Wikipedia:Reference desk/Archives/Mathematics/2021 December 25

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December 25

Rotating the other Platonic solids

Given the earlier question on the volume of the cube rotated on a vertex, I was thinking about the other platonic solids. Tetrahedron becomes a cone, Octahedron becomes a double cone, both should be pretty easy to calculate. Dodecahedron and Icosahedra *should* be composed of cones and partial codes, but each looks particularly ugly.Naraht (talk) 22:19, 25 December 2021 (UTC)

I'm not sure what the question is. Edges that pass through the axis sweep out cones, while those that don't and are not at right angles to it sweep out sections of a hyperboloid. The edges at the base of a tetrahedron, in two rings around the icosohedron, and in two rings around a dodecahedron, are at right angles to the axis, and sweep out part of a disk. Is there value in going into more detail than that? --RDBury (talk) 22:55, 25 December 2021 (UTC)

There isn't a pair of opposite vertexes on a tetrahedron, so you can't rotate it the same way as the other four solids.  Card Zero  (talk) 01:45, 26 December 2021 (UTC)

Each Platonic (or Archimedean or Catalan) solid has a well-defined center, and you can define an axis by that and one vertex rather than by two opposite vertices. —Tamfang (talk) 01:48, 29 December 2021 (UTC)

In any case, regardless of how or where you put the axis of rotation, the solid of revolution swept out is bounded by sections of a hyperboloid or cone (a degenerate hyperboloid). Note that in the most general case the resulting solid of revolution may not be convex; if the axis does not intersect the rotating solid, you even get a kind of doughnut.  --Lambiam 08:26, 29 December 2021 (UTC)