Discover Bumpy Space • EthnoPhysics

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 laboratory experience clearly shows 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.

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

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

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.

Any non-rotating particle has no orbital radius because $\textsl{\textsf{J}}=0$ when $N^{\mathsf{U}}=N^{\mathsf{D}}.$ So particles that are 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 energy by $\lambda = h c \! / \! E$ . So the orbital radius of a photon can be written as

$R \left( \gamma \right) = \dfrac{\lambda}{2\pi}$

Then a circular perimeter of $2 \pi R$ is the same as one wavelength.

Spatial Axes

Let events in the history of some particle 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 all scalar multiples of

$\widehat{x} \equiv \cos{\! 2\theta} \, \widehat{m} + \sin{\! 2\theta} \, \widehat{e}$

And similarly the ordinate axis is composed from multiples of

$\widehat{y} \equiv - \sin{\! 2\theta} \, \widehat{m} + \cos{\! 2\theta} \, \widehat{e}$

These new vectors together with $\widehat{z}$ are called a Cartesian basis after the work of René Descartes . Cartesian axes can be visualized by rotating the electric and magnetic axes by $2 \theta$ degrees around the polar axis. The factor of two means that $\widehat{x}$ and $\widehat{y}$ make two complete turns as $\theta$ goes through each cycle, one turn for quarks of each phase. These definitions can be rearranged to give $\widehat{m} = \cos{\! 2\theta} \, \widehat{x} - \sin{\! 2\theta} \, \widehat{y}$ and $\widehat{e} = \sin{\! 2\theta} \, \widehat{x} + \cos{\! 2\theta} \, \widehat{y}$ .

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.

Some Shapes

A particle described by its wavelength $\lambda$ and orbital radius $R$ may be classified by shape.

Shape	Definition
a point	$R = \lambda = 0$
a line segment	$\lambda > R = 0$
a plane section	$R > \lambda = 0$
a disc	$R \gg \lambda > 0$
a rod	$\lambda \gg R > 0$
a cylinder	$\lambda \sim R > 0$

Displacement

Let some particle P be characterized by a ordered chain of events

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

that is repetitive so that $\Psi$ may also be written as

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

where each orbital cycle $\mathsf{\Omega}$ is composed of $N$ sub-orbital events written as

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

The orbital radius of P is noted by $R$ and $\lambda$ is its wavelength. A spatial orientation for P is specified by $\delta_{\hat{m}}$ , $\delta_{\hat{e}}$ and $\delta_{z}$ . We use these characteristics to define the following numbers for describing sub-atomic events

$d \! x \equiv \delta_{\hat{m}} \dfrac{R\cos{\! 2\theta}}{\, N \,}$

$d \! y \equiv \delta_{\hat{e}} \, \dfrac{R\sin{\! 2\theta}}{\, N \,}$

$d \! z \equiv \delta_{z} \, \dfrac{\lambda}{\, N \,} = \dfrac{\lambda}{2\pi} \, d \! \theta$

where $\theta$ is the phase angle of P. Then using the Cartesian unit vectors $\hat{x}$ , $\hat{y}$ and $\hat{z}$ , a displacement vector for P is defined by

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

If we switch to implicitly using Cartesian basis vectors, we can express the displacement as an ordered set

$d \! \bar{r} = ( d \! x, \, d \! y, \, d \! z )$

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-atomic events may be large but not infinite. In principle $N$ is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. They are finite increments of displacement. Later we may assume that $N$ 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

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

where each event is characterized by $d \! \bar{r}$ its displacement. Then the position of event $\mathsf{P}_{k}$ may be defined by

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

where $\bar{r}_{\mathsf{o}}$ is arbitrary. Please notice that this algebraic vector has been defined entirely through a systematic description of sensation. So our ideas about position are based on an empirical approach that is scientific and consistent with the premise of EthnoPhysics. If all events are assigned a position, then $\Psi$ can be expressed as an ordered set of position vectors

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

Consider an ordered pair of events from $\Psi$ noted by $\left( \mathsf{P}_{i}, \, \mathsf{P}_{f} \right)$ . The separation vector between these two occurrences is defined by

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

And the norm of the separation 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 quantities $d \! x$ , $d \! y$ and $d \! z$ are finite increments of displacement. They are not infinitesimal. So we can just add them together in sums like other numbers. The 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 ordinate is defined as

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

And the $z$ -cooordinate 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 just a simple linear function of the wavelength. So if P is isolated then it moves in regular steps along the polar-axis, and can be described by

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

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

$\bar{r} = ( x, y, z )$

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So $N$ , the total number of quarks in a description, may be large but not infinite. In principle $N$ is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we assume that $N$ is large enough to make an approximation to spatial continuity. Then the use of calculus may be appropriate.

