Editing 24-cell (section)

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[quaternion]]s,{{Efn|In [[Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[William Rowan Hamilton|Hamilton]] did not see them as such when he [[History of quaternions|discovered the quaternions]]. [[Ludwig Schläfli|Schläfli]] would be the first to consider [[4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F<sub>4</sub> [[root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[ring (mathematics)|ring]]. This is the ring of [[Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D<sub>4</sub> root lattice is the [[dual lattice|dual]] of the F<sub>4</sub> and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[Hurwitz quaternion]]s, the [[#Hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}}