# 600-cell code[1]

## 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 in terms of quaternion coordinates yields the 120 elements of the binary icosahedral group \(2I\) [4].

## Protection

The 600-cell code is unique up to equivalence, which follows from saturating the Boroczky bound [5,6].

## Notes

The 600-cell code yields improved proofs of the Bell-Kochen-Specker (BKS) theorem [2].See post by J. Baez for more details.

## Parents

- Polytope code
- Universally optimal spherical code — The 600-cell is universally optimal, but it is not a spherical sharp configuration [7].
- Spherical design — The 600-cell code forms a spherical 11-design that is unique up to equivalence [6].

## 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.
- 24-cell code — Vertices of a 600-cell can be split up into vertices of five 24-cells [2,8,10].

## 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]
- L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
- [5]
- K. Boroczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hung. 32 (1978), 243–261.
- [6]
- P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
- [7]
- 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
- [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).

## 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