Take the vertices of a tetrahedron -- I think I was thinking of the tattoo image, start with hyperbolic helicoids between pairs of vertices, and the two fitting together (maybe touching or deforming one another like modular paper art)
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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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