# Talk:Icosahedron

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Field: Geometry

## regular icosahedron

It says

"The most symmetrical are the two kinds of regular icosahedron. Each has 20 equilateral triangle faces with five meeting at each of its twelve vertices."

This statement is mysterious, and unless I missed it regular icosahedron only mentions one kind. I guess the other one is some sort of non-convex imbedding, but that needs a description plus a citation for it being also called "regular icosahedron". McKay (talk) 05:00, 31 August 2015 (UTC)

The two kinds are described and illustrated immediately following the sentence you quote. This is one of those logical hiccups with language that mathematicians are so blind to. By common custom the name "regular icosahedron" is applied to the convex "Platonic" variety. But technically the description "regular icosahedron" includes both forms. This is not helped by the fact that in more theoretical formulations both are the same combinatorial or abstract construct and are morphs of each other: topologically, there is only one regular icosahedron. It would naturally help if the mathematical community were to more rigorously name the convex variety accordingly, but they do not care to see a problem. Consequently, citable sources are sorely lacking. For the encyclopedist of course, the problem is to make clear whether each instance of the phrase "regular icosahedron" is being used as a name or a description. I have amended the text to help clarify it a little. — Cheers, Steelpillow (Talk) 08:36, 31 August 2015 (UTC)