Outline
Definition of Length
Measurement is deeply interwoven with ideas about length. Historically, measurement is an important part of geometry. In 1637 the inventor of analytic geometry René Descartes wrote1Here is a translation of the foregoing text 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
... all points of those curves which we may call geometric are those which admit of precise and exact measurement ...
— René Descartes
These days 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 later. 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
— 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 and
. Then the length
of the distance between them is
where 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 fully three-dimensional quantities. A length is a distance that is well-defined, and potentially measurable. 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?
However this logical protection comes with a significant consequence. Photons are not atoms. So they cannot serve as an end-point for a length measurement. Thus the position of a solitary photon cannot be measured. And the only time we can precisely locate a photon, is when it has been absorbed by an atom. So the EthnoPhysics definition of length abandons any possibility of pinpointing the location of a free photon.
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 between
and
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 . To optically measure a length in meters, first measure the elapsed time in seconds for a photon to travel from
to
. Then
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 . The two increments are related as
where
is the Lorentz factor. Similarly, when
then the length is called a proper length, noted by
and given by
. So these lengths are related as
The Lorentz factor for a particle in motion is always greater than one, . So observations of moving atoms always measure a smaller length than between stationary atoms,
. 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 , to a new, almost-Euclidean space called
. In
, particles are rotating. Their electric and magnetic axes turn around the polar axis. And in
we can construct a Cartesian coordinate system as follows.
The algebraic vector space is a set of position vectors
for some collection of atoms
,
,
…
. This mathematical construction is generically written as
The following basis vectors are used to make general descriptions; the axis of the abscissa is directed by , the axis of the ordinate from
and the polar-axis by
Then, any position vector in
can be expressed in terms of
,
and
, its Cartesian coordinates as
The new space is closer than
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
is a Euclidean space.
A Linear Coordinate


Let the -axis be associated with the proton inside an atom of hydrogen
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
on the
-axis is specified by the numeric value
. 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 to distinguish it from other hydrogen atoms.
Both and
use the same basis vector
So they share the same
-axis. And this axis can be parameterized using the same
-coordinate that was used for 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.


Let us combine an atom of oxygen with the hydrogen atom
shown above, to make a hydroxide anion,
The description is again centered on the proton inside
The -axis is defined in
by sensation, but for
more conventional details are required. So in this Cartesian coordinate system, the direction of the unit vector
is chosen to align with the O–H chemical bond called
, and ultimately fixed by the material presence of atomic oxygen.
The -axis is also chosen to be orthogonal to
The position of oxygen is then described by the coordinates
and
where
notes the distance between hydrogen and oxygen atoms.
The key detail about this arrangement is that it involves two atoms. So is measurable and can meet the definition for being a length. Indeed
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 is functioning as a frame of reference.
A Three-Dimensional Cartesian System


Next we use three atoms to make a three-dimensional Cartesian system. Let us combine another atom of hydrogen with the hydroxide anion to make
a molecule of water. The
-axis,
-axis and spatial-origin are as before, but the
-axis still needs to be established. We choose it to be orthogonal to both the
and
-axes, and in the same plane as the chemical bonds in water.
There are two possible orientations, identified by The number
is called the handedness of the coordinate system. The water bonds are called
They make an angle of
with each other. So for example, the position of the new hydrogen atom might be given by the coordinates
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
and
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 described by a repetitive chain of events written as
where each repeated cycle is a space-time event. Let this particle be characterized by its orbital radius
and wavelength
. 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
-axis, similar to the one shown in the accompanying diagram. The area
of the circular cross section is defined by
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 of the cylinder is
We use this model to visualize one atomic event. Then it is easy to imagine the chain of events as a row of cylinders strung-out end-to-end, like a tube or wire.
Spheroidal Atomic Models
Spheroidal Shapes | |
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a perfect sphere | ![]() |
a prolate spheroid | ![]() |
a oblate spheroid | ![]() |
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 be described by its wavenumber
. Then we can model
as a spheroid that is mathematically represented in Cartesian coordinates using the equation
This shape is also known as an ellipsoid of revolution about the atom’s polar axis. If is in its ground-state then
and the sphere collapses into a circle
However if is an excited atom then its wavelength is
and its shape can be represented as
Traditional mensuration formulae give the volume enclosed by this curve as
So the spheroidal model has been scaled to give 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
Let the events of atom be described by their time coordinates
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
is represented as a rotating atomic clock with a phase angle
given by
such that is whirling about its polar axis with an angular frequency of
. The rotation supposedly averages-out variations in the electric and magnetic radii leaving an effective orbital radius
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


