such as alignment with the Earth’s magnetic field, or perhaps details about the construction of a battery.

\rule{200px}{2px}

Thus, with a fresh Cartesian point-of-view and some updated conventions, descriptions can methodically gloss-over color. Discussion becomes more objective and scientific. And the hypothesis of spatial isotropy becomes valid.2Glossing color also has another important consequence. As chromatic distinctions lose physical relevance, they can be used to convey other meanings. For example, in the laboratory red plastic insulation on a wire often means that it is connected to a positive electrode. But this meaning is obtained by convention, not from the quark content of electrodes or plastic. Instead, the ‘color of the atoms’ in the plastic insulation is explained by the red color of the photons they emit. But we are left with a reduced cast of particles. In three-dimensional isotropic space, mechanics is mostly about atoms, photons and their composites. There are also some important two-dimensional roles for protons and electrons. But seeds, quarks and field quanta are not usually given space-time descriptions.

Spatial isotropy due to averaging is suggested by the interlocking patterns in this 19th century Indonesian textile.
Tatibin, Paminggir people. Lampung region of Sumatra, Kota Agung district, 19th century, 91 x 39 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.

One-Dimensional Space

We can adapt the definition of time to make an example of a one-dimensional space. Recall that the ground-state model of the electron divided quarks into two groups that were distinguished from each other only by their relationship with the frame of reference. The sensory interpretation of this distinction was that the frame provided a backdrop that was dark for some quarks, and bright for others.

But this sensory quality was masked by shifting the description to using a time coordinate. The disguise was completed with an assumption that usually there are equal numbers of black and white sensations about. They supposedly cancel each other and fade from awareness, but the time coordinate was usefully retained. Calculus was not required for a quantitative version of this interpretation.

A collage of quark icons representing an electron.
A one-dimensional spatial model of the electron.

To make a similar example of a one-dimensional space recall that scalar multiples of the basis vector  \widehat{z} \equiv (0, 0, 1) are collectively called the polar axis. And remember that the phase  \delta_{\theta} characterizes black and white sensations in the background reference frame. Objectify any difference in  \delta_{\theta} as a variation in position along the polar axis. Put a number to this rudimentary notion of spatial structure using  d \! z the polar component of the displacement.

To illustrate, we use a rod to represent the polar axis. Then the ground-state model of an electron is modified slightly to make a one-dimensional spatial model as shown in the accompanying diagram. The  z-coordinate of an event is defined from a sum of displacements. Then an assumption about symmetry lets us drop explicit reference to sensation, but the  z-coordinate is kept and used to describe position on the polar axis. Recognizing this association between visual sensations and a spatial arrangement is the first step in defining a particle-centered Cartesian coordinate system.

Empty Space

Please notice that empty space has not been defined. The foregoing discussion of a one-dimensional space specifically considers an electron. And next, atoms are required to define fully three-dimensional spaces. Perfect vacuums and voids are dismissed due to deep respect for the work of experimental physicists. By insisting on sensible footings for theoretical physics we avoid paradoxes like those from Zeno of Elea. Moreover as Percy Bridgman has pointed out, considering empty space to be a physical object can lead to logical inconsistencies.3Percy Williams Bridgman, The Nature of Some of Our Physical Concepts, page 18. Philosophical Library Inc. New York, 1952. Indeed if a vacuum measuring instrument is put in an ostensibly empty space, then it is not empty.

Spacetime Events

Space-Time Events
 k\delta_{\hat{m}}\delta_{\hat{e}}\delta_{\theta}
1+10+1
20-1+1
3-10+1
40+1+1
5+10-1
60-1-1
7-10-1
80+1-1

Consider a particle P described by a repetitive chain of events that are written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3}  \, \ldots \, \right)

where each orbital cycle can be parsed into eight components

\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{P}_{1}, \, \mathsf{P}_{2} \, \ldots \, \mathsf{P}_{k} \, \ldots \, \mathsf{P}_{8} \right)

so that there is one component  \mathsf{P}_{k} for each combination of the phase  \delta_{\theta}, the magnetic polarity  \delta_{\hat{m}} and the electric polarity  \delta_{\hat{e}} as shown in the accompanying table.

