- What's special about √5 is that there just two finite groups generated by reflections in n-dimensional space (n > 2) that don't preserve a lattice L ⊂ ℝⁿ. These are the symmetry groups of the icosahedron in 3d and something called the 600-cell in 4d, and the matrices for these groups of linear transformations have entries that can be chosen to lie in ℚ[√5].
The former is important for 3-dimensional quasicrystals, and I'm seeing that the field automorphism sending √5 to -√5 plays a key though perhaps not well-studied role.
The latter would be important for 4-dimensional quasicrystals.
When you talk about eigenvectors/eigenvalues of a graph, I guess you mean of the graph Laplacian? Can you tell me a polytope whose 1-skeleton has eigenvalues that are roots of a cubic? I don't instantly know what I'd do with that, but an example might help me come up with ideas.
John Carlos Baez
npub1nf…3nqe4
2026-05-17 11:57:15 UTC
in reply to nevent1q…ydqc
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2026-05-17 11:57:15 UTCEvent JSON
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"content": "- What's special about √5 is that there just two finite groups generated by reflections in n-dimensional space (n \u003e 2) that don't preserve a lattice L ⊂ ℝⁿ. These are the symmetry groups of the icosahedron in 3d and something called the 600-cell in 4d, and the matrices for these groups of linear transformations have entries that can be chosen to lie in ℚ[√5]. \n\nThe former is important for 3-dimensional quasicrystals, and I'm seeing that the field automorphism sending √5 to -√5 plays a key though perhaps not well-studied role. \n\nThe latter would be important for 4-dimensional quasicrystals.\n\nWhen you talk about eigenvectors/eigenvalues of a graph, I guess you mean of the graph Laplacian? Can you tell me a polytope whose 1-skeleton has eigenvalues that are roots of a cubic? I don't instantly know what I'd do with that, but an example might help me come up with ideas.",
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