24-cell

In four-dimensional geometry, the 24-cell is the convex regular 4-polytope with Schläfli symbol. It is also called C₂₄, or the icositetrachoron, octaplex, icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.
The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual. The 24-cell and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.
The 24-cell does not have a regular analogue in three dimensions or any other number of dimensions, either [|below] or [|above]. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the cuboctahedron and its dual the rhombic dodecahedron.
Translated copies of the 24-cell can tesselate four-dimensional space face-to-face, forming the 24-cell honeycomb. As a polytope that can tile by translation, the 24-cell is an example of a parallelotope, the simplest one that is not also a zonotope.

Geometry

The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol, and the regular polygons with 7 or more sides. In other words, the 24-cell contains all of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but none of the pentagonal polytopes. The geometric relationships among all of these regular polytopes can be observed in a single 24-cell or the 24-cell honeycomb.
The 24-cell is the fourth in the sequence of six convex regular 4-polytopes. It can be deconstructed into 3 overlapping instances of its predecessor the tesseract, as the 8-cell can be deconstructed into 2 instances of its predecessor the 16-cell. The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.

Coordinates

The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

Squares

The 24-cell is the convex hull of its vertices which can be described as the 24 coordinate permutations of:
Those coordinates can be constructed as, rectifying the 16-cell with the 8 vertices that are permutations of. 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 and is inscribed in a 3-sphere of radius. 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 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.

Hexagons

The 24-cell is self-dual, having the same number of vertices as cells and the same number of edges as faces.
If the dual of the above 24-cell of edge length 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:
and 16 vertices with half-integer coordinates of the form:
all 24 of which lie at distance 1 from the origin.
[|Viewed as quaternions], these are the unit Hurwitz quaternions.
The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as unit radius coordinates to distinguish it from others, such as the radius coordinates used above.
The 24 vertices and 96 edges form 16 non-orthogonal great hexagons, four of which intersect at each vertex. 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.
The 12 axes and 16 hexagons of the 24-cell constitute a Reye configuration, which in the language of configurations is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.

Triangles

The 24 vertices form 32 equilateral great triangles, of edge length in the unit-radius 24-cell, inscribed in the 16 great hexagons. Each great triangle is a ring linking three completely disjoint great squares.

Hypercubic chords

The 24 vertices of the 24-cell are distributed at four different chord lengths from each other:,, and. The chords are the edges of central hexagons, and the chords are the diagonals of central hexagons. The chords are the edges of central squares, and the chords are the diagonals of central squares.
Each vertex is joined to 8 others by an edge of length 1, spanning 60° = of arc. Next nearest are 6 vertices located 90° = away, along an interior chord of length. Another 8 vertices lie 120° = away, along an interior chord of length. The opposite vertex is 180° = away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.
To visualize how the interior polytopes of the 24-cell fit together, keep in mind that the four chord lengths are the long diameters of the hypercubes of dimensions 1 through 4: the long diameter of the square is ; the long diameter of the cube is ; and the long diameter of the tesseract is. Moreover, the long diameter of the octahedron is like the square; and the long diameter of the 24-cell itself is like the tesseract.

Geodesics

The vertex chords of the 24-cell are arranged in geodesic great circle polygons. The geodesic distance between two 24-cell vertices along a path of edges is always 1, 2, or 3, and it is 3 only for opposite vertices.
The edges occur in 16 hexagonal great circles, 4 of which cross at each vertex. The 96 distinct edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.
The chords occur in 18 [|square great circles], 3 of which cross at each vertex. The 72 distinct chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers. The 72 chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.
The chords occur in 32 [|triangular great circles] in 16 planes, 4 of which cross at each vertex. The 96 distinct chords run vertex-to-every-other-vertex in the same planes as the hexagonal great circles. They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 edges apart on a great circle.
The chords occur as 12 vertex-to-vertex diameters, the 24 radii around the 25th central vertex.
The sum of the squared lengths of all these distinct chords of the 24-cell is 576 = 24². These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [|isoclinic rotations] rather than [|simple rotations].
The edges occur in 48 parallel pairs, apart. The chords occur in 36 parallel pairs, apart. The chords occur in 48 parallel pairs, apart.
The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes each forming a cuboctahedron. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees and 60 degrees apart. Each set of similar central polygons can be divided into 4 sets of non-intersecting Clifford parallel polygons. Each set of Clifford parallel great circles is a parallel fiber bundle which visits all 24 vertices just once.
Each great circle intersects with the other great circles to which it is not Clifford parallel at one diameter of the 24-cell. Great circles which are completely orthogonal or otherwise Clifford parallel do not intersect at all: they pass through disjoint sets of vertices.

Constructions

Triangles and squares come together uniquely in the 24-cell to generate, as interior features, all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions. Consequently, there are numerous ways to construct or deconstruct the 24-cell.

Reciprocal constructions from 8-cell and 16-cell

The 8 integer vertices are the vertices of a regular 16-cell, and the 16 half-integer vertices are the vertices of its dual, the tesseract. The tesseract gives Gosset's construction of the 24-cell, equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. The 16-cell gives the reciprocal construction of the 24-cell, Cesàro's construction, equivalent to rectifying a 16-cell. The analogous construction in 3-space gives the cuboctahedron which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.
We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract. This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.