Press "Enter" to skip to content

Length

Outline

Length measurements determine Cartesian coordinates which were invented by René Descartes who is pictured in this engraving.

Historically, measurement is an important part of geometry. In 1637 the inventor of analytic geometry René Descartes wrote1Here is another translation by David Eugene Smith and Marcia L. Latham.

“I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and then classify them in order, is by recognizing the fact that all points of those curves which we may call geometric, that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by means of a single equation.

If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse ; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class ; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely”.

From The Geometry of René Descartes, Open Court Publishing Company, page 48. La Salle Illinois, 1952.
that

An icon indicating a quotation.... all points of those curves which we may call geometric are those which admit of precise and exact measurement ...

— René Descartes

Nowadays mathematicians are less constrained and many non-Euclidean geometries are studied. But Descartes is clear about his geometry, it presumes measurement. This also seems to have agreed with Sir Isaac Newton as he set down the laws of motion fifty years after the publication of the text shown in the photograph below. He wrote2Isaac Newton, Mathematical Principles of Natural Philosophy, page xvii in the preface to the first edition. Translated by Andrew Motte and Florian Cajori. University of California Press, 1934. that

An icon indicating a quotation.... geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring ...

— Sir Isaac Newton

Thus mensuration is an essential notion for both Cartesian geometry and Newtonian mechanics. So to ensure that measurement is theoretically well founded, EthnoPhysics defines a length as the distance between two atoms. Let these two atoms be noted by   \mathbf{A} and   \mathbf{B}. Then the length  \ell of the distance between them is

\ell \equiv \left\| \, \bar{r}^{\, \mathbf{B}} - \bar{r}^{\, \mathbf{A}} \rule{0px}{10px} \right\|

where  \bar{r} notes the position. Recall that the distance between any two events is determined by their positions. So this definition of length just adds the requirement that positions are anchored by atoms thus guaranteeing that they are well-defined three-dimensional quantities. A length is a measurable distance. And this implies that some minimum amount of sensory detail is required to logically discuss length: There have to be at least enough quarks involved to make a couple of atoms.

This restrictive definition of length is important because as the mathematician Benoit Mandelbrot has remarked; if the scale of a length measurement is not limited, then as it is made smaller and smaller every approximate length tends to increase steadily without bound.3Benoit B. Mandelbrot, The Fractal Geometry of Nature, page 26. W. H. Freeman and Company, New York 1977. So the foregoing definition safeguards the possibility of answering questions like: How long is the coast of Britain?

Measuring Length

Length has been measured at least since ancient Egyptians stretched cords and knotted ropes to survey agricultural fields and construct pyramids. For the last few hundred years, calibrated measurement techniques have usually required some kind of a measuring rod. An ideal measuring rod is rigid so its own length is presumably constant.

To measure the length  \ell between \mathbf{A} and \mathbf{B} count the least number of rods that fit between them. Lengths are conventionally expressed in metres and abbreviated as (m). The requirement for a least number is based on the historical practice of stretching a rope or surveyor’s chain.

Length Contraction

More recently an optical method has been adopted to measure length. It requires a clock to determine an elapsed time  \Delta t. To optically measure a length in meters, first measure the elapsed time in seconds for a photon to travel from \mathbf{A} to \mathbf{B}. Then

\ell = 299,792,458  \cdot \Delta t

The elapsed time depends on the frame of reference F. So the length depends on the frame too. If F is chosen so that the atoms being measured are at rest, then the elapsed time is the proper elapsed time and noted by \Delta t^{\ast}. The two increments are related as \Delta t^{\ast} = \gamma \Delta t where  \gamma is the Lorentz factor. Similarly, when  \gamma =1 then the length is called a proper length, noted by \ell ^{\ast} and given by \ell^{\ast} \equiv \, 299,792,458 \cdot \Delta t^{\ast}. So these lengths are related as

\ell^{\ast} = \gamma \ell

The Lorentz factor for a particle in motion is always greater than one, \gamma > 1. So observations of moving atoms always measure a smaller length than between stationary atoms, \ell < \ell^{\ast}. This effect is called length contraction.

Cartesian Coordinate Systems

We have previously defined Cartesian coordinates. Now we discuss how to assemble these numbers into coordinate systems.

