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.
Comments are closed.
Enter your email address to follow this blog and receive notifications of new posts by email.
Join 540 other followers
Brent's blogging goal