Editing 24-cell (section)

==== Hexagons ====
The 24-cell is [[self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

<math display="block">\left( \pm 1, 0, 0, 0 \right)</math>

and 16 vertices with ''half-integer'' coordinates of the form:

<math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math>

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br>
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)
{{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br>
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[Reye configuration]], which in the language of [[Configuration (geometry)|configurations]] is written as 12<sub>4</sub>16<sub>3</sub> to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.}}