nLab 24-cell




…one of the regular polytopes in dimension 4…

…hence a higher dimensional analog of the Platonic solids


The 24-cell is the regular polyhedron in the Cartesian space/Euclidean space 4\mathbb{R}^4 whose vertices are, under the identification 4 \mathbb{R}^4 \simeq_{\mathbb{R}} \mathbb{Q} with the space of quaternions, the 8 unit quaternions ±1\pm 1, ±i\pm i, ±j\pm j, ±k\pm k and the 16 unit quaternions given by 12(ε 01+ε 1i+ε 2j+ε 3k)\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k) where (ε 0,,ε 3){1,1} 4(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4.

(These 24 quaternions form a group under quaternion multiplication, and this group is isomorphic to the binary tetrahedral group.)


Symmetry group

The finite rotation group inside O(4) which is the symmetry group of the 24-cell is the Coxeter group F4.


See also

Last revised on October 10, 2021 at 16:57:57. See the history of this page for a list of all contributions to it.