all Platonic solids have an even number of vertices, so take pairs of them and link with hyperbolic helicoids (maybe they all have uniform charge distributions so repel one another?)
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and each helicoid might be distorted a little bit to make room for the others it's intercalated with. ... weaving with rotini
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I would love to see the object you describe! Though as soon as you distort the surfaces they are no longer really helicoids.
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