mr_e_man wrote:So we could have a characteristic polynomial t⁶+t³+1, for a triple rotation by 2π/9,4π/9,8π/9. But I don't know any specific example of a 6D tiling or symmetry group containing this rotation. Crystallographic restriction is a necessary condition, not a sufficient condition, for a tiling to exist with such a symmetry.
⌈ -1⌉ ⌈-1 -w +1 -w⌉ ⌈-1 +1 -w +w⌉
A = |+1 | C = |+1 -w +1 +w| C⁻¹ = (2-√2)/8 |-w -w +1 +1|
| -1 | , |-w +1 +w +1| , |+1 +1 +w +w|
⌊ -1 ⌋ ⌊+w +1 +w -1⌋ ⌊-w +w +1 -1⌋
⌈+1/√2 -1/√2 ⌉
D = C⁻¹AC = |+1/√2 +1/√2 |
| -1/√2 -1/√2|
⌊ +1/√2 -1/√2⌋
⌈cos(θ) -sin(θ) ⌉
D(θ) = |sin(θ) cos(θ) |
| cos(3θ) -sin(3θ)|
⌊ sin(3θ) cos(3θ)⌋
quickfur wrote:I finally found a way to orient the tesseract according to the plane you describe, but I'm not seeing a 45°/135° double rotation symmetry here.
⌈-1 -w +1 -w⌉ ⌈ cos(θ) sin(θ) 0 0 ⌉
|+1 -w +1 +w| |-sin(θ) cos(θ) 0 0 |
apacs[1 1 w w] * |-w +1 +w +1| * | 0 0 cos(3θ) sin(3θ)|
⌊+w +1 +w -1⌋ ⌊ 0 0 -sin(3θ) cos(3θ)⌋
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