Cuboidal Prefixes
Chung van Gog, 02 September 2022
Metric prefixes are hopeless.
- Inconsistent capitalization (k, but M, G, & T)
- Nonconformity to 10^(3n) (c, d, da, h)
- Contain Greek characters (μ)
- Etymological hotchpotch (pico, femto, atto)
- Unmemorable and uninspiring (yotta, zetto, femto)
So I propose something different. Specifically, to steal something different.
Namely, the multiplier and power prefixes from the Primel Metrology. Now, Primel Metrology is an ingenious yet hopelessly obscure duodecimal metrology developed by John Kodegadulo. As such, Kodegadulo’s prefixes are intended to be used in a dozenal numerical system rather than in our familiar ten-finger system.
While I am sympathetic toward the dozenal cause, it would be convenient to have access to a similar set of prefixes for decimal SI units. So in the present cuboid I would like to propose a decimal equivalent of Table 1 below, to be exclusively used for SI units. (You can read more about this table by Kodegadulo here).
Without further ado, this is what I propose. For no good reason whatsoever, I will call them cuboidal prefixes. Positive powers are capitalized, negative powers aren’t.
| N | Root Form | 10^(3N) | Abbr. | 10^(-3N) | Abbr. |
|---|---|---|---|---|---|
| 0 | nil | Nilkilo- | NK | nilmilli- | nm |
| 1 | un | Unkilo- | UK | unmilli- | um |
| 2 | bi | Bikilo- | BK | bimilli- | bm |
| 3 | tri | Trikilo- | TK | trimilli- | tm |
| 4 | quad | Quadkilo- | QK | quadmilli- | qm |
| 5 | pent | Pentkilo- | PK | pentmilli- | pm |
| 6 | hex | Hexkilo- | HK | hexmilli- | hm |
| 7 | sept | Septkilo- | SK | septmilli- | sm |
| 8 | oct | Octkilo- | OK | octmilli- | om |
| 9 | enn | Ennkilo- | EK | ennmilli- | em |
So one kilometer (1 km) is now one Unkilometer (1 UKm), a Megawatt (MW) a Bikilowatt (BKW), and a nanogram (ng) a trimilligram (tmg). (To improve readability, cuboidal prefixes are typeset in bold, though they normally wouldn’t be.) For good measure, the “un” is optional, so Kilometer and milligram are still allowed—but do notice the capitalization. Instead of Nilkilo-/nilmilli-, one may also omit nil- altogether, since 1 Nilkilo = 1 = 1 Nilmilli. The differene between Nilkilo and nilmilli is akin to the difference between C♯ and D♭ in music theory—context determines which one is more appropriate.
The only thing to remember…
…is how to count to ten in prefixes: Nil, Un, Bi, Tri, Quad, Pent, Hex, Sept, Oct, Enn, Unnil.
A note on capitalization
Without ambiguity, the first letter of a prefix may be capitalized if grammer so requires (e.g., title case, first word in a sentence), even if it is normally written in lowercase. So we have 1 bmg, but “Bmg is short for bimilligram”. Caveat: the “m” for “milli” is never capitalized.
X, Y, and Z as placeholders
I quite like how $x$ and $y$ are used as placeholders in arithmatic. So why not borrow this concept? A BKx is a Bikilo of some unspecified unit $x$. And an XKy is an X-kilo of some unspecified unit $y$. The capitalization of $x$ and $y$ follows the usual rules.
Stand-alone prefixes
We may even use stand-alone prefixes like 1 BK and 2 tm (synonymously, 1BK and 2tm) as quantities (1 million and 2 billionth respectively). 1 BKg = 1K Kg.
Nullx
Mathematics works much better with a concept of “zero”. So we can use Nullx = nullx = ∅x to indicate that there is 0 of $x$. So a nullmeter (∅m) is zero meter. And 5 ∅m is still zero meter.
Hecto, deca, deci, centi, nil
Sometimes we are interested in distinguishing quantities that are one order of magnitude apart instead of three orders of magnitude apart. In this case, we may prefix cuboidal prefixes with hecto (h) & deca (da) (in combination with kilo-) or deci (d), or centi (c) (in combination with milli-). So we could have an NKm (m), daNKm (dam), hNKm (hm), UKm (Km), daUKm (daKm), hUKm (hKm), BKm, etc. and an nmm (m), dnmm (dm), cnmm (cm), umm (mm), dumm (dmm), cumm (cmm), bmm, etc. Caveat: hecto and deca cannot be used in combination with milli- because othersie hecto- would be indistinguishable from hex-. For the same reason, hecto- is never capitalized, also not at the beginning of a sentence.
Prefixes as categories
Not only can a stand-alone prefix indicate a quantity, a prefix plus unit can also indicate a range of quantities. For example, UKm includes distances between 1 and 1,000 kilometers. If we want to specify distances between 100 and 1000 kilometers, we can use hUKm. For distances between 10 and 100 Km there is daUKm.
For distances between 1 and 10 kilometers we may use the prefix nil- (n). So nUKm indicates a distance between 1 and 10 kilometers.
SI-fication
I quite like the project of expressing any quantity under the Sun with cuboidal prefixes and (if applicable) SI units (with gram, not kilogram, as the unit of mass). From (U)k€ to hK (hecto-Kelvin) and much more.
The SI system has the following defining constants:
| Constant | Value |
|---|---|
| $\Delta v_{Cs}$ | 9.192631770 TKHz |
| $c$ | 2.99792458 hBKm/s |
| $h$ | 6.62607015 duumJs |
| $e$ | 1.602176634 dhmC |
| $k$ | 1.380649 csmJ/K |
| $N_A$ | 6.02214076 hSmol-1 |
| $K_{cd}$ | 6.83 hlm/W |
Seeing the world through cuboidal prefixes
Astronomical distances become easy. The radius of the Earth is a bit less than 7 BKm. The distance to the Moon somewhat more than 1/3 TKm. That to the Sun about 150 TKm. And from the Sun to Saturn some ten times more, 1.5 QKm. And so on. Travel 1 EKm (= 1 Ronnameter) and you are outside the observeable universe. The age of the universe is 430 Pentseconds. Earth weighs about 6 OKg. I will be writing a series of blog posts through the “cuboidal point of view”.
Effortless extension beyond yotto and yocta
The great thing about this system is that we can effortlessly go beyond enn. After enn comes unnil, unun, unbi, …, binil, biun, …, and eventually ununnil and beyond. The mass of the Sun is slightly less than two Ununkilogram (2 UUKg). That of Sgt A* slightly more than 8.5 Untrikilogram (8.5 UTKg). No need for clumsy expressions such as “8.5 * 10^9 queccagrams”, or worse, on giving up on metric prefixes altogether. (To emphasize: even though “unnil” designates N = 12 in dozenal, in the combination with “kilo”, “milli” and SI-units, a clearly decimal context, “unnil” unambiguously designates N = 10)
Etymology
A brief note about etymology. The root forms nil to enn are taken from Kodegadulo, who in turn has based them on the IUPAC roots for the systematic element names. “Kilo” and “milli” designate 10^3 and 10^-3, and these are shorthand for 10^(3N) and 10^(-3N), the power prefixes we are interested in.
A googol…
… is simply 1 UNN.