As we objectify descriptions, and apply the hypothesis of spatial isotropy, we stop referring directly to chromatic visual sensation. Physics becomes color-blind by changing the descriptive framework from quark space  \mathbb{Q}, to a new, almost-Euclidean space called  \mathbb{E}. In  \mathbb{E}, particles are rotating. Their electric and magnetic axes turn around the polar axis. And in  \mathbb{E} we can construct a Cartesian coordinate system as follows.

The algebraic vector space  \mathbb{E} is a set of position vectors  \overline{r} for some collection of atoms \mathbf{A}_{1}, \mathbf{A}_{2}, \mathbf{A}_{3}\mathbf{A}_{N}. This mathematical construction is generically written as

\mathbb{E} = \left\{ \rule{0px}{12px} \, \bar{r}_{1}, \, \bar{r}_{2}, \, \bar{r}_{3} \, \ldots \, \bar{r}_{N} \right\}

The following basis vectors are used to make general descriptions; the axis of the abscissa is directed by  \widehat{x} \equiv (1, 0, 0), the axis of the ordinate from  \widehat{y} \equiv (0, 1, 0) and the polar-axis by  \widehat{z} \equiv (0, 0, 1). Then, any position vector in  \mathbb{E} can be expressed in terms of  x,  y and  z, its Cartesian coordinates as

\bar{r} = x \widehat{x} + y \widehat{y} + z \widehat{z}

The new space  \mathbb{E} is closer than  \mathbb{Q} to the ordinary space of everyday experience. And we use a Euclidean metric to determine vector norms. But some other details such as spatial continuity have to be assessed before we could say that  \mathbb{E} is a Euclidean space.

A one-dimensional coordinate system.

A Linear Coordinate

Let the  z-axis be centered on the proton inside an atom of hydrogen  \mathrm{H} . When in its ground-state, the proton is at rest in any inertial frame of reference. And it is extremely stable. So it is a good place to start constructing a coordinate system. The location of  \mathrm{H} on the  z-axis is specified by the numeric value  z=0. This position is called the spatial-origin. So by definition, this special hydrogen atom is always located at the spatial-origin.

We use the same spatial-origin in other coordinate-systems to be discussed next. So the special atom is called  \mathrm{H}_{\mathsf{o}} to distinguish it from other hydrogen atoms. Both  \mathbb{Q} and  \mathbb{E} use the same basis vector  \widehat{z} \equiv (0, 0, 1). So we could use this  z-coordinate to parameterize the one dimensional space discussed earlier.

A Cartesian Plane

As descriptions are objectified, we stop referring directly to sensations. This is done partly by shifting the focus to particles that are larger than quarks. For example, we next use two atoms to define a two-dimensional space, the Cartesian plane.

The length between two atoms is illustrated in this schematic diagram of a hydroxide anion.
A two-dimensional Cartesian coordinate system defined by the two atoms that form a hydroxide ion. The red ball represents oxygen.

Let us combine an atom of oxygen  \mathrm{O} with the hydrogen atom  \mathrm{H}_{\mathsf{o}} shown above, to make a hydroxide anion,  \mathrm{OH}^{-} . The description is again centered on the proton inside  \mathrm{H}_{\mathsf{o}} .

The  x-axis is defined in  \mathbb{Q} by sensation, but for  \mathbb{E} more conventional details are required. So in this Cartesian coordinate system, the direction of the unit vector  \widehat{x} \equiv (1, 0, 0) is chosen to align with the O–H chemical bond called  \mathbb{B} \mathsf{(2)}, and ultimately fixed by the material presence of atomic oxygen.

The  x-axis is also chosen to be orthogonal to  \widehat{z}. The position of oxygen is then described by the coordinates  z=0 and  x = \ell where  \ell notes the distance between hydrogen and oxygen atoms.

The key detail about this arrangement is that it involves two atoms. So  \ell is measurable and can meet the definition for being a length. Indeed  \ell is observed4Computational Chemistry Comparison and Benchmark Database Edited by Russell D. Johnson III, National Institute of Standards and Technology, Standard Reference Database Number 101, Release 18, Department of Commerce USA, October 2016. to be 96.4 ± 0.1 (pm).

Note that the foregoing coordinate system uses the atom of hydrogen to furnish a descriptive context for the atom of oxygen, and also for any other atoms that may be included. So  \mathrm{H}_{\mathsf{o}} is functioning as a frame of reference.

A Three-Dimensional Cartesian System

A three-dimensional Cartesian coordinate system defined by the three atoms that make a molecule of water.

