The New Metric Fables, by Leonard Cottrell

Postscript: human-scale Planck units compared with old metric units

(still in draft)

Fundamental Constants: Comparison With Old Metric Values

In the preface to these fables, values in human-scale natural units are given for five fundamental physical constants. These are the gravity constant, the speed of light, the frequency coefficient of temperature, the voltage coefficient of frequency, and the elementary charge. Algebraically, they span the full set of Planck quantities.

In the units used in the Forbidden City, one of two employed in the Fables, the values of these constants are a millionth, a billion, a trillion, a quadrillionth, and a quintillionth. As one might expect, since the metric system is antiquated, the corresponding metric values are messy. For a comparison, we can take a brief visit to the NIST (National Institute of Standards and Technology) website.

Main index for the "NIST Reference on Constants, Units, and Uncertainty": Under the heading "Fundamental Physical Constants" select "Values of the constants" then select "All values"

"Newtonian constant of gravitation":
6.673(10) × 10-11 m3 s-2 kg-1
Compare our G = 10-6 finger3/trice2 per billet (i.e. value is a millionth).

"Speed of light in vacuum":
299792458 meters per second
Compare our c = 109 finger per trice.

"Kelvin-hertz relationship":|search_for=all!
(2.083 6644 × 1010 Hz, must be multiplied by 2 pi to put in angular format)
1.309205 × 1011 per second per kelvin.
Our k/h-bar = 1012 per trice per grade = a trillion per trice per grade.

"Planck's constant over 2 pi in eVs":|search_for=all!
(6.582 118 89 × 10-16 eV s, replace eV by V to get photoelectric constant)
6.582 118 89 × 10-16 volt second
Our h-bar/e = 10-15 tao trice = a quadrillionth of a tao trice.

Elementary charge:|search_for=all!
1.602 176 462(63) × 10-19 coulomb
Compare e = 10-18 quint = a quintillionth of a quint.

Voltage coefficient of frequency, the photoelectric constant:
Historically the photoelectric effect first revealed that the energy in light is bundled in the form of quanta (Einstein's 1905 paper). The boost of energy which light of a certain color or frequency can give to the electrons in a photoelectric tube is measured by the voltage barrier which the light can cause them to overcome. The tube voltage (and associated energy) is proportional to the frequency of the light. The phototube is actually measuring the energy carried by an individual quantum of light of a certain frequency. The photoelectric constant is intuitively the most accessible form of Planck's constant and the phototube the clearest way of exhibiting the bundledness of energy in light. Our value of a quadrillionth means that the unit voltage of one tao corresponds to light of frequency one quadrillion per trice. An individual quantum in light of that frequency has enough energy to boost an electron by unit voltage.

Frequency coefficient of temperature:
What the NIST site calls the "Kelvin-hertz relationship" is the proportionality linking every temperature to the keynote frequency (kT/h-bar) in the thermal glow of things at that temperature. It is a bit simpleminded and "metricentric" for them to call the ratio "Kelvin-hertz relationship" of course. As if one were to call the speed of light the "meter-second relationship" or Planck's constant the "joule-hertz" relationsip. I'd expect real physicists at NIST to be better-spoken than that. Anyway the mix of frequencies in the thermal glow (and the keynote frequency, which Planck identified as crucial to describing the mix) is characteristic of the temperature, probably the temperature's clearest single attribute. This forms the basis for modern techniques of thermometry which tell temperature from the thermal glow and are applied over the widest possible range of temperatures from near absolute zero to the surface temperature of stars. Our value of a trillion means that unit temperature of one grade is characterized by a keynote frequency of one trillion per trice. The angular form of frequency is used throughout.

Alternative form of G

Because of equivalences within any system of units G can always be written several different ways. In an earlier collection called Planckian Fables we used a form of G related to the one here. That form was a ratio of quartic speed to force and it directly relates the fourth power of a uniformly round planet's surface orbit speed ("skimspeed") with what the planet would weigh in its own surface gravity. I am going to put our G into that alternative form to help establish continuity with the earlier fables.

