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Length and Atoms

Historically, measurement is an important part of geometry. In 1637 the inventor of analytic geometry René Descartes wrote that1Here 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.

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, La Géométrie, Paris 1637.

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 accompanying photograph. 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 is relevant 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 of length 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 \ge  1. So observations of moving atoms always measure a smaller length than between stationary atoms, \ell \le \ell^{\ast}. This effect is called length contraction.

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.