User talk:Tomruen/Uniform honeycombs
The special lattices that I know of, in dimensions 1 to 8 (inclusive), are as follows:
1 dimension:
- Z (integer lattice)
2 dimensions:
- A2 = A*2 (hexagonal lattice)
3 dimensions:
- A3 = D3 (FCC lattice) / A*3 = D*3 (BCC lattice)
4 dimensions:
- A4 lattice / A*4 lattice
- D4 = D*4 lattice (4-dimensional checkerboard lattice)
5 dimensions:
- A5 lattice / A*5 lattice
- D5 lattice / D*5 lattice
6 dimensions:
- A6 lattice / A*6 lattice
- D6 lattice / D*6 lattice / D6+ lattice
- E6 lattice / E*6 lattice
7 dimensions:
- A7 lattice / A*7 lattice
- D7 lattice / D*7 lattice
- E7 lattice / E*7 lattice
8 dimensions:
- A8 lattice / A*8 lattice
- D8 lattice / D*8 lattice
- E8 lattice
Asterisks indicate dual lattices.
-- Calcyman (talk) 20:15, 4 June 2010 (UTC)
Where is the F4 lattice? Tom Ruen (talk) 20:41, 4 June 2010 (UTC)
These are the lattices with simply laced Dynkin diagrams. See ADE classification. -- Calcyman (talk) 20:45, 4 June 2010 (UTC)
I see F4=D4. What is D6+? Tom Ruen (talk) 20:48, 4 June 2010 (UTC)
D6+ is the lattice formed by two copies of the D6 lattice, where one copy is translated by (½,½,½,½,½,½) relative to the other. In 4 dimensions, this yields the basic Z4 lattice; in 8 dimensions, this produces the E8 lattice. I can't guarantee that this is a uniform lattice, though. -- Calcyman (talk) 10:58, 5 June 2010 (UTC)
- E8 is self-dual (unimodular). 'Dual' here means 'dual lattice', not dual in the sense that the triangular and hexagonal lattices are (geometrically) dual. Apparently, the terms 'polar' and 'reciprocal' are also used as well as 'dual' for the first definition. For a lattice L, the dual (L*) consists of all points such that the inner product (dot product) of any point in L with any point in L* is an integer.