# Talk:3 31 honeycomb

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Field: Geometry

## Voronoi and duals

Its vertex arrangement is called the E7 lattice. The Voronoi cell of the dual E7* lattice is the 132 polytope.

If I'm reading this correctly, and the E7* lattice is the set of vertices of the dual of 331 — shouldn't its Voronoi cells be the (hyper)cells of 331? —07:17, 16 June 2012 (UTC)

I added a "related honeycomb" section, and put E7* there. Its constructed from the union of 2 E7 lattices, but dual in the polytope sense was wrong. I'll have to read more. I do see this sort of union construction allows for higher symmetry than pure reflections can generate, hence the <[3^(3,3,1)}> symmetry doubling. Tom Ruen (talk) 22:15, 19 June 2012 (UTC)
p.s. Apparently a dual lattice, marked by a * superscript, is the union (or compound) of all "symmetric copies" of a given uniform honeycomb. So like {4,4} represents the C2 lattice, and the C2* lattice {{4,4}} = + , but in the An lattice, Dn, and E6 lattice there are more symmetry positions to be unionized or compounded. Tom Ruen (talk) 20:03, 27 June 2012 (UTC)