Next we use three atoms to make a three-dimensional Cartesian system. Let us combine another atom of hydrogen with the hydroxide anion  \mathrm{OH}^{-} to make  \mathrm{H}_{\mathsf{2}} {\mathrm{O}} a molecule of water. The  z-axis,  x-axis and spatial-origin are as before, but the  y-axis still needs to be established. We choose it to be orthogonal to both the  x and  z-axes, and in the same plane as the chemical bonds in water.

There are two possible orientations, identified by \varpi = \pm 1 . The number \varpi is called the handedness of the coordinate system. The water bonds are called  \mathbb{B} \sf{(aqueous)} . They make an angle of  \vartheta with each other. So for example, the position of the new hydrogen atom might be given by the coordinates \bar{r} ( \mathrm{H} ) = \ell ( 1-\cos\vartheta, \; \varpi\sin\vartheta, \; 0 ) .

The material presence of three atoms ensures that the three-dimensional framework is scientifically well-founded: Lengths and angles can actually be measured. In fact they are reported5Computational Chemistry Comparison and Benchmark Database Edited by Russell D. Johnson III, National Institute of Standards and Technology, Standard Reference Database Number 101, Release 18, Department of Commerce USA, October 2016. to be

\ell = 95.8 \pm 0.1 \; \mathsf{\text{(pm)}} and

\vartheta = 104.4776 \pm 0.0019 \; \mathsf{\text{(degrees)}}

Thus a physical three-dimensional coordinate system is established in principle. And, it may be extended to include other atoms just by making more measurements. Different atoms are assigned different coordinates, that algebraically represent different geometric positions.

This water-based coordinate system is not very practical. But it demonstrates that we can finally put aside some concerns about Pauli’s exclusion principle. From now on, when considering a bundle of particles, we assume that Pauli’s principle is satisfied if they all have different Cartesian coordinates.

Atomic Models

Here are some models of atoms that bring the notion of shape to their description. Aggregations of chemical and thermodynamic quarks are represented as cylinders, spheres and even a morsel of pasta.

Cylindrical Atomic Models

Consider an atom  \mathbf{A} described by a repetitive chain of events written as

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

where each repeated cycle \mathsf{\Omega} is a space-time event. Let this particle be characterized by its orbital radius  R and wavelength  \lambda. These properties are related to position and we can use them to make a simple geometric model of P that is shaped like a cylinder. Or to be more exact; like a finite section of a right circular cylinder with its ends closed to form two circular surfaces, oriented along the  z-axis, similar to the one shown in the accompanying diagram. The area  A of the circular cross section is defined by

A \equiv \pi R^{2}

This is just a statement of ancient knowledge about circles going back at least to Archimedes . And to restate another traditional mensuration formula, the volume  V of the cylinder is

V \equiv \lambda A = \lambda \pi R^{2}

We use this model to visualize one atomic event. Then it is easy to imagine the chain of events \Psi as a row of cylinders strung-out end-to-end, like a tube or wire.

Spheroidal Atomic Models

The French mathematician René Descartes certainly thought that atoms were like little balls spinning around and bumping into each other.6For example he writes that; “The material, as I said, is composed of many small balls which are in mutual contact; and we have sensory awareness of two kinds of motion which these balls have. One is the motion by which they approach our eyes in a straight line, which gives us the sensation of light; and the other is the motion whereby they turn about their own centers as they approach us.

If the speed at which they turn is much smaller than that of their rectilinear motion, the body from which they come appears blue to us; while if the turning speed is much greater than that of their rectilinear motion, the body appears red to us. But the only type of body which could possibly make their turning motion faster is one whose tiny parts have such slender strands, and ones which are so close together (as I have shown those of the blood to be), that the only material revolving round them is that of the first element.

The little balls of the second element encounter this material of the first element on the surface of the blood; this material of the first element then passes with a continuous, very rapid, oblique motion from one gap between the balls to another, thus moving in an opposite direction to the balls, so that they are forced by it to turn about their centres.”

