Note: The context of the follwing is nimbers.
I found the following nimber addition mnemonic on the Wikipedia page for Fano planes. Inspired by the Fano plane mnemonic, I decided to see if I could make a nimber multiplication mnemonic & much to my surprise I was able to figure 1 out!
Addition mnemonic
My 1st multiplication mnemonic
How it works Find a line (for example from 6-2), the midpoint (11) is the product. The previous example works all along the outer edge (eg. the line from 11-9 has 2 a midpoint). For lines thru the origin, the product is 1 (eg. the line from 7-11 has $\bigstar \equiv* \equiv 1$ as a midpoint).
My initial graph was based on the fano plane, but I'm not sure how to create that in Quiver (aside what can I use to create graphs like the addition mnemonic?). While trying to rearrange my diagram I stumbled upon a another configuration:
Block Nimonic
This works similar to the previous nimonic, however it doesn't go around the corners ♀️
I wasn't able to find much online about nimonics, but I was able to produce more of both types & figure out some patterns, which led me to creating this nimonic:
There's a lot going on (& there are still lines that can be added )..
TLDR Different line types emerging from the midpoints point $\mapsto$ towards the endpoints (eg. the squiggly lines from the midpoint 2 point at $\mapsto$ the endpoints 5 & 4).
Aside In this process of doing all of this I got tired of using the multiplication table on Wikipedia, so I had ChatGPT help me make a nimber multiplier (available beneath interactive Quiver chart - works up to 15x15).
Does something like this already exist? Or is there some other concept this is possibly related to? I'm a bit perplexed..
Edit I found 2 numonics (new nimonics) that feel more natural based on the "graph of $G_2$ as a subgroup of F4 and E8 projected into the Coxeter plane" (whatever that means..)
& "The A2 Coxeter plane projection of the 12 vertices of the cuboctahedron" (again no clue..)
Aside I find it interesting that both graphs overlap (w/ a $90^o$ rotation) however the 2nd numonic isn't just a rotation of the first.
These are essentially the same layout as the previous large numonic, just slightly rearranged. I think there is a probably some optimal configuration that I haven't found yet ♀️
Bonus question: I have been able to get multiple fano planes that embed into the larger nimonic. How many unique nimonics exist? How many unique smaller nimonics can be embedded into any given larger nimonic?
Double bonus: What does $G_2$ have to do with nimbers &/or combinatorial games?