Talk:The Seven Limit Symmetrical Lattices: Difference between revisions

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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>
+A = \begin{bmatrix}
+& -1 & -1 \\
+-1 &  0 & -1 \\
+-1 & -1 &  0 \\
+\end{bmatrix}
+</math>
+Computing explicitly:
+:<math>
+\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.
+– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 22:00, 26 April 2025 (UTC)