From A Description of the Human Body published in The Philosophical Writings of Descartes, Volume I, page 323. Translated by John Cottingham, Robert Stoothoff and Dugald Murdoch. Cambridge University Press, 1985.
So also let atom  \mathbf{A} be described by its wavenumber  \kappa. Then we can model  \mathbf{A} as a spheroid that is mathematically represented in Cartesian coordinates using the equation

\dfrac{x^{2} + y^{2}}{R^{2}} + \left( \dfrac{2\kappa}{3\pi} \rule{0px}{14px} \right)^{2} z^{2} = 1

This shape is also known as an ellipsoid of revolution about the atom’s polar axis. If  \mathbf{A} is in its ground-state then  \kappa=0 and the sphere collapses into a circle

x^{2} + y^{2} = R^{2}

However if  \mathbf{A} is an excited atom then its wavelength is \lambda =2 \pi /\kappa and its shape can be represented as

\dfrac{x^{2} + y^{2}}{R^{2}} + \left( \dfrac{4}{3\lambda} \rule{0px}{14px} \right)^{2} z^{2} = 1

Spheroidal Shapes
a perfect sphere3 \lambda = 4 R
a prolate spheroid3 \lambda > 4 R
a oblate spheroid3 \lambda < 4 R

Traditional mensuration formulae give the volume enclosed by this curve as

V = \lambda \pi R^{2}

So the spheroidal model has been scaled to give  \mathbf{A} exactly the same volume as the cylindrical atomic model. A variety of spheroidal shapes are specified in the accompanying table.

A Rotini Model of an Atom

Principia Philosophiae by Rene Descartes, page 271. Amsterdam 1644.

Let the events of atom  \mathbf{A} be described by their time coordinates  t. Our first spatial conception of such an atom was as a compound quark in quark space. But to implement the hypothesis of spatial isotropy our next view is set in a Cartesian coordinate system where  \mathbf{A} is represented as a rotating atomic clock with a phase angle  \theta given by

\theta \! \left( t \right) = \theta_{\mathsf{o}} + \omega t

such that  \mathbf{A} is whirling about its polar axis with an angular frequency of  \omega. The rotation supposedly averages-out variations in the electric and magnetic radii leaving an effective orbital radius  R that is then used to represent the atom as a rotating cylinder. This rotating cylinder model smooths out some rough edges, but it is still amiss because the electromagnetic part of the terrestrial metric is larger than the other non-polar components. So one radial direction is predominant and the atom is shaped more like a piece of rotini pasta than a solid cylinder. This corkscrew spiral can be approximated by a geometric curve called a helicoid . It is described mathematically by radii of

\rho_{x} = R \cos{\! 2 \theta}

\rho_{y} = R \sin{\! 2 \theta }

\rho_{z} = \dfrac{ \lambda \theta}{2\pi}

When moving, the rotini model looks a lot like a machine called the Archimedean screw . Humans have been thinking about screw conveyor mechanisms like this for thousands of years. They were reportedly used to irrigate the Hanging Gardens of Babylon as early as 600 BC. This atomic model is good for understanding the Euclidean metric of the ordinary spaces in our laboratories and classrooms.

Atomic Displacement

Consider an atom  \mathbf{ A} that is described by some repetitive chain of events written as

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

where each orbit  \mathsf{\Omega} is represented by a bundle of quarks that can be parsed into eight sub-atomic events

\mathsf{\Omega}^{\mathbf{A}} = \left(\mathsf{P}_{1}, \mathsf{P}_{2}, \mathsf{P}_{3}, \mathsf{P}_{4}, \mathsf{P}_{5}, \mathsf{P}_{6},  \mathsf{P}_{7},  \mathsf{P}_{8} \right)

Let \Delta r \! \left( \mathsf{\Omega}^{\mathbf{A}} \right) note the distance that  \mathbf{ A} moves during one complete orbit. We can assess this quantity as follows. Since there are eight sub-atomic events, the phase angle  \theta of the  k^{\mathsf{th}} sub-atomic event is given by

\theta_{k} = \theta_{0} + \dfrac{k\pi}{4}

Without loss of generality let \theta_{\mathsf{0}} = 0 so that

\sin{2\theta} = \begin{cases} \; +1 \; &\mathsf{\text{if}} \; k = 1 \; \sf{\text{or}} \; 5 \\ \; \; 0 \; &\mathsf{\text{if}} \; k = 2, \; 4, \; 6  \; \mathsf{\text{or}} \; 8 \\ \; -1 \; &\mathsf{\text{if}} \; k = 3 \; \mathsf{\text{or}} \; 7 \end{cases}

and

\cos{2\theta} = \begin{cases} \; +1 \; &\mathsf{\text{if}} \; k = 4 \; \sf{\text{or}} \; 8 \\ \; \; 0 \; &\mathsf{\text{if}} \; k = 1, \; 3, \; 5  \; \mathsf{\text{or}} \; 7 \\ \; -1 \; &\mathsf{\text{if}} \; k = 2 \; \mathsf{\text{or}} \; 6 \end{cases}

