Janus Physical Units

The Metric System is great, but it's based on decimal numbers, so the Janus numbers we just showed you don't work well with it. But that's not the only problem with the Metric System :

The last problem is the most important: our languages and our units predispose us to think of real numbers as integers, which they're not. Integers use trailing zeroes to indicate the decimal place of the first digit - the magnitude. Real numbers use trailing zeroes to show precision via significant digits. 5 kilometers is not the same as 5000 meters, but we conflate the two because we like to think of units as things, like 5 cows. Metric measurements favor integers; Janus favors real numbers.

To solve these problems, we use Janus metrics, based on natural units. Natural units normalize certain physical constants, defining them as opposed to measuring them. The list of constants that have been normalized in various systems includes :

There are also two important constants that can't be normalized:

A fundamental relationship unites several of these constants : α h c = ke e². Because of this, these physical constants can't all be normalized, and different systems of natural units must choose some not to normalize.

For example, the Planck units normalize c ħ ke G kB to 1, but not e. But it seems odd to leave the elementary charge unnormalized, since it is the only one which corresponds directly to a fundamental unit: all charges (except quarks) are integer multiples of the elementary charge. And the Dirac constant is only useful when angles are measured in radians, not cycles as we do in Musa. Worst, the gravitational constant G is only known to four significant digits, so the actual values of the Planck units are not precisely known. And finally, many of the Planck units are far too large or too small to be useful.

Stoney units differ only in normalizing the elementary charge but not the Planck or Dirac constant (unknown when Stoney proposed his units in 1881). Hartree atomic units normalize e ke, ħ and me but not c. All of these systems are designed for specific domains, for instance Planck units for quantum-level physics, and atomic units for atomic-level physics. And that's as it should be.

But Janus units are intended to be universal, not anthropic. Planck units are sometimes called God's units, but the elementary charge is more fundamental than the Coulomb constant, and radians are less fundamental than cycles. So the Janus units start by normalizing e c h, which seem indisputably fundamental: the universal unit of charge, the universal upper limit of speed, and the universal relationship, via wavelength and frequency, between distance and time on the one hand (united by c) and energy and mass on the other (also united by c). Janus units also normalize u and kB, since the relationships uniting energy to mass and temperature are almost trivial.

But we don't normalize the force constants ke or G. The first is easily calculated to be α / τ, and the second is not precise enough to define a system of units. Instead, we normalize me, the electron mass, just as we normalize the electron charge e. The electron mass isn't an elementary mass of which all other masses are multiples, as is the case with the elementary charge, but since the electron is the lightest charged particle, we treat it as if it were and normalize it.

Nor do we normalize these constants to 1 - we normalize them to other values in order to derive base units at useful scales. The Metric system addresses the scale problem with a base unit, the mole, which consists of Avogadro's number (NA) of molecules, where NA = 6.022141×10²³. In like manner, Janus Metrics introduce a dimensionless unit called the  Dodekit (δ), whose value is 12¹², or 8,916,100,448,256 (8.9×10¹²). We then use the Dodekit to normalize several constants.

Janus Units

Using these normalized constants, Janus derives five fundamental units for charge, mass, time, distance and temperature, and another 20 physical and electromagnetic units, plus a few more social units which will be explained on the next page. All of these have been given English names based on Greek and Latin roots, and other languages will use similar names. Unit names are never plural, and each unit is abbreviated by its first two letters, which are always written before the value.

The other units are straightforward. Here are the other mechanical units, which all have Greek names:

And here are the electromagnetic units, which all have Latin names based on the name of the dimension:

In addition to the physical units, Janus defines a purely mathematical unit for angles. We don't measure angles in degrees or radians. Instead, we use fractions of the unit circle, which we define as 1  Torit. For example, an angle of 90° is ¼ of a circle, or To0.25. As explained above, Janus considers τ to be fundamental, not π, so 1 Torit equals τ radians or 360 degrees.

The Torit is extended into a unit of solid angle, the  Sferit: 1 Sferit is the entire surface of a sphere. A spherical triangle with three 90° angles subtends 1/8 Sferit.

There are many other possible composite units, such as "Man Chronit" (a measure of the size of a task) or "Macrit / Sterit" (a measure of gas mileage), but they are all based on the units above.

Note that most of these units are around the small end of the range we use in everyday life. That permits us to describe common measures using Musa magnitide notation with small positive magnitudes. For example, the Gravit is about one-fourteenth of a gram, almost the same size as the grain, an old unit of weight, and 12+6 Gravit is about a ton.


ChronitChelapsed time
DensitDeflux density
FluxitFlmagnetic flux
SpinthitSpelectric charge
SferitSfsolid angle
ToritToplane angle

Note: the abbreviations are based on the Greek or Latin root, not the English name, so that they're universal across languages.

Conversion Tables

The Janus column shows conversion factors, in English decimal numbers, between common and Janus units. The second shows the same factors in Janus numbers using the Musa script.

Credit for help with this section is due to Neil Basescu, the Web sites of Erik Max Francis and Eric Weisstein, and numerous Wikipedia pages.

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