Some Esoteric Unit Systems
I don't want to waste too much time writing an intro for this one; if you've clicked on this blog you've probably already heard of Jan Misali's a joke about measurement, which covers a lot of the introductory stuff regarding what coherence is and constructing coherent systems of units. If you have any questions about the blog, I'm just going to redirect you there.
Methods
So, I'm going to test if a system is coherent by seeing if I can convert the given set of units into any (m,l,t) set of units (mass, length, time), mainly because I know that SI units are coherent. I'm also probably just going to add in the more niche units of measurement (like the coulomb and moreso the candela) despite the fact that some of these unit systems are made to fit an aesthetic. Unless it's important to the specific case, I'm going to leave out those niche units that are required for a wholly coherent system, but just assume they're there pl0x.
lets see if i can get this entire thing completed in under a week lmao
The GOOD: Planck Units
Planck Units are used as a kind of "standards-neutral" unit system. There aren't any metal kilograms that we have to keep in pristine conditions, nor are there shaky and fluctuating defintions such as 1/60 of 1/60 of 1/24 of a day. No; rather than weak, malleable, awful bases such as a rock, Planck units are based off 4 fundamental constants of the universe:
| Unit Name | Value | SI Base units |
|---|---|---|
| c, the speed of light in a vacuum | 299792458 m/s | m1 s-1 |
| G, the Gravitational constant | 6.67430 * 10-11 N m2 kg-2 | m3 kg-1 s-2 |
| kb, the Boltzman constant | 1.380649 * 10-23 J/K | kg1 m2 s-2 K-1 |
| h/2π, the reduced Planck constant | 6.62607015 * 10-34 J/Hz | kg1 m2 s-1 |
The Wikipedia article for planck units provides a good overview of the rationale behind them - essentially boiling down to the "normalization" of universal constants by turning physical relations into expressions with dimensionless quantities (thereby setting the physical constant to 1). How that works in greater detail is out of the scope of this blog post, though; what we're concerned with is how the esoteric choice of basis for this unit system has made its mark on the world of physics. The main reason that this unit system is good is because of its "objectivity" (using that word in this context makes my skin crawl, but it's apt). These are extensively well-studied numbers that we're working with, that don't change at the arbitrary whims like "atmospheric pressure" or "the changing weight of a well-kept rock." This is what runs through my head when people endlessly dunk on Fahrenheit for being arbitrary compared to celsius.
Like, if I could rant for a second, the way that people talk about Fahrenheit being arbitrary can be applied the exact same way to celsius. Because guess what? Both temperature systems have already been redefined to be based on Planck units, because that's the actual "objective" system of measurement. Fahrenheit is basically just as objective as celsius, and works well within temperature ranges that we see in daily life, so stop basing your superiority complex off the fact that you use a system of measurement that uses water to gague temperature. That's not even mentioning the base-10 centrism that I constantly see coming from celsius supremacists. Complete delusion. If you held me at gunpoint and forced me to pick which temperature system was better, I would go with Kelvin.
uhhh that last part was kinda dry
The WET: Water Units
These units are an original creation based off various properties of water. The system also has both comically large and small units, which is always fun. Let's get into it.
| Unit Name | Value | SI unit |
|---|---|---|
| C, the specific heat of water | 4.184 J * g-1 * K-1 | m2 s-2 K-1 |
| p, the density of water* | 0.998 g * mL-1 | kg1 m-3 |
| ~, niagra-years** | 917.7 cm | m1 |
| B, the boiling point of water | 373.14 K | K1 |
*at STP
**the amount of land historically (it's lower now) eroded by niagra falls in a year; silly, I know
Constructing the "SI Base"
For this to be close to coherent, we need to have accurate descriptions for units of mass, length, and time. Now, we already have length, and the mass is trivially easy to calculate given that we have the density of water. So let's do it!
We set 0.998p = (x) g * 1~-3, which gives us that one flood is equal to 771.3166034 MEGA-grams. To put that in perspective, 771 Mg is the mass of roughly 3.8 million blue whales, which, if strewn about on a straight line from our Earth's surface, would make it about 30% of the way there to the moon.
turtles all the way down? nah, blue whales all the way upNow that we have mass, 2 out of the 3 that are needed for the "SI Base" have been determined. To reach the second, note that the SI unit for C = m2 s-2 K-1. This is yet another simple algebra problem, and we can determine that the flow is equal to 23.22570952 s. This is the first unit with interesting water base units: C-0.5 B-0.5 ~1. If anyone has any guess at a physical interpretation of this unit, I would absolutely love to hear it.
Now that we've constructed the base SI units, we've essentially created a coherent system, all we need to do is add the coulomb for electrical units, the molar mass of water for molar "units," and the lumen for brightness. We've just created an esoteric 7-unit coherent system of measurement! I'd like to dedicate the rest of this section to naming out various fun units (I've really enjoyed giving names like flood to my units for an aesthetically unifying calculation experience (although it's a bit like how we give literally each electrical unit a unique name, but I digress)).
Water Epilogue: Naming some (interesting) Units
| Name | Water Units | Quantity in SI Units | Comments |
|---|---|---|---|
| river (energy) | C1 p1 ~3 B1 | 1.20319 * 1012 J | The hoover dam outputs roughly 11958 rivers each year |
| ebb (frequency) | C0.5 B0.5 ~-1 | 0.04305573524 s-1 | wordplay is fun |
| WATER (current) | q1 C0.5 B0.5 ~-1 | 0.04305573524 A | I've named it water in reference to the pipe analogy for electricity |
Small linear algebra detour, I decided to calculate for myself the matrix that converts the basis of units by planck units to SI units, borrowing ideas from this youtube video, water units are actually somewhat easy to build a proper transformation matrix, since the joke that I made was of fairly low effort (i.e. I included some S.I. base units already) and was a petty jab at fahrenheit haters.
We want to construct a linear transformation W : R7 -> R7, which can be done by constructing a matrix W whose columns are the SI representations of each base unit quantity. In sympy, that matrix looks like this:
W = Matrix([[-2, 0, 0, 0, 0, 0, 0], [2, -3, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1]])
We can invert this matrix to get the conversion between SI units and water units, allowing us to skip some of the more tedious calculation required for calculating the units for something like capacitance - which, in our case, is the unit C-2 p-1 ~-1 A2 B-2. Looking at the unit system in this way reveals the low effort put into deriving it, since there are too many 0s involved in converting between SI and Wet units. The jab at celsius is still valid, though.