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<blockquote data-quote="orsal" data-source="post: 2290053" data-attributes="member: 16016">platonic solidsA bit more: there are three kinds of symmetry a polyhedron can have (well, there are more, but three that I'm discussing here): Vertex-transitivity, meaning that the geometry of the solid itself can't distinguish any vertex from any other; Edge-transitivity, meaning that pure geometry can't distinguish any edge from any other; Face-transitivity (you get the pattern). By &quot;pure geometry&quot;, I mean the shape of the die, ignoring labels. You can tell the difference between the face of your d6 labelled &quot;6&quot; and the face labelled &quot;2&quot; because, well, one is labelled &quot;6&quot; and the other is labelled &quot;2&quot;. But if you were able to obliterate the labels, the die would look exactly the same with the &quot;6&quot; on top as with the &quot;2&quot; on top.&nbsp;The five platonic solids are the only polyhedra which are both face-transitive and vertex-transitive. It's not hard to show, with a little high-school level mathematics, that there are only those five, and you end up getting edge-transitivity for free. But if you demand only one kind of transitivity, there are infinitely many possibilities. All your more exotic dice (by &quot;more exotic&quot; I mean &quot;not included way back when&quot;, so the d10 counts) are face-transitive but not vertex-transitive or edge-transitive. If you think a little about it, face-transitivity is what you want in a die. It guarantees that all faces are equally likely. Vertex-transitivity doesn't really matter, and I don't understand the purists who think the lack of vertex-transitivity is a point against the d10.</blockquote>

[QUOTE="orsal, post: 2290053, member: 16016"] [b]platonic solids[/b] A bit more: there are three kinds of symmetry a polyhedron can have (well, there are more, but three that I'm discussing here): Vertex-transitivity, meaning that the geometry of the solid itself can't distinguish any vertex from any other; Edge-transitivity, meaning that pure geometry can't distinguish any edge from any other; Face-transitivity (you get the pattern). By "pure geometry", I mean the shape of the die, ignoring labels. You can tell the difference between the face of your d6 labelled "6" and the face labelled "2" because, well, one is labelled "6" and the other is labelled "2". But if you were able to obliterate the labels, the die would look exactly the same with the "6" on top as with the "2" on top. The five platonic solids are the only polyhedra which are both face-transitive and vertex-transitive. It's not hard to show, with a little high-school level mathematics, that there are only those five, and you end up getting edge-transitivity for free. But if you demand only one kind of transitivity, there are infinitely many possibilities. All your more exotic dice (by "more exotic" I mean "not included way back when", so the d10 counts) are face-transitive but not vertex-transitive or edge-transitive. If you think a little about it, face-transitivity is what you want in a die. It guarantees that all faces are equally likely. Vertex-transitivity doesn't really matter, and I don't understand the purists who think the lack of vertex-transitivity is a point against the d10. [/QUOTE]

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