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 one direction is just about the same as another. That is why it is called a hypothesis of spatial isotropy

Sensory interpretation: The phase can be explained as a representation of black and white perceptions, the magnetic polarity depends on red and green sensations, and the electric polarity is defined from blue and yellow perceptions. 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 plan for systematically glossing-over visual sensations 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 a Euclidean space that has 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.

Second, Objectify Visual Details

The second step toward isotropy is to objectify visual sensations as experimental observations. Vast quantities of complex visual information are described in the scientific literature of experimental physics. There are millions of pages carefully recording how laboratory conditions are controlled, samples prepared, equipment arranged and data presented. Laboratory reports all have some dependence on visible objects like dials, meter-needles, cathode-ray tubes, printer output, computer screens, etc. These visual details are then included in analysis that ultimately produces a measurement of something.

Thus the experimental process objectifies visual sensations and often reports them as some property of a particle. For an isotropic space, talk of dynamic quarks is replaced by discussing the dynamics of larger particles.

Third, Describe Atoms using Lengths

Next we step-back and refocus the description on particles that are larger than quarks, starting with atoms. Then lengths can be well-defined and used to determine Cartesian coordinates. Also sub-atomic variations in leptonic quark distributions may be averaged, and cancelled-out. Leptonic quarks represent chromatic sensations, so colors are smeared-out of the description of atoms. That is, red and green sensations cancel each other, and likewise for yellow and blue. When considered all together, the quarks in atoms are collectively greyish or colorless. So for isotropic descriptions, colors are not attributed to atoms based on their dynamic quarks. Instead, dynamic characteristics like velocity and acceleration are determined by measuring lengths. And chromatic terms are free 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.

Fourth, Describe Photons using Wavelengths

For EthnoPhysics, the color of a photon is fundamentally known by the personal experience of seeing one. We also have a more communal and categorical method of describing colors, but the procedure is coarse and still partly subjective. On the other hand, we can objectively measure photon wavelengths using prisms and gratings. And wavelengths are correlated with colors.

So to do physics in an isotropic space, quantitative measurements of wavelength supersede the categorical description of a photon’s color. For instance, absorbing a Balmer-alpha photon is an archetypal example of seeing red. But this perception is not attributed to the colors of leptonic quarks in the photon. Instead, the color of Balmer-alpha is explained by the common human experience that any photon with a wavelength of about 650 (nm) looks like seeing blood.

Finally, 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 and obelisks. 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.

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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. 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 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. Objectify any difference in $\delta_{\theta}$ as a variation in direction on 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 is based on a specific particle. And later, atoms are required to define fully three-dimensional spaces. Perfect vacuums and voids are not considered out of 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.¹Percy 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	+1	0	+1
2	0	-1	+1
3	-1	0	+1
4	0	+1	+1
5	+1	0	-1
6	0	-1	-1
7	-1	0	-1
8	0	+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 cycle can be parsed into eight sub-orbital 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 ensures that P has a fixed relationship with the electric, magnetic and polar axes. It provides a logically sufficient array of sensation to make an account of events that is fully three-dimensional. We can assign $\bar{r}$ , a well-defined position, and $t$ , the time of occurrence to these events without making further assumptions.

Compound events like $\mathsf{\Omega}$ are called space-time events. 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.

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 sub-orbital components. The eight components may be composed from some miscellaneous collection of quarks beyond the bare minimum required to establish a spatial orientation. So sub-orbital events are shown as different pie-shaped 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. Here is a short movie looking around all eight sub-orbital events in an atomic-cycle.

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 sub-orbital events 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 sub-orbital events 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 sub-orbital events have characteristics that are 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 sub-orbital components. Specifically we require that each octant has the same lepton number $L$ , baryon number $B$ and charge $q$ . So this is a lot like spatial isotropy but it is even more stringent because it requires that sub-orbital events are similar all to each other, not just similar in pairs. 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 sub-orbital event

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

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 lepton and baryon numbers

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

and

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

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. The variation in their radii is presumably similar in all directions so 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.