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Unsupportable Finite Cells

An unexpected discovery I made while writing Great Stella, is that some finite cells for some Archimedean stellations, can't be fully supported by other finite cells. In other words, they are partly supported by infinite cells. Infinite cells which have some finite faces pointing away from the centre of the model. It's hard to explain, but it's very surprising for anyone who knows what I'm talking about.

Another way of saying this is that the final stellation is re-entrant. Here, the final stellation is the one consisting of all finite stellation cells, and re-entrant means not fully supported. Fully supported means that any ray from the centre of the model out to infinite will only cross the model's surface once.

This situation arises for the following Archimedean/dual models:

  • Snub Cube.
  • Snub dodecahedron.
  • Pentagonal hexecontahedron.
It also happens for various non-convex uniform solids. Interesting that it happens for the snub dodecahedron and its dual, but the snub cube's dual is not included in the list.

If you want to see what those unusual cells look like, you can use the "Stellation>Fill All Unsupportable Cells" menu item in Great Stella, which will only have an effect for the few models that demonstrate this weird behaviour.

As far as I can tell, no one had discovered this before.

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