Then by substitution the displacement of  \mathbf{ A} is given by

d \! x = \delta_{\hat{m}} 	\dfrac{R}{8} \begin{cases} \; +1 \; &\mathsf{\text{if}} \; k = 1 \; \mathsf{\text{or}} \; 5 \\ \; \; 0 \; &\mathsf{\text{if}} \; k = 2, \; 4, \; 6  \; \mathsf{\text{or}} \; 8 \\ \; -1 \; &\mathsf{\text{if}} \; k = 3 \; \mathsf{\text{or}} \; 7 \end{cases}

d \! y = \delta_{\hat{e}} 	\dfrac{R}{8} \begin{cases} \; +1 \; &\mathsf{\text{if}} \; k = 4 \; \mathsf{\text{or}} \; 8 \\ \; \; 0 \; &\mathsf{\text{if}} \; k = 1, \; 3, \; 5  \; \mathsf{\text{or}} \; 7 \\ \; -1 \; &\mathsf{\text{if}} \; k = 2 \; \mathsf{\text{or}} \; 6 \end{cases}

and

d \! z = \dfrac{\lambda}{\, 8 \,}

where  \lambda is the wavelength,  R is the orbital radius,  \delta_{\hat{m}} is the magnetic polarity and  \delta_{\hat{e}} is the electric polarity. These displacements can be used to evaluate changes in the position of  \mathbf{ A}. The variation in the abscissa during one atomic cycle is

    \begin{align*} \Delta x \! \left( \mathsf{\Omega}^{\mathbf{A}} \right) &= \sum_{k=1}^{8} d \! x_{k} \\ &= \delta_{\hat{m}} \frac{R}{8} \left(1+0-1+0+1+0-1+0 \rule{0px}{11px} \right) \\ &=0 \end{align*}

The change in the ordinate is

    \begin{align*} \Delta y \! \left( \mathsf{\Omega}^{\mathbf{A}} \right) &= \sum_{k=1}^{8} d \! y_{k} \\ &= \delta_{\hat{e}} \frac{R}{8} \left(0-1+0+1+0-1+0+1 \rule{0px}{11px} \right) \\ &=0 \end{align*}

And the change in position along the polar axis is

    \begin{align*} \Delta z \! \left( \mathsf{\Omega}^{\mathbf{A}} \right) &= \sum_{k=1}^{8} d \! z_{k} \\ &= \delta_{z} \frac{\lambda}{8} \left(1+1+1+1+1+1+1+1 \rule{0px}{11px} \right) \\ &= \delta_{z} \lambda \end{align*}

So over one full atomic cycle, the separation vector between initial and final sub-atomic events is

\Delta \bar{r} \! \left( \mathsf{\Omega}^{\mathbf{A}} \right) = \left( \Delta x, \, \Delta y, \, \Delta z \rule{0px}{10px} \right) = \left( 0, \, 0, \, \delta_{z} \lambda \rule{0px}{10px} \right)

The distance between events is given by the norm of their separation, \Delta r \equiv \left\| \, \Delta \bar{r} \, \right\|. So consecutive atomic cycles are separated from each other by one wavelength

\Delta r \! \left( \mathsf{\Omega}^{\mathbf{A}} \right) = \lambda

Melethallia, Ernst Haeckel, Kunstformen der Natur. Chromolithograph 32 x 40 cm, Verlag des Bibliographischen Instituts, Leipzig 1899-1904. Photograph by D Dunlop.

The Euclidean Metric

This article extends an earlier, more general discussion of metrics that considered a particle’s shape as described by its radius vector  \overline{\rho}. Initially, the radius was established from a particle’s quark content, and a vector space  \mathbb{S} was defined from sets of particles and their radii.  \mathbb{S} was characterised using statistical averages  \widetilde{\rho}, standard deviations  \delta \! \rho and correlation coefficients noted by  \chi _{\alpha \beta}. These quantities have already been assessed for generic compound quarks.

Now let us consider that  \mathbb{S} might be filled with a more restricted set of particles that have special attributes. Specifically we examine collections of atoms so that radius vectors may be expressed in Cartesian coordinates. Atomic aggregates are characterized using their orbital radius  R, a wavelength  \lambda and their phase angle  \theta. We say that  \mathbb{S} has a Euclidean metric if every particle satisfies the following conditions.

