Note

I originally tried to ask a variation of this question on math.stack; however 1 commenter pointed out that math.stack is not a puzzle site, which made me think maybe the fine folks of puzzling could help me out! If this isn’t an appropriate question for this site, I apologize. Also, I tried to post links to relevant information, however I don’t have enough reputation to post more than 8 items in a post. See the variation post for relevant links.

Background

The following is a variation of Nim-triplet addition mnemonic that is shown in On Numbers & Games by John Conway. The nim-triplets are also listed: (1,2,3), (1,4,5), (1,6,7), (2,4,6), (2,5,7), (3,4,7), (3,5,6). Any 2 numbers from a triplet sum to the 3rd (eg. 3+5=6, 3+6=5 & 5+6=3).

Note: the nim sum is equivalent to bitwise xor.

Inspired by the octonion multiplication mnemonic I wanted to see if I could create a multiplication mnemonic for nimbers. Much to my surprise I was able to create 1!

How it works Find a line (for example from 6-2), the midpoint (11) is the product. It 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 ★≡∗≡1
as a midpoint).

I was trying to reconfigure my layout & stumbled upon a ‘block’ style mnemonic:

This 1 works similar to the previous except it doesn’t work around the corners 🤷‍♀️

I found a few other ‘simple’ mnemonics & a pattern to create block mnemonics from them. Then I was able to piece together a larger mnemonic:

There’s a lot going on (& there are still lines that can be added 😅)..

TLDR Different line types emerging from the midpoints point ↦
towards the endpoints (eg. the squiggly lines from the midpoint 2 point at ↦
the endpoints 5 & 4).

I eventually cleaned things up / moved things around a bit to come up with the following mnemonic:

This layout was inspired by a graph of $G_2$, specifically “the A2 Coxeter plane projection of the 12 vertices of the cuboctahedron” (full disclaimer: I don’t know anything about $G_2$ or A2 Coxeter plane projections).

The following is an “alternative representation of the Fano plane” for order 2 & order 3. I think a mnemonic could be developed for nimber multiplication based on the order 3 (or something similar), possibly with 13 & 14 omitted

I’m not sure what the optimal configuration is for nimber multiplication up thru 15×15 (forms a closed field with no result larger than 15). Note: including 0, the 2nd degree fano plane has 8 digits & can be seen to correspond with octonions. I’m curious if this 16 digit field (including 0) might relate to sedonions somehow (super theoretical).

Which brings me to my questions:

• Primary What is the optimum configuration for a nimonic (nimber nmemonic) which contains digits 1-15? (I suspect it may be 3d but have not yet been able to explore that avenue)
• Secondary Given a larger nimonic, how can we determine how many smaller sub-nimonics (graph minors?) are embeded within it?
• Bonus Does this structure/math relate to something else besides nimbers?