600-cell code 

Description

Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. See [1; Table 1][2; Table 3] for realizations of the 120 codewords. A realization in terms of quaternions yields the 120 elements of the binary icosahedral group \(2I\) [3].

Figure I: Projection of the coordinates of the \(600\)-cell.

Protection

The 600-cell code is unique up to equivalence, which follows from saturating the Boroczky bound [4,5].

Realizations

Improved proofs of the Bell-Kochen-Specker (BKS) theorem [1].

Notes

See post by J. Baez for more details.

Parents

Child

  • 24-cell code — Vertices of a 600-cell can be split up into vertices of five 24-cells [1,6,9].

References

[1]
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
[2]
S. Mamone, G. Pileio, and M. H. Levitt, “Orientational Sampling Schemes Based on Four Dimensional Polytopes”, Symmetry 2, 1423 (2010) DOI
[3]
L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
[4]
K. Boroczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hung. 32 (1978), 243–261.
[5]
P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
[6]
H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
[7]
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
[8]
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
[9]
Schoute, P. H. (1903). Mehrdimensionale Geometrie, Vol. 2 (Die Polytope).
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Zoo Code ID: 600cell

Cite as:
“600-cell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/600cell
BibTeX:
@incollection{eczoo_600cell, title={600-cell code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/600cell} }
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Permanent link:
https://errorcorrectionzoo.org/c/600cell

Cite as:

“600-cell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/600cell

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/600cell/600cell.yml.