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Short description: Regular object in four dimensional geometry
  1. REDIRECT Template:Infobox 4-polytope

In geometry, the 24-cell is the convex regular 4-polytope[1] (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, or the icositetrachoron,[2] octaplex (short for "octahedral complex"), icosatetrahedroid,[3] 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.[lower-alpha 1] It and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.[lower-alpha 2]

The 24-cell does not have a regular analogue in 3 dimensions. It is the only one of the six convex regular 4-polytopes which is not the four-dimensional analogue of one of the five regular 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 tile 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.


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,[lower-alpha 3] and the polygons {7} and above. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or its honeycomb.

The 24-cell is the fourth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[lower-alpha 4] It can be deconstructed into 3 overlapping instances of its predecessor the tesseract (8-cell), as the 8-cell can be deconstructed into 2 overlapping instances of its predecessor the 16-cell.[5] 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.[lower-alpha 5]



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

[math]\displaystyle{ (\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 }[/math].

Those coordinates[6] can be constructed as CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, rectifying the 16-cell CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 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[7] 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.[lower-alpha 2]

The 24 vertices form 18 great squares[lower-alpha 6] (3 sets of 6 orthogonal[lower-alpha 8] 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[lower-alpha 7] great squares which intersect at no vertices.[lower-alpha 11]


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 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:

(±1, 0, 0, 0)

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

1/2, ±1/2, ±1/2, ±1/2)

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[lower-alpha 2] in this coordinate system. We refer to the system as unit radius coordinates to distinguish it from others, such as the 2 radius coordinates used above.[lower-alpha 12]

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,[lower-alpha 13] four of which intersect[lower-alpha 11] at each vertex.[lower-alpha 15] 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.[lower-alpha 16]

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.[8]


The 24 vertices form 32 equilateral great triangles[lower-alpha 17] inscribed in the 16 great hexagons.[lower-alpha 18]

Hypercubic chords

Vertex geometry of the radially equilateral[lower-alpha 2] 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.

The 24 vertices of the 24-cell are distributed[9] at four different chord lengths from each other: 1, 2, 3 and 4.

Each vertex is joined to 8 others[lower-alpha 19] by an edge of length 1, spanning 60° = π/3 of arc. Next nearest are 6 vertices[lower-alpha 20] located 90° = π/2 away, along an interior chord of length 2. Another 8 vertices lie 120° = 2π/3 away, along an interior chord of length 3. The opposite vertex is 180° = π away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center can be treated[lower-alpha 21] as a 25th canonical apex vertex,[lower-alpha 22] which is 1 edge length away from all the others.

To visualize how the interior polytopes of the 24-cell fit together (as described below), keep in mind that the four chord lengths (1, 2, 3, 4) are the long diameters of the hypercubes of dimensions 1 through 4: the long diameter of the square is 2; the long diameter of the cube is 3; and the long diameter of the tesseract is 4.[lower-alpha 23] Moreover, the long diameter of the octahedron is 2 like the square; and the long diameter of the 24-cell itself is 4 like the tesseract. In the 24-cell, the 2 chords are the edges of central squares, and the 4 chords are the diagonals of central squares.


Stereographic projection of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.

The vertex chords of the 24-cell are arranged in geodesic great circle polygons.[lower-alpha 25] The geodesic distance between two 24-cell vertices along a path of 1 edges is always 1, 2, or 3, and it is 3 only for opposite vertices.[lower-alpha 26]

The 1 edges occur in 16 hexagonal great circles (in planes inclined at 60 degrees to each other), 4 of which cross[lower-alpha 15] at each vertex.[lower-alpha 14] The 96 distinct 1 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.[lower-alpha 16]

The 2 chords occur in 18 square great circles (3 sets of 6 orthogonal planes[lower-alpha 10]), 3 of which cross at each vertex.[lower-alpha 29] The 72 distinct 2 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.[lower-alpha 30] The 72 2 chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 1 edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,[lower-alpha 24] 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 3 chords occur in 32 triangular great circles in 16 planes, 4 of which cross at each vertex.[lower-alpha 32] The 96 distinct 3 chords[lower-alpha 17] run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.[lower-alpha 18] They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 1 edges apart on a great circle.[lower-alpha 33]

The 4 chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.[lower-alpha 22]

The sum of the squared lengths[lower-alpha 34] of all these distinct chords of the 24-cell is 576 = 242.[lower-alpha 35] 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 corresponding to diagonal isoclinic rotations rather than simple rotations.[lower-alpha 36]

The 1 edges occur in 48 parallel pairs, 3 apart. The 2 chords occur in 36 parallel pairs, 2 apart. The 3 chords occur in 48 parallel pairs, 1 apart.[lower-alpha 37]

The central planes of the 24-cell can be divided into 4 central hyperplanes (3-spaces) 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.[lower-alpha 40] Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).[lower-alpha 41] Each set of Clifford parallel great circles is a parallel fiber bundle which visits all 24 vertices just once.

