Talk:Quasiregular polyhedron

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[edit] Inconsistencies

By the definition here, "A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular", all regular polyhedra are quasiregular. By the definition at Polyhedron, "vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive", no regular polyhedron is quasiregular. Thus these two definitions are inconsistent. Furthermore, the enumeration of convex quasiregular polyhedra given here, which includes the octahedron but excludes the other regular polyhedra, is inconsistent with both definitions.

In fact the two external sites cited disagree as to whether the octahedron should be regarded as quasiregular. But both agree that to be quasiregular, faces with n sides must alternate with faces with m sides at each vertex -- implying an even number of faces at each vertex. Then the octahedron is quasiregular or not, depending on whether you allow n=m or not, but the other regular polyhedra (having 3 or 5 faces at each vertex) are not.

(posted by 24.58.33.52 on 06:06, 3 January 2008).

Hi, you are quite right about the inconsistencies. The literature on quasiregular polyhedra is pretty sparse, but it appears to be one of those many areas where geometers define one thing and then describe something else. One of the difficulties is that there are several interesting quasiregular properties, but nobody has ever methodically figured out which properties are fundamental and which are consequences of these. Feel free to make clarifications as to the nature of the muddle.
As for the octahedron, it can be seen as quasiregular by the definition at Polyhedron, if we colour alternate faces black and white so there are two kinds - a figure sometimes also called the tetratetrahedron.
And as for other web sites, they are for the most part a terrible load of rubbish. Mathworld perpetuates many shameful myths about polyhedra. George Hart is a lot more reliable, though his classification of quite so many star polyhedra as quasiregular is unusual (I happen to agree with him, but that is a long and continuing story).
HTH -- Steelpillow (talk) 12:55, 3 January 2008 (UTC)

[edit] Missing entries?

If non-convex polyhedra with non-convex vertex figures count, there's a number missing, firstly the edge-sharing forms (is there an "official" term?) of the pictured five:

Also, these have only singly symmetrical but nevertheless alternating vertex figures. Dunno if that's an additional problem.

Didn't want to add these directly for two reasons - the section for non-convex quasiregulahedrons is titled with "examples", and I notice the discussion on what the authorities consider quasiregular anyway... --Tropylium (talk) 19:28, 20 February 2008 (UTC)

You have the right idea. The first list are indeed quasiregular (though the second list are not - they do not have edges all wthin a single symmetry orbit). We can also find quasiregular examples among the apeirohedra - plane tessellations and infinite skew polyhedra. Trouble is, none of this seems to have been published in an adequate reference (It's on George Hart's website but AIUI personal websites are deemed inadequate). A few months ago I submitted a paper on exactly this to the Mathematical Intelligencer, so we shall have to wait and see. -- Steelpillow (talk) 20:58, 22 February 2008 (UTC)