Quasiperiodicty in 1, 2, 3, and 4 dimensions: Excellent talk by Dan SHECHTMAN on interference and light in soap bubbles and on quasicrystals
Quasiperiodicty in 1, 2, 3, and 4 dimensions: Excellent talk by Dan SHECHTMAN on interference and light in soap bubbles and on quasicrystals. Question and comment on "The Science and Aesthetics of Soap Bubbles - Agora Talks | Lindau Mediatheque https://mediatheque.lindau-nobel.org/recordings/42235"
Excellent talk by Dan SHECHTMAN on interference and light in soap bubbles and on quasicrystals:
Quasiperiodicity (≈ 40:00)
in one dimension: FIBONACCI Series described by Leonardo di Pisa FIBONACCI;
in two dimensions: Roger PENROSE’s PENROSE Tilings;
in three dimensions: Dan SHECHTMAN’s Quasicrystals.
It’s of interest that the heptad motif of coiled coil helices* is a second quasiperiodicity in one dimension described by Édouard LUCAS: LUCAS series are only slightly different at the starting and reach appoximately the golden ratio Phi, too. In addition they can be found in the seed heads of sunflowers sometimes in addition to FIBONACCI series.
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!)
What's Your point of view?
Yours Stefan Geier, Haidholzen
*GEIER Stefan, GEIER-NOEHL Michèle: Coiled Coil Helices Including Alpha-Keratin and Leucine Zippers are Related to the Golden Ratio Concept by the Omega Constant Ω and are Related to Tetrahedra Helices and to Quantum Physics. Research Gate, June 2024, DOI:10.13140/RG.2.2.11482.35525; https://www.researchgate.net/publication/381659767_Coiled_Coil_Helices_Including_Alpha-Keratin_and_Leucine_Zippers_are_Related_to_the_Golden_Ratio_Concept_by_the_Omega_Constant_O_and_are_Related_to_Tetrahedra_Helices_and_to_Quantum_Physics?utm_source=twitter&rgutm_meta1=eHNsLVVId2p0QTlHeWNUUVRzck9CV2NzNXNkMXdVWjRSSDZmcW42MWdaWUxKaE43cndOeEh6bEhkUmo2V0FuTXZ5U0VuQ3BZQ2l0WnVzUXVNNnprMTQ3QVZsUT0%3D
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.
AntwortenLöschenMay I argue that the the Boerdijk–Coxeter helix 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? (?)
Yours Stefan Geier, Haidholzen