600-cell code[1]
Alternative names: Hexacosichoron code.
Description
Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. See [2; Table 1][3; Table 3] for realizations of the 120 codewords. A realization of the 600-cell can be done in terms of icosians, which are quaternion coordinates of the 120 elements of the binary icosahedral group \(2I \cong 2.A_5\) (a.k.a. icosian group) [5][4; Ch. 8, pg. 207].
Protection
The 600-cell code is unique up to equivalence, which follows from saturating the Boroczky bound [6,7].Notes
The 600-cell code yields improved proofs of the Bell-Kochen-Specker (BKS) theorem [2].See post by J. Baez for more details.Cousins
- Dual polytope code— The 600-cell and 120-cell are dual to each other.
- 120-cell code— Vertices of a 120-cell can be split up into vertices of five 600-cells [8,9]. The 600-cell and 120-cell are dual to each other.
- Icosahedron code— A realization of the 600-cell can be done in terms of icosians, which are quaternion coordinates of the 120 elements of the binary icosahedral group \(2I \cong 2.A_5\) (a.k.a. icosian group) [5][4; Ch. 8, pg. 207].
- Witting polytope code— The 120 vertices of the 600-cell are the unit icosians, and these icosian units, together with their multiples by \((1-\sqrt{5})/2\), form the 240 minimal vectors of a version of the \(E_8\) lattice, i.e., the Witting polytope [4; Ch. 8, pg. 210].
- 24-cell code— Vertices of a 600-cell can be split up into vertices of five 24-cells [2,8,10].
Member of code lists
Primary Hierarchy
References
- [1]
- L. Kollros, “An Attempt to determine the twenty-seven Lines upon a Surface of the third Order, and to divide such Surfaces into Species in Reference to the Reality of the Lines upon the Surface”, Gesammelte Mathematische Abhandlungen 198 (1953) DOI
- [2]
- M. Waegell and P. K. Aravind, “Critical noncolorings of the 600-cell proving the Bell–Kochen–Specker theorem”, Journal of Physics A: Mathematical and Theoretical 43, 105304 (2010) arXiv:0911.2289 DOI
- [3]
- S. Mamone, G. Pileio, and M. H. Levitt, “Orientational Sampling Schemes Based on Four Dimensional Polytopes”, Symmetry 2, 1423 (2010) DOI
- [4]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [5]
- L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
- [6]
- K. Boroczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hung. 32 (1978), 243–261.
- [7]
- P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
- [8]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [9]
- M. Waegell and P. K. Aravind, “Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell”, Foundations of Physics 44, 1085 (2014) arXiv:1309.7530 DOI
- [10]
- Schoute, P. H. (1903). Mehrdimensionale Geometrie, Vol. 2 (Die Polytope).
- [11]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
Page edit log
- Victor V. Albert (2022-11-16) — most recent
Cite as:
“600-cell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/600cell