# Digits for a base-210 system

My own contribution to the field of making up improbable alien number systems involves counting in base 210. Rather than try to define 210 distinct symbols, or to make a 14-upon-15 or 15-upon-14 digit pair system, I chose to break each digit down into three components mod 5, mod 6, and mod 7.

Mod 6 is the fusion of mod 2 and mod 3 to try to make things a little less unwieldy.

This means that when counting in ones, all three components change, but they wrap around at different points, and you can derive the overall magnitude via Chinese Remainder Theorem.

It also implies that you can determine, at a glance, the remainder of a value divided by 2, 3, 5, and 7. And probably a few other things, too…

## Practicality and utility

Can a human learn 210 distinct digits and also break these digits down into components which reveal useful properties for arithmetic? Can these be used to formulate new tricks for arithmetic that avoid having to learn a multiplication table with 44100 entries? I don’t know. Who cares? This is for aliens or whatever, right?

In our decimal world we use a lot of formal techniques and clever tricks for breaking problems down into something easier. This system needs many completely different approaches.

### addition

Breaking the whole digit apart into modulos 5, 6, and 7 you can simply add the individual parts with their respective modulos and stick them back together.

This gives you the least significant digit easily, but I’m not sure how to carry to the next digit.

TODO: figure that out.

### multiplication

TODO: try to coalesce my half-baked thoughts on this

### divisibility tests

TODO: try to coalesce my half-baked thoughts on this

### error resilience

A criticism levelled at metric is that because the units involve nothing but moving a decimal point, mistakes are harder to notice. When you convert from inches to feet, however, more digits change and the perturbation is more apparent.

Maybe this works here, too. Because small changes are very chaotic.

## Making a thing

My peculiar numerical specification says nothing about shapes.

Shapes are hard.

They have to be easy to distinguish from each other, which is a hard thing to judge because when they’re novel they all look the same and you have to develop some familiarity before you can decide if they’re still genuinely confusing or not.

They have to be easy to decode in the face of some distortions. Looking at them at an angle without a reference point (we have this with 6 and 9, but we get by because an error of 180 degrees is extreme), or sloppy handwriting, or a font that didn’t take up all the ink it should have, or a pen that didn’t land soon enough during the stroke, etc..

And while it’s not essential if you just learn all the characteristics of all 210 digits individually, I wanted the components to be separable again to reveal the original underlying remainders.

These are the segments I ended up with on my first attempt:

In this scheme, the absense of a feature (an arc, or a ring, or an elbow, or a line descending from the centre) represents divisibility by a corresponding prime. Removing the feature also means that I need one fewer variations of that feature. So I managed to squeeze five states out of four positions of the ring, for example. It might be clearer to put the ring in the centre for the fifth state. I don’t know.

I may have had some other sub-patterns in mind when I planned these, but I don’t recall what they all were. This is left as an exercise for the reader.

Combining these segments gave me the following skeleton for a set of 210 digits:

That’s just a skeleton. The thing to do next would be to round these out into
more plausible, coherent, and distinct glyphs. For example, if they were
written with a pen, what path would that pen *actually* trace? When we learn
to write we learn to trace specific paths (with some variations). Even though
other paths through the same letters would theoretically have the same
outcomes, once that becomes slurred it gets harder to read. For example, we
write 5 starting in the top left corner, and then come back for the top stroke
after finishing the bottom loop. If we didn’t do that it would end up coming
out like an S.

Stroke order is even more important in Chinese. Not only does it ensure that a slurred version of the character is slurred in a way everybody expects and understands, but it defines the character’s place in a sorted list.

And the combination of that stroke order and slurring helps to give each digit a more unique character to make them easier to learn and distinguish at a glance. It also reduces mirror symmetry where the skeleton itself is symmetrical.

But if this is some alien system then they might not use a pen or a brush or a clay-poking tool. They might use stencils or plants or toenail clippings fixed in place with snot.

But I’m *not* satisfied with this skeleton. I think that maybe I can encode
more clues as to the underlying number theory in the relations between the
segments and the other segments with which they fuse.

Sticking with placing things around a ring, here’s another attempt:

This also neglects to address symmetries and how confusing they are (again, this might be addressed in a subsequent pass), and it has worse problems with the discrimination of small angles. It comes out like this:

This time the angles between each component say something about how many cycles have passed, and so we have a notion of the overall magnitude of the number. Distinguishing angles is a dubious prospect, so maybe they could be filled in with tick marks or somesuch. Those ticks could then be evolved into something both easier to draw and more distinct than just ticks.

Clearly there doesn’t need to be any circle at all. I just got hung up on that because of clock analogies, or whatever, and then I ran out of energy for exploring a fairly unconstrained space or figuring out how to better meet the constraints that I’ve given myself, because everything I have right now is way too symmetrical.

It’d probably help if I didn’t start with a circle.

Random thoughts:

- No circle.
- Maybe reserve symmetries for round numbers, or numbers with other interesting properties.
- In that second clock-face example the angle suggests magnitude mod 30 because two hands on a clock can’t help but show that off – how does one work in a third term where the difference between that angle and something else shows the larger magnitude mod 210?
- Figuring out magintudes given the remainders seems difficult, and may need an additional cue it the design of the digits. Why does the discrete logarithm spring to mind? That just feels like it’s more of exactly the same problem. But maybe…
- On divisibility by primes, for some reason Miller-Rabin springs to mind, but I don’t have a whole thought on that.