This arrangement guarantees that P has a fixed relationship with the electric, magnetic and polar axes. It provides a sufficient array of sensory detail to ensure that P is described by three independent classes of sensation. So P has a well-defined spatial orientation that may be described using three independent parameters. Aggregates like this are called three-dimensional particles.

A diagram showing the mutual orientation of space-time octants.
Orientation of a space-time event in quark space. Red numbers indicate values for k.

This image shows the relationship between spatial axes and P’s orbital components. The eight components may be made up from some miscellaneous collection of quarks beyond the bare minimum required to establish a spatial orientation. So orbital components are shown as different wedges. Events  \mathsf{P}_{5} through  \mathsf{P}_{8} are out-of-phase with events  \mathsf{P}_{1} through  \mathsf{P}_{4} so they are depicted in a lower tier on the polar axis.

Compound events like \mathsf{\Omega} are called space-time events because we can assign  \bar{r}, a well-defined three-dimensional position, and  t, a time of occurrence to \mathsf{\Omega} without making further assumptions. A particle must contain dozens of quarks before it can be logically described in space-time. A full octet of correctly polarized orbital-components are necessary. Atomic hydrogen with 48 quarks is the smallest stable particle that satisfies this requirement. So space-time is usually used to describe atoms, molecules and larger Newtonian particles.

Chains of space-time events like  \Psi are called trajectories. Particle trajectories are generically written as  \Psi \!  \left( \bar{r}, t \right) to emphasize that their events have space-time coordinates.

Coherent Interactions

Consider some generic particles \mathbb{X}, \mathbb{Y} and \mathbb{Z} that are objectified from space-time events like the ones defined above. And let these particles interact with each other as \mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z}. If this process preserves relationships between components such that

\mathsf{P}_{1}^{ \, \mathbb{X}} + \mathsf{P}_{1}^{ \, \mathbb{Y}} \leftrightarrow  \mathsf{P}_{1}^{ \, \mathbb{Z}}

\mathsf{P}_{2}^{ \, \mathbb{X}} + \mathsf{P}_{2}^{ \, \mathbb{Y}} \leftrightarrow  \mathsf{P}_{2}^{ \, \mathbb{Z}}

\mathsf{P}_{3}^{ \, \mathbb{X}} + \mathsf{P}_{3}^{ \, \mathbb{Y}} \leftrightarrow  \mathsf{P}_{3}^{ \, \mathbb{Z}}

\mathsf{P}_{4}^{ \, \mathbb{X}} + \mathsf{P}_{4}^{ \, \mathbb{Y}} \leftrightarrow  \mathsf{P}_{4}^{ \, \mathbb{Z}}

\ldots etc \ldots

\mathsf{P}_{8}^{ \, \mathbb{X}} + \mathsf{P}_{8}^{ \, \mathbb{Y}} \leftrightarrow  \mathsf{P}_{8}^{ \, \mathbb{Z}}

then we say that the interaction is coherent. That is, relationships that determine the phase and orientation do not get mixed-up when a particle is formed or decomposed. Alternatively, we say that an interaction is incoherent if information about phase and orientation gets scrambled during the process.

