Editing 24-cell (section)

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = <small>{{sfrac|{{pi}}|2}}</small> away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{CDD|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[fundamental region]] of its [[Coxeter group|symmetry group]] [[F4 polytope|F<sub>4</sub>]], by reflection of that 4-[[orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[orthoscheme]] is a [[chiral]] irregular [[simplex]] with [[right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}