Talk:Octahemioctacron

Question: Is it possible that the prisms of this octahemioctacron may only go out in one direction? This figure is the dual of the octahemioctahedron, which has tetrahedral symmetry, so this dual figure should also have tetrahedral symmetry. One end of each of three edges meet at each of the eight corners of a cube. These 24 edges then go out to infinity, forming the four hexagonal prisms. All eight corners of the cube are actual vertices, which along with the four infinite vertices form the twelve vertices of the dual. These correspond to the eight triangular and four hexagonal faces of the octahemioctahedron.

It is said that this figure is indistinguishable from the hexahemioctacron. It bothers me that the duals of two different polyhedra would be the same. The cubohemioctahedron has octahedral symmetry, so its dual, the hexahemioctacron would also have octahedral symmetry, and so each of its four prisms must extend to infinity in both directions. But it seems that the four octahemioctacron prisms should each only go out in one direction.

Paul Develet pauld28@netzero.net


 * The octahemioctahedron has octahedral symmetry. It cannot be constructed using the Wythoff construction using octahedral symmetry, only by using tetrahedral symmetry, although the polyhedron itself has octahedral symmetry. As for the duals being the same, the difference is where their true vertices are, and where the faces merely pass through each other (like the medial and great triambic icosahedra). Double sharp (talk) 13:54, 11 August 2011 (UTC)

Doesn't Exist
The octahemioctahedron has faces passing through its centre, and so does not have a dual. This needs to be made clear. The object illustrated here is an attempt to visualise something that doesn't really exist. Auximines (talk) 12:45, 15 January 2014 (UTC)

Does reading this take too long?
 * Since the octahemioctahedron has four hexagonal faces passing through the model center, the octahemioctacron has four vertices at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting infinite prisms passing through the model center, cut off at a certain point that is convenient for the maker.


 * I did read that section. Please don't be sarcastic, I was trying to help. My point is that, as mentioned in the article Dual polyhedron, in normal Euclidean space polyhedra with faces passing through its centre do not have duals. Auximines (talk) 11:18, 16 January 2014 (UTC)
 * I copied a longer paragraph from Dual hemipolyhedra if that helps. I'd just tend to call the polyhedron degenerate, but the Dual models book is the only source that described it at all. Tom Ruen (talk) 23:19, 16 January 2014 (UTC)
 * Wenninger's approach may or may not be valid, but it is the only published source describing these forms. Norman Johnson rejects these as proper polyhedra: but then he also rejects the fissary small hexagonal hexecontahedron and small hexagrammic hexecontahedron. In practice you are far more likely to see a list of uniform polyhedra and duals giving these infinite figures (and using Wenninger, Coxeter, or Bowers names, instead of Johnson's). The only site I have seen that uses Johnson's names is orchidpalms.com (George Hart uses them inconsistently). Double sharp (talk) 07:25, 30 March 2014 (UTC)