My supposition or question: Is the 600 tetrahedron cell {3,3,5} polytope a four dimensional quasicrystal?
I do think so, because its circumsphere is described by a radius of exactly the golden ratio Phi (Jean-Francois SADOC, Eur Phys J, 575-582, 2001).
Or: Are the tetrahedra of the {3,3,5} polytope more like crystal twins? (I don’t think so!)
Hm, ChatGPT tells me the {3,3,5} polytope is not a quasicrystal becaues it lacks aperiodicity. But I'm not shure this argument is correct in the given context.
May I argue that the the Boerdijk–Coxeter helix related to and tesselating the {3,3,5} polytope is not rotationally repetitive in 3-dimensional space but is in 4D, but lacks translational symmetry by integers in 3D and 4D. However, the Hopf fibration allows to state a form of periodicity indeed. Can we describe the 10 or 12 rings of the Hopf fibration by translation with integers? (?)
What's the role of translations in 4D?
What's Your point of view? Thank You in advance!
Yours Stefan Geier, Haidholzen
Please, compare with:
SHECHTMAN Dan: The Science and Aesthetics of Soap Bubbles. Lindau Nobel Laureate Meetings 2024. https://mediatheque.lindau-nobel.org/embed/42235
and
SHECHTMAN Dan, BLECH Ilan, GRATIAS Denis, CAHN John Werner: Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Phys. Rev. Lett. 1984, 53, 1951-1953
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