 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 units aren't based on anything - they're arbitrary.
• There are no metric units for several important qualities, like speed and acceleration. We talk about land in hectares, but real estate in square meters. And we measure electrical energy in kilowatt-hours, not joules.
• Time is not metric: a minute is not 100 seconds, and an hour is not 100 minutes or one-tenth of a day.
• Angles aren't metric, either: a circle isn't divided into 100 or 1000 degrees.
• The prefixes (kilo- milli- etc.) are a good idea, but we don't use them as much as we should. We say that A4 paper is 210mm wide, not 2.1dm (decimeters). We say San Francisco is 5000km from New York, not 5Mm (megameters). In other words, the prefixes aren't the metric version of scientific notation, as they should be.

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 :

• the elementary charge e
• the speed of light c, which appears in the formula Energy = u Mass c² (Einstein)
(the u is a conversion factor in order to convert the units: in SI units of joules, kilograms, meters and seconds, its value is 1.)
• the Planck constant h, which appears in the formula Energy = h Frequency (Planck)
(when frequency is expressed in radians, the formula uses the reduced version, the Dirac constant ħ = h / τ)
• the Coulomb constant ke, which appears in the formula for electrostatic force FE = ke Charge Charge / Distance² (Coulomb)
• the gravitational constant G, which appears in the similar formula for gravitational force FG = G Mass Mass / Distance² (Newton)
• the Boltzmann constant kB, which appears in the formula relating macroscopic temperature (θ) to atomic energy E = kB Θ
• the electron mass me

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

• the fine structure constant α, a dimensionless number whose value is roughly 0.0072973525698 or 1/137.03599907 no matter which units are used.
• the circle constant τ = 2π ≈ 6.2831853, the ratio of the circumference to the radius (not the diameter!) of a circle.
(For a discussion of why Janus chooses 2π, or τ, for its circle constant, please visit Pi is Wrong and The Tau Manifesto).

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.

• We start the derivation by normalizing e to 1 / δ, and thus defining the Janus unit of charge, the  Spinthit [Sp] as a Dodekit of elementary charges.

1 Spinthit = eδ = 1.428516719 × 10-6 Coulombs

However, the sign of charges is reversed : the electron carries a positive charge in Janus!

• We normalize me to 1 / δ², and thus define the Janus unit of mass, the  Gravit [Gr], as a Dodekit Squared (1224) of the mass of the electron.

1 Gravit = meδ² = 7.241672213E × 10-5 kilograms

• We normalize c to δ, and thus define the Janus unit of speed, the  Tachit [Ta], as One-Dodekith (12-12) of the speed of light.

1 Tachit = c/δ = 3.36237192 × 10-5 meters / second

• We also normalize u to δ, and then calculate the value of the Janus unit of energy, the  Ergit [Er], using Einstein's formula E = u m c² :

1 Ergit = 7.29970511 × 10-1 Joules

• We then normalize the Planck constant h to 1 / δ³, and then calculate the value of the Janus unit of time, the  Chronit [Ch], using the Planck formula E = h ν, where ν (frequency) = 1 / time :

1 Chronit = 6.43391816709006 × 105 seconds

• The Janus unit of length, the  Macrit [Ma], is set to the distance you'd cover running at 1 Tachit for 1 Chronit:

1 Macrit = 2.16332257927855 × 101 meters

• We can then introduce a unit of Acceleration, the  Archit [Ar] = 1 Tachit / Chronit :

1 Archit = 5.22600977285115 × 10-11 meters / second2

• The Janus unit of Force is the  Dynit [Dy] = 1 Gravit Archit :

1 Dynit = 3.78450497576255 × 10-15 newtons

• Finally, we normalize the Boltzmann Constant to 1 / δ², and define the Janus unit of temperature, the  Thermit [Th], using the formula E = kB Θ :

1 Thermit = 6.65077402 × 10-4 degrees Kelvin

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

• The  Sychnit [Sy] is the unit of frequency: one Sychnit means once per Chronit
• The  Platit [Pl] is the unit of area: a Platit is one square Macrit
• The  Sterit [St] is the unit of volume: a Sterit is one cubic Macrit
• The  Barit [Ba] is the unit of pressure, 1 Dynit per Platit
• The  Pyknit [Py] is the unit of density, 1 Gravit per Sterit
• The  Kinit [Ki] is the unit of momentum, 1 Gravit Tachit
• The  Rhomit [Rh] is the unit of power, 1 Ergit per Chronit

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

• The  Currit [Cu] is the unit of current, 1 Spinthit per Chronit.
• The  Potit [Po] is the unit of potential, 1 Rhomit per Currit.
• The  Resit [Re] is the unit of resistance, 1 Potit per Currit.
• The  Condit [Co] is the unit of conductance, the reciprocal of a Resit.
• The  Capit [Ca] is the unit of capacitance, 1 Spinthit per Potit.
• The  Fluxit [Fl] is the unit of magnetic flux, 1 Potit Chronit.
• The  Densit [De] is the unit of magnetic flux density, 1 Fluxit per Platit.
• The  Indit [In] is the unit of inductance, 1 Fluxit per Currit.

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.

### Recap

Archit Ar Barit acceleration   pressure   capacitance   elapsed time   conductance   current   flux density   quantity   force   energy   magnetic flux   mass   inductance   momentum   length   area   potential   density   resistance   power   electric charge   solid angle   volume   frequency   speed   temperature   plane 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.

• Units of distance :

EnglishMetricJanusMusa
0.0394in1mm4.6225×10-5 Ma 
0.394i1cm4.6225×10-4 Ma 
1 inch2.54cm1.1741×10-3 Ma 
1 foot30.5cm1.4089×10-2 Ma 
1 yard91cm4.2268×10-2 Ma 
3ft3in1 meter4.6225×10-2 Ma 
71.0ft21.633m1 Macrit 
5/8 mile1km4.6225×101 Ma 
1 mile1.6km7.4392×101 Ma 

• Units of mass :

EnglishMetricJanusMusa
0.00255 oz7.24167cg1 Gravit 
0.0353 oz1g1.38090×101 Gr 
1 oz28.35g3.91478×102 Gr 
1 lb454g6.26364×103 Gr 
2.20 lbs1kg1.38090×104 Gr 
1 ton907kg1.25273×107 Gr 
1.10 tons1 tonne1.38090×107 Gr 

• Units of time :

EnglishJanusMusa
1 second1.5542×10-6 Ch 
1 minute9.3256×10-5 Ch 
1 hour5.5953×10-3 Ch 
1 day1.3429×10-1 Ch 
1 week9.4002×10-1 Ch 
7 days 10 hours
43min 12sec
1 Chronit 
1 month4.0873×100 Ch 
1 year4.9048×101 Ch 
1 century4.9048×103 Ch 
1 millennium4.9048×104 Ch 

• Units of charge :

GaussianMetricJanusMusa
1 esu3.33564×10-10 C2.33504×10-4 
4.28259×103 esu1.42852×10-6 C1 Spinthit 
2.99792×109 esu1 Coulomb7.00027×105 

• Units of temperature :

FahrenheitCelsiusKelvinJanusMusa
-459.7°F-273.15°C0°K0 Thermit 
1.19714×10-3°F6.65077×10-4°C
(relative)
6.65077×10-4°K 1 Thermit
-459°F-272.15°C1°K1.504×103 Th 
0°F-17.778°C255°K3.834×105 Th 
32°F0°C273°K4.104×105 Th 
212°F100°C373°K5.608×105 Th 

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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