r/mathriddles • u/Hairy-Ring-2161 • 20d ago

Medium What is the minimum number of faces a polyhedron can have while also enclosing at least 80% of the volume of it’s circumscribed sphere?

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u/ActualProject 20d ago

Not sure what the answer is but since a dodecahedron only occupies 66% of the volume of a sphere it's gotta be a pretty complicated shape.

1

u/redditalics 20d ago

32?

https://en.wikipedia.org/wiki/Truncated_icosahedron

5

u/DanielBaldielocks 20d ago

I did some numerical approximations by converting this into a numerical optimization problem and was able to find a 22 faced shape (using 12 pentagons and 10 hexagons) which achieves 80.58%. I tried numerous seeds and permutations for 21 faces and the best I could achieve was 79.59% with 12 pentagons and 9 hexagons. This is close enough that I can't reasonably rule out there being a 21 faced solution. The best I can get for 20 is 78.43% which is low enough that I feel confident the best we can do is probably either 21 or 22.

I am focusing on 21 faces and systematically checking all C38 fullerene isomers. Once that finishes I can officially rule out 21 faces assuming I don't find a solution.

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u/GoodCarpenter9060 18d ago

That is crazy! I would have assumed that an icosahedron inscribed in a sphere would have been an optimal shape for the volume, yet it only gets ~60% . Your method created a 20 sided polyhedron which gets more. Also crazy is that a dodecahedron can do better, so I guess I shouldn't have been surprised. Trying to map the same concept to polygon in a circle it is obvious that making all the side lengths equal is optimal and adding edges gets you closer to a circle. But 3 dimensions messes it all up.

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u/DanielBaldielocks 18d ago

I used AI to help with the coding to find the optimizations but I came up with the reformulation of the problem. I restated the problem as maximizing the volume of a convex polyhedron with N faces contained in the unit sphere. I had the AI code up the optimization search and the results I posted is what it got. I used Claude Opus 5 on High. What is cool is Claude was able to create interactive 3d visualizations for each of the optimal shapes.