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### Meta

This is totally cheating but I think it’s the only way to make it work.

Right, of course it is! But I found it interesting that it is possible to cheat in a “principled” sort of way that ends up producing something nice and symmetric.

Wow! Thank you for doing this. I agree, it does end up with something with a symmetry that, visually speaking, is aesthetically pleasing. Even on a node-by-node analysis it is very satisfying, but as I mentioned in my post on #14, on a deeper analysis it’s the difficulty to rationally justify which two pairs to duplicate that frustrates it. If nothing else, it really does reinforce the elegance and beauty of the n = 5 case.

Very well said.

Why are there duplicates here and not in n=5?

When n = 5 the number of circles with one dot is C(5,1) = 5, and with two, three, and four dots it is, C(5,2) = 10, C(5,3) = 10, and C(5,4) = 5 respectively. Decagon and pentagon rings can be placed inside each other to form a diagram with rotational symmetry. However, when n = 4, we have C(4,1) = 4, C(4,2) =6, and C(4, 3) = 4. You cannot have square and hexagon rings with rotational symmetry, so Brent replaced the hexagon by duplicating two of the circles with 2 dots to make an octagon ring.

Alternately, there’s no symmetrical choice of where to put nodes that correspond to opposing colors so they have to be duplicated. Since 5 is prime, this issue doesn’t come up at any level except x=0 and x=5, which are omitted and the central point respectively. (In my version the “outer” point is “at infinity” and represented by a circle around the whole diagram.)

Well, let me try to defend the solution: the ‘cheating’ (node duplication) has a well-defined mathematical basis! 😉 If you think of whitespace as the + operator from the algebra of graphs, then this diagram is equal to the ‘correct’ one without node duplication.