  1. Atoms and molecules must be fully three dimensional like cylinders or rods so that R \ne 0 and \lambda \ne 0. Wavelengths and radii are all presumably positive, not nil.
  2. Particles must exhibit some variation in their shape so that \delta \! \rho \ne 0. The standard deviation for any component of the radius is presumably positive.
  3. The space  \mathbb{S} must be well-stirred. We require spatial homogeneity at an atomic level so that \delta \! \rho_{x} = \delta \! \rho_{y} = \delta \! \rho_{z}.
  4. The particles in  \mathbb{S} cannot all have an extremely low temperature, or all be resonating like the atoms in a laser. We assume that \delta \theta \ne 0 so that there is some variation among atomic phase angles.
  5. Finally, atoms must interact incoherently so that the variation among phase angles is random. Then we can presume that large sums over odd powers of circular functions of  \theta add-up to zero.

For a specific example of a Euclidean space consider a big collection of  N atoms, and larger particles that are composite atoms, noted by  \mathsf{P}^{1}, \; \mathsf{P}^{2}, \; \mathsf{P}^{3} \ldots \mathsf{P}^{N}. Let these particles be described using the rotini model where atomic shapes are like a corkscrew spiral. They approximate a geometric curve called a helicoid so that atomic radii can be mathematically written as

\rho_{x} = R \cos{\! 2\theta}

\rho_{y} = R \sin{\! 2\theta}

\rho_{z} = \dfrac{ \lambda \theta}{2\pi}

The radius vector is expressed in Cartesian coordinates as \, \overline{\rho} = \rho_{x} \hat{x} + \rho_{y} \hat{y} + \rho_{ z} \hat{z} so a space like  \mathbb{S} can be generically represented as

\mathbb{S} = \left\{ \overline{\rho}^{ 1 }, \, \overline{\rho}^{2}, \, \overline{\rho}^{3} \, \ldots \, \overline{\rho}^{\, i} \, \ldots \, \overline{\rho}^{\, N} \right\}

The average radii of the particles in  \mathbb{S} are given by

\displaystyle \widetilde{\rho}_{x} = \frac{R}{N} \sum_{i=1}^{N} \cos{2\theta^{i}}

\displaystyle \widetilde{\rho}_{y} = \frac{R}{N} \sum_{i=1}^{N} \sin{2\theta^{i}}

\displaystyle \widetilde{\rho}_{z} = \frac{\lambda}{2\pi} \sum_{i=1}^{N} \theta^{i} = \frac{\lambda \widetilde{\theta}}{2\pi}

As noted above, we presume that the atoms in  \mathbb{S} have phase angles that are completely incoherent so that sums over the circular functions add-up to zero. Then

\widetilde{\rho}_{x} \approx 0

and

\widetilde{\rho}_{y} \approx 0

and so the variations in particle radii are

\delta \! \rho_{x} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{x}^{i} \rule{0px}{11px} \right)^{2} \; } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \cos{2\theta^{i}} \rule{0px}{11px} \right)^{2} \; } \approx \dfrac{R}{2}

\delta \! \rho_{y} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{y}^{i} \rule{0px}{11px} \right)^{2} \; } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \sin{2\theta^{i}} \rule{0px}{11px} \right)^{2} \; } \approx \dfrac{R}{2}

\delta \! \rho_{z} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{z}^{i} - \widetilde{\rho}_{z} \right)^{2} \; } = \dfrac{\lambda}{2\pi} \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \theta^{i} - \widetilde{\theta} \rule{0px}{11px} \right)^{2} \; } = \dfrac{\lambda}{2\pi} \delta \! \theta

We presume that all the particles in  \mathbb{S} are three-dimensional so that R \ne 0 and \lambda \ne 0. And as noted above, we also assume that phase angles vary. So \delta \theta \ne 0 and the standard deviations in particle radii must be positive

\delta \! \rho_{x} \ne 0

\delta \! \rho_{y} \ne 0

\delta \! \rho_{z} \ne 0

We also presume that  \mathbb{S} has spatial homogeneity at the atomic-level. Then any variations cannot depend on direction, \delta \! \rho_{x} = \delta \! \rho_{y} = \delta \! \rho_{z}. But by their definition, correlation coefficients and standard deviations are related as  \chi _{\alpha \alpha} = \delta \! \rho_{\alpha} so that