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 Motion
Consider an atom that is described by some repetitive chain of events written as
where each orbit is represented by a bundle of quarks that can be parsed into eight orbital components
Let note the distance that
moves during one complete orbit. We can assess this quantity as follows. Since there are eight orbital components, the phase angle
of the
component is given by
Without loss of generality let so that
and
Then by substitution the displacement of is given by
and
where is the wavelength of
is its orbital radius,
is the magnetic polarity,
is the electric polarity and
is the helicity. These displacements can be used to evaluate changes in the position of
The variation in the abscissa during one atomic cycle is
The change in the ordinate is
And the change in position along the polar axis is
So over one full atomic cycle, the separation vector between initial and final events is
The distance between events is given by the norm of their separation, . So consecutive atomic cycles are separated from each other by one wavelength


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 . Initially, the radius was established from a particle’s quark content, and a vector space
was defined from sets of particles and their radii.
was characterised using statistical averages
, standard deviations
and correlation coefficients noted by
. These quantities have already been assessed for generic compound quarks.
Now let us consider that might be filled with a more restricted set of particles that have special attributes. Specifically we examine collections of atoms and their aggregates. Then radius vectors may be expressed in Cartesian coordinates. Particles are characterized using their orbital radius
, a wavelength
and their phase angle
. An atomic vector space like this is noted by
We say that
has a Euclidean metric if almost every particle that it contains can satisfy the following conditions.
- Atoms and molecules must be fully three dimensional like cylinders or rods so that
and
. Wavelengths and radii are all presumably positive, not nil.
- Particles must exhibit some variation in their shape so that
. The standard deviation for any component of the radius is presumably positive.
- The space
must be well-stirred. We require spatial homogeneity at an atomic level so that
.
- The particles in
cannot all have an extremely low temperature, or all be resonating like the atoms in a laser. We assume that
so that there is some variation among atomic phase angles.
- 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
add-up to zero.
For a specific example consider a big collection of atoms, and larger particles that are composite atoms, noted by
. 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
The radius vector is expressed in Cartesian coordinates as so a space like
can be generically represented as
The average radii of the particles in are given by
As noted above, we presume that the atoms in have phase angles that are completely incoherent so that sums over the circular functions add-up to zero. Then
and so the variations in particle radii are
We presume that all the particles in are three-dimensional so that
and
. And as noted above, we also assume that phase angles vary. So
and the standard deviations in particle radii must be positive

We also presume that has spatial homogeneity at the atomic-level. Then any variations cannot depend on direction,
. But by their definition, correlation coefficients and standard deviations are related as
so that
The other correlation coefficients are given by
These coefficients must all be nil because, as noted above, we presume that large sums over odd powers of circular functions of add-up to zero. Then recall that the metric of
is given by the ratios
where and
. So overall
and
Then we say that the metric of
is Euclidean.
The Euclidean Metric | |
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Footnotes


1 | Here is a translation of the foregoing text 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. |
---|---|
2 | Isaac 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. |
3 | Benoit B. Mandelbrot, The Fractal Geometry of Nature, page 26. W. H. Freeman and Company, New York 1977. |
4, 5 | Computational 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. |
6 | For 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. |