Talk:7-limit symmetrical lattices
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Some notes
Interesting page, actually. I'm trying to work through it bit by bit to make sure I understand it. The page doesn't really explain what is "symmetric" about these lattices. The original lattice is of course already full of symmetries since it's just the usual cubical lattice.
For the first part, there is some bit about coming up with various norms and then comes to the conclusion that you can make a cuboctahedron out of the 7-odd limit. I feel like you can explain this quite simply by considering the transformation:
- [math]\displaystyle{ A = \begin{bmatrix} 0 & -1 & -1 \\ -1 & 0 & -1 \\ -1 & -1 & 0 \\ \end{bmatrix} }[/math]
Computing explicitly:
- [math]\displaystyle{ \begin{aligned} A[1,0,0]^\mathsf{T} &= [0, -1, -1]^\mathsf{T} \\ A[0,1,0]^\mathsf{T} &= [-1, 0, -1]^\mathsf{T} \\ A[0,0,1]^\mathsf{T} &= [-1, -1, 0]^\mathsf{T} \\ A[1,-1,0]^\mathsf{T} &= [1,-1,0]^\mathsf{T} \\ A[1,0,-1]^\mathsf{T} &= [1,0,-1]^\mathsf{T} \\ A[0,1,-1]^\mathsf{T} &= [0,1,-1]^\mathsf{T} \\ \end{aligned} }[/math]
Which are the vertices of a cuboctahedron, as we wanted.