Talk:Dihedron

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Polyhedra

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truncation or what[edit]

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From a Wythoff construction on dihedral symmetry, a truncation operation on a regular {n,2} dihedron transforms it into a 4.4.n n-prism.

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Cantellation? —Tamfang (talk) 07:07, 31 May 2012 (UTC)[reply]

Yup, it should be rr{n,2} = {n}×{}: t{n,2} is just {2n,2}. Double sharp (talk) 13:49, 3 March 2015 (UTC)[reply]

Hmmm... Cantellation isn't usually applied across 2 branches. t{2,n} works better, or a bitruncation from a dihedron as 2t{2,n}. Tom Ruen (talk) 09:01, 4 March 2015 (UTC)[reply]

Perhaps so, but it does work: if you imagine taking r{n,2} and bevelling the edges, each edge turns into a square, and the hemispherical polygons shrink, creating an n-gonal prism. We should probably list both constructions. Double sharp (talk) 10:06, 4 March 2015 (UTC)[reply]

r{n,2}={n,2}, another reason rr is problematic. Tom Ruen (talk) 11:07, 4 March 2015 (UTC)[reply]

You're right: we ought to use t_0,2 then, I guess? Double sharp (talk) 08:13, 5 March 2015 (UTC)[reply]

I'd stick with t{2,n}, truncated hosohedra as Coxeter expresses, and just leave the dihedrons in peace. Or say a dihedron is a degenerate prism with height of zero (it already says that).

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= n-gon. Tom Ruen (talk) 09:11, 5 March 2015 (UTC)[reply]

As a degenerate prism[edit]

It seems to me that if a dihedron is seen as a prism with zero height, then there are two degeneracy interpretations possible. Each side face become either a digon or single edge. So the regular form has 2 faces, n edges and vertices, while a uniform solution could have 2 n-gons and n 2-gonal faces, and 2n edges, and n vertices. This sort of topological problem also exists in alternation of polygons with even number of sides where squares are reduced either to digons or edges. There can be reasons to keep digons, and the Euler characteristic is unchanged since you're adding or removing n faces and n edges.

Anyway, I admit I've not read anything to support this specifically, or it is here I guess Digon#Theoretical significance. And it could apply in any polyhedron, replacing any edge with 2 edges and a digon between. It is important for hosohedra which do have digon faces since Coxeter uses t{2,n} = for a special case wythoff construction of uniform prisms for both spherical tilings and polyhedra. But you could argue better antipodal point digons are more real since they make spherical lunes. Tom Ruen (talk) 23:51, 13 February 2017 (UTC)[reply]