\chi_{xx} = \chi_{yy} = \chi_{zz} > 0

The other correlation coefficients are given by

\chi_{xy} = \sqrt{  \frac{1}{N} \sum_{i=1}^{N} \rho_{x}^{i} \rho_{y}^{i} \; } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \cos{2\theta^{i}} \right) \left(\sin{2\theta^{i}} \right) \; } \approx 0

\chi_{xz} = \sqrt{  \frac{1}{N} \sum_{i=1}^{N} \rho_{x}^{i} \left( \rho_{z}^{i} - \widetilde{\rho}_{z}   \right) \; } = \sqrt{ \frac{R}{N} \sum_{i=1}^{N} \cos{2\theta^{i}} \left( \rho_{z}^{i} -  \widetilde{\rho}_{z} \right) \; } \approx 0

\chi_{yz} = \sqrt{  \frac{1}{N} \sum_{i=1}^{N} \rho_{y}^{i} \left( \rho_{z}^{i} - \widetilde{\rho}_{z}   \right) \; } = \sqrt{ \frac{R}{N} \sum_{i=1}^{N} \sin{2\theta^{i}} \left( \rho_{z}^{i} -  \widetilde{\rho}_{z} \right) \; } \approx 0

These coefficients must all be nil because, as noted above, we presume that large sums over odd powers of circular functions of  \theta add-up to zero. Then recall that the metric of  \mathbb{S} is given by the ratios

k_{\alpha \beta}^{ \mathbb{S}} \equiv \dfrac{\chi_{\alpha \beta}^{\mathbb{S}}}{\chi_{zz}^{ \mathbb{S}}}

where \alpha and \beta \in \{ x, y, z \}. So overall k_{xx} = k_{yy} = k_{zz} = 1 and k_{xy} = k_{xz} = k_{yz} = 0 . Then we say that the metric of  \mathbb{S} is Euclidean.

The Euclidean Metric
k_{zz} \equiv 1k_{xy}  = 0
k_{xx} = 1k_{xz}  = 0
k_{yy} = 1k_{yz}  = 0

Next we look at EthnoPhysics faviconNewtonian Mechanics.

Under Construction

Under construction iconContinuous Space (Under Construction)

Under construction iconEuclidean Space (Under Construction)

Under construction iconOrbital Angular Momentum (Under Construction)

René Descartes, La Géométrie, book 2 page 319, Paris 1637. Roman font with archaic ligatures.
References
1Here is another translation by David Eugene Smith and Marcia L. Latham.

“I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and then classify them in order, is by recognizing the fact that all points of those curves which we may call geometric, that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by means of a single equation.

If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse ; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class ; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely”.

From The Geometry of René Descartes, Open Court Publishing Company, page 48. La Salle Illinois, 1952.

2Isaac Newton, Mathematical Principles of Natural Philosophy, page xvii in the preface to the first edition. Translated by Andrew Motte and Florian Cajori. University of California Press, 1934.
3Benoit B. Mandelbrot, The Fractal Geometry of Nature, page 26. W. H. Freeman and Company, New York 1977.
4, 5Computational Chemistry Comparison and Benchmark Database Edited by Russell D. Johnson III, National Institute of Standards and Technology, Standard Reference Database Number 101, Release 18, Department of Commerce USA, October 2016.
6For example he writes that; “The material, as I said, is composed of many small balls which are in mutual contact; and we have sensory awareness of two kinds of motion which these balls have. One is the motion by which they approach our eyes in a straight line, which gives us the sensation of light; and the other is the motion whereby they turn about their own centers as they approach us.

If the speed at which they turn is much smaller than that of their rectilinear motion, the body from which they come appears blue to us; while if the turning speed is much greater than that of their rectilinear motion, the body appears red to us. But the only type of body which could possibly make their turning motion faster is one whose tiny parts have such slender strands, and ones which are so close together (as I have shown those of the blood to be), that the only material revolving round them is that of the first element.

The little balls of the second element encounter this material of the first element on the surface of the blood; this material of the first element then passes with a continuous, very rapid, oblique motion from one gap between the balls to another, thus moving in an opposite direction to the balls, so that they are forced by it to turn about their centres.”

From A Description of the Human Body published in The Philosophical Writings of Descartes, Volume I, page 323. Translated by John Cottingham, Robert Stoothoff and Dugald Murdoch. Cambridge University Press, 1985.