Spatial Homogeneity

Consider some particle P that is described by a repetitive chain of events written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \, \ldots \, \right)

And let each orbital cycle  \mathsf{\Omega} be parsed into eight components as required to establish a three-dimensional spatial description

\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{P}_{1}, \, \mathsf{P}_{2} \, \ldots \, \mathsf{P}_{k} \, \ldots \, \mathsf{P}_{8} \right)

Then we define quark-level homogeneity as an assumption that for almost any particle in a description, these eight components have characteristics that are much the same as each other. Whole particles may differ from each other, but for perfect spatial homogeneity, every particle is presumed to have an all-around similarity between its eight components. Specifically we require that each octant has the same lepton number  L, baryon number  B, charge  q, and strangeness  S. For example, a presumption of spatial homogeneity constrains the charge as

q \! \left(\mathsf{P}_{1} \right) =q \! \left(\mathsf{P}_{2} \right) = q \! \left(\mathsf{P}_{3} \right) = \, \ldots \, = q \! \left(\mathsf{P}_{8} \right)

Then recall that charge is conserved so the total charge on P is eight times the charge of one orbital component

\displaystyle q^{\mathsf{P}} = \sum_{\it{k} \mathrm{=1}}^{8} q \! \left( \mathsf{P}_{k} \right) = 8 \, q \! \left(\mathsf{P}_{1}\right)

But charge is defined by multiples of the fraction one-eighth as

q \equiv \frac{1}{8} \left( \Delta n^{\mathsf{T}} - \Delta n^{\mathsf{B}} + \Delta n^{\mathsf{C}} - \Delta n^{\mathsf{S}} \right)

where \Delta n notes a particle’s quark coefficients. So in a perfectly homogeneous space

q^{\mathsf{P}} = \Delta n^{\mathsf{T}} \! \left(\mathsf{P}_{1}\right) - \Delta n^{\mathsf{B}} \! \left(\mathsf{P}_{1}\right) + \Delta n^{\mathsf{C}} \! \left(\mathsf{P}_{1}\right) - \Delta n^{\mathsf{S}} \! \left(\mathsf{P}_{1}\right)

The quark coefficients of  \mathsf{P}_{1} are always integers so  q^{\mathsf{P}} is determined by a sum or difference of integers. Therefore quark-level homogeneity constrains the charge on P as

q^{\mathsf{P}} = 0, \, \pm1, \, \pm2, \, \pm3 \, \ldots

Similar reasoning leads to the same result for strangeness, lepton and baryon numbers

S = 0, \, \pm1, \,  \pm2 \, \ldots

L = 0, \, \pm1, \,  \pm2 \, \ldots

B = 0, \, \pm1, \,  \pm2 \, \ldots

The notion of homogeneity depends on the extent or scale of a description. By the foregoing definition, a space must be filled with particles that are at least as large as atoms to achieve quark-level homogeneity. But almost any ordinary space is full of atoms and molecules, and  q,  L and  B are invariably observed to be integers.

At the atomic level, particles are described using Cartesian coordinates and similarity conditions are expressed in terms of  x,  y and  z. Then atomic-level homogeneity requires that particles are distributed so that their shapes are not oriented in any special direction. Their radial variation  \delta \! \rho , is presumably similar in all directions such that   \delta \! \rho_{x} = \delta \! \rho_{y} = \delta \! \rho_{z} \, . And at the macroscopic level, further conditions like well-stirred mixture assumptions may have to be included to make a complete specification of spatial homogeneity.

Spatial homogeneity is suggested by the uniform distribution of woven motifs in this Indonesian textile.
Tampan, Paminggir people. Sumatra circa 1900, 43 x 46 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.
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EthnoPhysics faviconBig 3-Dimensional Atoms

Atoms are big enough to be three-dimensional. Atomic bonds are modeled using chemical quarks. Results are compared with experiments.
References
1 Red and green sensations cancel each other, and likewise for yellow and blue.
2 Glossing color also has another important consequence. As chromatic distinctions lose physical relevance, they can be used to convey other meanings. For example, in the laboratory red plastic insulation on a wire often means that it is connected to a positive electrode. But this meaning is obtained by convention, not from the quark content of electrodes or plastic. Instead, the ‘color of the atoms’ in the plastic insulation is explained by the red color of the photons they emit.
3 Percy Williams Bridgman, The Nature of Some of Our Physical Concepts, page 18. Philosophical Library Inc. New York, 1952.