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WikiProject Mathematics (Rated Stub-class, Low-importance)
WikiProject Mathematics
Mathematics rating:
Stub Class
Low Importance
 Field: Geometry

Schläfli symbol?[edit]

The symbol Schläfli symbol {3,5,3} is apparently shared with a hyperbolic honeycomb: Order-3_icosahedral_honeycomb

Can anyone help explain this? I just saw an article about this in Discover Magazine, although it didn't list this symbol.

Tom Ruen 05:57, 12 March 2007 (UTC)

The symbol {3,5,3} as you probably know means [3-gons, 5 per vertex, 3 of these things around an edge]. This important information is not at all a complete description of the geometric object it describes. Even the Schläfli symbol for the icosahedron, {3,5}, is shared by the hemi-icosahedron -- a triangulation of the projective plane by 6 triangles that is the relevant "height 3" object in the abstract regular polytope called the 11-cell, also known as the "hendecatope". (Some people avoid the "-choron" suffix in favor of the more general "-tope" simply because "-tope" uses only one syllable.)
As for the {3,5,3} hyperbolic honeycomb, this is a way of tessellating (all of) hyperbolic 3-space by regular hyperbolic icosahedra of the unique size such that their dihedral angles are all 120 degrees, enabling them to fit 3 around each edge (as the right-hand 3 in {3,5,3} requires).
But for the 11-cell, the symbol {3,5,3} signifies that there are 3 hemi-icosahedra around each edge. Because the projective plane is not the boundary of any 3-dimensional manifold, it is not clear (at least to me) whether there is any natural embedding of the 11-cell (in a larger space) that preserves its metric and symmetry. It is the union of 11 hemi-icosahedra with their 110 triangles identified in pairs, resulting in a total of 11 vertices, 55 edges, 55 triangles, and 11 hemi-icosahedra with its symmetry group transitive on flags, and therefore containing 660 symmetries; this group is isomorphic to PSL(2,11).Daqu 22:37, 19 May 2007 (UTC)
You can embedd it in 10-space as a hendecaxennon, but then the hemi-icosahedral cells are skew. Not sure if it's possible to embedd it in a lower-dimensional Euclidean space. Professor M. Fiendish, Esq. 09:39, 8 September 2009 (UTC)

"constructed by pasting hemi-icosahedra together"[edit]

How do you paste hemi-icosahedra together? Surely they can't be made in the real world, which has the wrong topology for that (at the human scale, at least)! Double sharp (talk) 09:13, 19 August 2012 (UTC)

Error in diagram?[edit]

In the diagram "Hemi-icosahedron_coloured.svg", I think there is a mistake. The six cells surrounding the "t" vertex are coloured red, blue, oink, yellow, orange and grey. However, the diagram for the yellow cell has a purple face adjacent to the "t" vertex. By elimination, I think it should be blue. Apt1002 (talk) 19:20, 7 September 2013 (UTC)