With the present system, unit speed is one didge per mil, exactly one billionth of the speed of light. Unit force turns out to be about 27 pounds, rather close to the traditional British unit, the tod. For us here, tod force is that able to give unit acceleration (1 fingerwidth per trice2) to unit inertia (1 billet). A tod is, in effect, equal to 1 billet finger/trice2. A little algebra then shows that G = 10-6 (finger/trice)4 per tod.

This means that a skimspeed of 1 finger/trice corresponds to a planet weight of one million tod.

In an earlier set of fables we used a ton force measure equal to a hundred tod or 2700 pounds, and a dime speed measure equal to a tenth of the finger/trice unit speed. In those terms what I just said is the same as saying that a skimspeed of 10 dimes corresponds to a planet weight of 10 thousand tons. We know from the earlier fables that you can always take the fourth power of the speed in dimes and get the weight in tons. Dimes and tons are nice units for this reason. But the present units, with speed in finger/trice and force in tods, is well-adapted for a wider range of uses so we go with them.

Short summary of units

Most of what needs saying has already been said. I should mention that the unit energy, a todfinger, is very close to two metric joules. It is the energy delivered by pushing with tod force for a distance of one finger (about the thickness of my little finger.) The unit of charge, a quint, is the charge on a quintillion electrons, or, to be a little more exact, the positive version of that. The tao unit voltage provides unit energy to a unit of charge. This said, we can list a few basic units which are used in the fables.

finger/trice — one billionth of the speed of light in vacuum
trice — roughly a thousandth of a minute, 1113 trice make a minute
tod — a force of about 27 pounds, named after a traditional British 28 pound unit.
chi—unit power, toddidge/mil, about 36 watts, the exertion of unit force at units speed, the delivery of unit energy per unit time ("tchee" as in qi gong, the discipline. The same character is used in the modern Chinese word for automobile: powerwagon=vital breath wagon.)
grade — 141.7 Kelvin
quint — a quintillion times the elementary (proton or electron) charge
tao — unit voltage, todfinger/quint, slightly over 12 conventional volts.

The tao unit derives from nature since it is 10-27 of the Planck voltage. I'm also using a Chinese word for power, namely "chi" as a nickname for the todfinger/trice unit of power. One chi is approximately 36 watts and is delivered by something pushing with tod force at unit speed of a finger per trice. "Chi" is easier to say than "todfinger per trice" and perhaps it's more in keeping with the fables some of which were inspired by the light-hearted Taoist stories of Chuangtze.

ancient systems of units

It wouldn't be a bad idea to notice the role units play in joining human language to the natural world, and to have some acquaintance with systems of units used in other civilizations. Here is a website that summarizes the systems of units used in a number of classical civilizations including Egyptian, Persian, Chinese, Greek and Roman. It may be interesting to note that all these civilizations had a fingerwidth unit, as did the English later on among others. The fingerwidth unit was in the ballpark of 1033Planck lengths (1.616 centimeter). A mile of of 1038Planck lengths (1616 meter) is within half a percent of the US mile still in use—a distance corresoponding to a thousasand paces also used in many ancient systems. This is one reason moving to Planck units for practical measurement can have a somewhat retro feel.

earlier text from draft preface

The universe has a set of special quantities (technically known as Planck quantities) which serve as natural scales of measurement: a top speed, a main force, a core frequency, a full power, and so on. We can attune our own quantity-senses to the natural units by using familiar-sized, power-of-thousand versions of them — a set of human-scale Planck units. This has both practical and cultural advantages.

The foot unit which came down to us from the Romans just happened to be rather close to a billionth of a light-second so that the speed of one foot per second is close to a billionth of the speed of light in vacuum.

Because our basic units will be the same as nature's (though sometimes scaled up or down by powers of thousand to give them handy sizes), you can expect the most commonly used physical constants to be numerically simple and you can expect to be able to use them without getting bogged in arithmetic — indeed for the most part instead of calculating we merely need to move the decimal point and swap units.

