The New Metric Fables, by Leonard Cottrell

These fables, and the preceding set called Planckian Fables, are intended to illustrate mainstream physics ideas (often pre-/circa-1900) with a twist and to suggest practical and cultural advantages of human-scale versions of the Planck units. The practical-sized units are equal to the Planck units conveniently scaled by powers of ten. In terms of such units the fundamental constants of nature commonly take on values which are powers of ten. Which powers of ten depend to some extent on choice.

Earlier this year (January 24) while working on a draft version of the preface for these fables, I noticed that the set of units I was using had an interesting property. Five of what are probably among the most intuitively accessible fundamental constants take on an especially simple and memorable series of values. The values of the gravitational constant, the speed of light, the frequency coefficient of temperature, the voltage coefficient of frequency, and the elementary charge turned out, respectively, to be a millionth, a billion, a trillion, a quadrillionth, and a quintillionth.

That is to say that *G*, *c*, *k*_{B}/h-bar,
h-bar/*e*, and *e* have the values 10^{-6}, 10^{9},
10^{12}, 10^{-15}, and 10^{-18}.

On the other hand I did not at the time have a particularly handy choice of
*names* for the units. They were basically just "time unit",
"length unit", and likewise with the others.

In a closely related set of units, called ton-pace-minute units, the same
series of fundamental constants had the values 1, 10^{10},
10^{15}, 10^{-20}, and 10^{-20}. The ton-pace-minute
units are specially chosen to make the fundamental physical constants have
values which are all powers of 10^{5} rather than just any old powers
of ten, and also to make the gravitation constant have value one. They have a
some fairly obvious names relating to the sizes that they turn out to be. The
time unit is minute-sized, the force unit is ton-sized, and so on.

The first set, in which the constants have values where are powers of a thousand (the exponents as you see are all multiples of three) was chosen in such a way as to make the units have sizes that are reasonably handy to work with—the force is a hundredth of a ton, the length is the breadth of a finger, the time interval is a thousandth of a minute, the voltage unit is twelve volts, to mention a few.

So here we have two pretty good systems of units for mainstream physics: the
practical powers-of-thousand set and the ton-pace-minute TPM (powers of
10^{5}) set. Incidentally both have the same temperature
scale—they use a step called a *grade* that is 10^{-30} of
Planck temperature. Since 30 is a multiple both of five and three, and so fits
both.

To compare them, I need a set of generic unit names for the unnamed set and for the time being I'm just going to make use of the METRIC SYSTEM'S NAMES for everything except the unit of time. The "length unit" will be meter, and the "mass unit" will be kilogram.They won't be the same sizes as common metric units but they will play the same roles. The time unit, which is smaller than a second, will be called a trice.

**Short list of constants in Generic ("New Metric") units**

*G*—universal gravitational constant—10^{-6}
meter^{3} per trice^{2} per kilogram

*c*—speed of light in vacuum—10^{9} meter per trice

*k*_{B}/h-bar—ratio of characteristic radiant frequency to
temperature—10^{12} per trice per kelvin.

h-bar/*e*—ratio of photoelectric voltage to frequency of incoming
light—10^{-15} volt trice

*e*—elementary charge (on proton, for example)—10^{-18}
coulomb.

**Short list of constants in TPM units**

*G*—universal gravitational constant—1 pace^{4} per
minute^{4} per ton

*c*—speed of light in vacuum—10^{10} pace per minute

*k*_{B}/h-bar—ratio of characteristic radiant frequency to
temperature—10^{15} per minute per grade.

h-bar/*e*—ratio of photoelectric voltage to frequency of incoming
light—10^{-20} grand minute

*e*—elementary charge (on proton, for example)—10^{-20}
score

That is to say that *G*, *c*, *k*_{B}/h-bar,
h-bar/*e*, and *e* have the values 1, 10^{10},
10^{15}, 10^{-20}, and 10^{-20}.

In case you are curious as to sizes, roughly speaking the Plank minute, pace, ton, score, and grand are (in that order) ninety percent of an ordinary minute, a two-step pace, about 1.2 metric ton force, sixteen conventional metric coulombs and some twelvehundred ordinary volts.

As fractions of these, the trice and the new meter, newton, coulomb, and volt are (in that order) 1/1000, 1/100, 1/100, 1/100, 1/100.

The temperature step in either case—new metric kelvin, or TPM grade—is 141.7 ordinary kelvin.

The pace/minute speed,10^{-10} of the speed of light, is sometimes
called a DIME of speed. TPM unit power is sometimes called a TONDIME (the power
delivered by pushing with unit force at unit speed, about half a horsepower or
360 watts).

A postscript has been attached at the end of the fables which gives a brief comparison of the values of the fundamental constants in these units and in their old metric counterparts.

**Conditions for parts of culture to evolve**

Units are a part of language and thus of our culture. A system of units is an intersection of language with the physical world since each unit has a dual nature: it is both an idea and a particular physical quantity. So the set of units is an important point of contact between language and the world. In my opinion and that of many other people, it generally works better to let culture evolve rather than trying to legislate it. But for culture to evolve there must be some diversity. I think it is healthy, for instance, that there are several temperature scales in use. It lets people compare the experience of each and form opinions about what they like to have in a temperature scale. Uniformity might be efficient in certain respects but it interferes with evolution.

Understandably, experimental scientists, who depend on the continuity and reliability of measurement, cling to their units and modify them only with grudging caution. Poets and mathematicians, by contrast, are sometimes more inclined to transform language and teachers are occasionally willing to try changes in language if there is some chance of improved understanding in the long run. No one, I think, is entirely one thing or another — all share responsibility for inventing and choosing language.

**Names for the generic units in the Forbidden City**

In the stories taking place in the Forbidden City everybody uses the Emperor's units. These turn out to be the set to which metric names were given earlier. The length unit is equal to the width of the Emperor's little finger (1.616 centimeters). Anyone familiar with Planck quantities will recognize that 1.616 centimeters as a power of thousand multiple of the Planck length.

Metric names sound out of place in the Forbidden City so other names are used such as finger, tod, quint, and tao (for units of length, force, charge, and voltage.) The length unit, as you may have guessed, is the 1.616 centimeter width of the Emperor's little finger.

Copyright 2002 Leonard Cottrell. All rights reserved.

Table of contents for the New Metric Fables