Talk:Triakis icosahedron
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A triakis icosahedron is a catalan solid, the dual polyhedron to the truncated icosahedron. ... This is the first stellation of the icosahedron ...
It can't be both. A Catalan, being the dual of a convex solid, is convex; a stellation is not. Each consists of 60 obtuse isosceles triangles, but in the Catalan they lie on 60 distinct planes (being dual to 60 distinct points), while in the stellation they are all on the twenty planes of the icosahedron (that's the nature of a stellation). —Anton Sherwood 06:53, 12 January 2006 (UTC)
- You're right, there is a difference!
- They're the same topology, but different geometric coordinates. However the name triakis icosahedron would still seem to apply to both, since both have an icosahedron with faces replaced by short tetrahedron pyramids.
- I verified that Wenninger calls it by this name in his book, Polyhedron Models.
- Obviously this difference needs to be noted in the article, but both sources should remain. Tom Ruen 07:27, 12 January 2006 (UTC)
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- Wenninger also uses the name Triakisicosahedron (one word) for the Catalan in Dual Models!
- How about this: "The name triakis icosahedron has been applied to two different polyhedra whose surfaces each consist of 60 obtuse isosceles triangles. Its more usual meaning is a Catalan solid, bla bla bla. The same name has also been applied (e.g. by Magnus Wenninger) to the first stellation of the regular icosahedron ..."
- —Anton Sherwood 07:52, 12 January 2006 (UTC)
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- I'm content with that - when in doubt, state the facts as known!
- Interestingly, I just noticed my (quick&dirty) program generates dual polyhedron by connecting dual vertices at face centers and this can cause concave faces on nonregular polyhedra, and in this case "my" face-centered dual of the truncated dodecahedron looks like the First stellation of icosahedron! SO it is due to different (possible) definitions of dual geometry, even if the formal one is defined to make the duals convex.
- Worse for me, my definition can cause nonplanar faces if not triangles!
- With a bit more (unavailable) time I could confirm my guess that the vertices of the First stellation of icosahedron match the face centers of a truncated dodecahedron!
- Tom Ruen 08:05, 12 January 2006 (UTC)
- I'm content with that - when in doubt, state the facts as known!
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- Okay probably different vertices anyway.
- ALSO interesting. A Triakis octahedron is topologically identical to the Stellated octahedron BUT a polyhedron generated with verices as the face-centers of a regular Truncated cube are a bit too short to match vertices of the Stellated octahedron.
- Tom Ruen 08:16, 12 January 2006 (UTC)
... A BIT CRAZY, but I added 3 related stellations with images. The "interpretation" is a little messy. Both Stellations and nonconvex uniform polyhedra have different interpretations of their surfaces. The "pure" approach introduces no new vertices on the intersections, while a "physical" model" approach computes the actual intersections and only shows the external surfaces. So the "external intersected surfaces" are topologically related.
This difference is similar to star polygons. For example, a "pure" pentagram has 5 vertices and is self-intersecting, while its "surface" can be considered a concave decagon polygon by computing vertices at the intersections and removing the interior edges.
A great dodecahedron is NOT a Triakis icosahedron, but its intersected surface IS! Tom Ruen 09:57, 12 January 2006 (UTC)