This illustrates what was earlier described as the practical advantages of using human-scale versions of the natural units. The cultural advantages have to do with being in step with nature, and the possibilities this affords for lightness, quickness, and enriched metaphor: values relevant to poetry. It was this potential I intended to explore in the fables.


trice: 1042 Planck time

The names given units matter less than their sizes. Our suggested time unit is a third division of the hour, which extends the usual series of divisions: hour, minute, second, trice—and is accordingly called a trice. The idea of a trice is to be the 1042 multiple of the Planck interval of time. So as to have an exact conversion factor, it is defined so that there are exactly 1113 mil in an ordinary minute. This is the same as making it exactly 1/18.55 of an ordinary second.

finger: 1033 Planck length

The unit speed of 1 finger/trice is defined to be a billionth of the speed of light, which makes the finger unit length come out to be about a fingerwidth. Many civilizations have used a traditional fingerwidth as unit of length. The unit was called a "finger" in English, "digitus" in Latin, and "daktylos" in Greek. It's about 1.616 cm length and 1033 Planck lengths. Twenty fingerwidths is approximately a foot, a hundred is a traditional pace, and a thousand such paces is very close to (within half a percent of) the familiar US statute mile. Since there are a hundredthousand fingerwidths in a mile, the speed of light can also be expressed as tenthousand miles per trice. Since the earth's speed in its orbit around the sun is very close to a tenthousandth of light speed, the earth's speed is right about one mile per trice.

tod: 1042 Planck force

Another traditional unit is the tod of force. For us the tod is a 27 pound force and the idea is for it to be a 1042-th part of the natural unit, the Planck force. The traditional tod is defined in Webster's and on-line at unit dictionary web-sites as 28 pounds, so our 27 pound tod is a minor variant of the traditional unit.


When they are expressed in these terms, the Planck quantities (and some other closely allied fundamental contants) have comparatively simple values. I will show the table of values first without unit-names because the values matter more than the name. In a second table, further down the page, the unit names will be inserted.

 Planck time, tP  10-42
 Planck's h-bar  10-33
 Planck length, lP  10-33
 Boltzmann's k  10-21
 Elementary charge e  10-18
 Planck mass, mP  10-9
 Gravitational constant G  10-6
 Stefan-Boltzmann sigma/(pi2/60)  10-3
 Planck momentum, mPc   1
 Planck energy, EP  109
 Speed of light in vacuum, c  109
 Coulomb constant kC/alpha  1012
 Planck current, IP  1024
 Planck voltage, VP  1027
 Planck temperature, TP  1030
 Planck force, FP  1042
 Planck acceleration, c/tP  1051
 Planck power, PP  1051

Here is what the table looks like with unit-names inserted. The basic unit-names used here are trice (time), finger (length), tod (force), grade (temperature), and quint (charge.) Some auxilliary units derived from these basic units, and defined in terms of them, are: finger/trice (speed), billet (mass), todfinger (energy). A quint is the charge on a quintillion electrons. A billet is a billion times the Planck mass.

In definition, a todfinger is analogous to a "footpound"—it's a quantity of energy corresponding to pushing with tod force for finger distance—and in metric terms it comes to just under two joules. The billet, which is a mass of about 22 kilograms, is called that as a reminder that it is a billion times the Planck mass. A chi (36 watts), the unit of power, is that delivered by pushing with unit force at unit speed—pushing, in other words, with tod force at one finger per trice.

 Planck time, tP  10-42 trice
 Planck's h-bar  10-33 todfinger trice
 Planck length, lP  10-33 finger
 Boltzmann's k  10-21 todfinger/grade
 Elementary charge e  10-18 quint
 Planck mass, mP  10-9 billet
 Gravitational constant G  10-6tod finger2/billet2
 Stefan-Boltzmann sigma/(pi2/60)  10-3 chi/finger2grade4
 Planck momentum, mPc  1 billetfinger/trice
 Planck energy, EP  109 todfinger
 Speed of light in vacuum, c  109 finger/trice
 Coulomb constant kC/alpha  1012 tod finger2/quint2
 Planck current, IP  1024 quint/trice
 Planck voltage, VP  1027 todfinger/quint
 Planck temperature, TP  1030 grade
 Planck force, FP  1042 tod
 Planck acceleration, c/tP  1051 finger/trice2
 Planck power, PP  1051 todfinger/trice

In earlier fables we often needed to refer to a speed which is one tenth of unit speed, and we called that DIME speed. A force of 100 tod is roughly ton-sized and since there are many different tons (short ton, long long, metric ton, ship's ton) we simply call our hundred tods a TON.

Copyright © 2002 by Leonard Cottrell. All rights reserved.