24-cell
Squares

The 24-cell is the convex hull of its vertices which can be described as the 24 coordinate permutations of:

(± 1, ± 1, 0, 0) ∈ R 4. {\displaystyle (\pm 1,\pm 1,0,0)\in \mathbb {R} ^{4}.}

Those coordinates can be constructed as, rectifying the 16-cell with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the octahedron; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length √ 2 and is inscribed in a 3-sphere of radius √ 2. Remarkably, the edge length equals the circumradius, as in the hexagon, or the cuboctahedron. Such polytopes are radially equilateral.

The 24 vertices form 18 great squares (3 sets of 6 orthogonal central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of completely orthogonal great squares which intersect at no vertices.
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