Each great circle intersects[lower-alpha 11] with the other great circles to which it is not Clifford parallel at one 4 diameter of the 24-cell.[lower-alpha 42] Great circles which are completely orthogonal[lower-alpha 7] or otherwise Clifford parallel[lower-alpha 24] do not intersect at all: they pass through disjoint sets of vertices.[lower-alpha 43]


Triangles and squares come together uniquely in the 24-cell to generate, as interior features,[lower-alpha 21] all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the 5-cell and the 600-cell).[lower-alpha 44] Consequently, there are numerous ways to construct or deconstruct the 24-cell.

Reciprocal constructions from 8-cell and 16-cell

The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular 16-cell, and the 16 half-integer vertices (±1/2, ±1/2, ±1/2, ±1/2) are the vertices of its dual, the tesseract (8-cell). The tesseract gives Gosset's construction[17] 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.[lower-alpha 45] The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,[18] equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described above). The analogous construction in 3-space gives the cuboctahedron (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.[19]

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.[20] This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.


We can facet the 24-cell by cutting[lower-alpha 46] through interior cells bounded by vertex chords to remove vertices, exposing the facets of interior 4-polytopes inscribed in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes (above) are only some of those planes. Here we shall expose some of the others: the face planes[lower-alpha 47] of interior polytopes.[lower-alpha 48]


Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by 1 edges to remove 8 cubic pyramids whose apexes are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,[lower-alpha 49] and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a tesseract. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.[lower-alpha 33] They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume. They do share 4-content, their common core.[lower-alpha 50]


Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by 2 chords to remove 16 tetrahedral pyramids whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the 1 edges, exposing 2 chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,[lower-alpha 51] and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a 16-cell. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell. They overlap with each other, but all of their element sets are disjoint:[lower-alpha 52] they do not share any vertex count, edge length,[lower-alpha 53] or face area, but they do share cell volume. They also share 4-content, their common core.[lower-alpha 50]

Tetrahedral constructions

The 24-cell can be constructed radially from 96 equilateral triangles of edge length 1 which meet at the center of the polytope, each contributing two radii and an edge.[lower-alpha 2] They form 96 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 2, where the three vertices of each triangle are located 90° = π/2 away from each other on the 3-sphere. They form 48 2 tetrahedra (the cells of the three 16-cells), centered at the 24 mid-edge-radii of the 24-cell.[lower-alpha 53]

The 24-cell can be constructed directly from its characteristic simplex CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, a fundamental region of its symmetry group F4, by reflection of that 4-orthoscheme in its own cells (which are 3-orthoschemes).[lower-alpha 54]

Relationships among interior polytopes

The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.[lower-alpha 50] The tesseracts and the 16-cells are rotated 60° isoclinically[lower-alpha 55] with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are 3 (120°) apart.[lower-alpha 33]

The tesseracts are inscribed in the 24-cell[lower-alpha 56] such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell[lower-alpha 57] such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior[lower-alpha 58] 16-cell edges have length 2.

Kepler's drawing of tetrahedra inscribed in the cube.[21]

The 16-cells are also inscribed in the tesseracts: their 2 edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.[22] This is reminiscent of the way, in 3 dimensions, two tetrahedra can be inscribed in a cube, as discovered by Kepler.[21] In fact it is the exact dimensional analogy (the demihypercubes), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.[23][lower-alpha 53]

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable[4] 4-dimensional interstices[lower-alpha 59] between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are 4-pyramids,[lower-alpha 60] alluded to above.

Boundary cells

Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.[lower-alpha 62] Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.[lower-alpha 61]

As we saw above, 16-cell 2 tetrahedral cells are inscribed in tesseract 1 cubic cells, sharing the same volume. 24-cell 1 octahedral cells overlap their volume with 1 cubic cells: they are bisected by a square face into two square pyramids,[25] the apexes of which also lie at a vertex of a cube.[lower-alpha 63] The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.[lower-alpha 62]

As a configuration

This configuration matrix[26] represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

[math]\displaystyle{ \begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix} }[/math]

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

Symmetries, root systems, and tessellations

The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the F4 group, as shown in this F4 Coxeter plane projection

The 24 root vectors of the D4 root system of the simple Lie group SO(8) form the vertices of a 24-cell. The vertices can be seen in 3 hyperplanes,[lower-alpha 38] with the 6 vertices of an octahedron cell on each of the outer hyperplanes and 12 vertices of a cuboctahedron on a central hyperplane. These vertices, combined with the 8 vertices of the 16-cell, represent the 32 root vectors of the B4 and C4 simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the root system of type F4.[28] The 24 vertices of the original 24-cell form a root system of type D4; its size has the ratio 2:1. This is likewise true for the 24 vertices of its dual. The full symmetry group of the 24-cell is the Weyl group of F4, which is generated by reflections through the hyperplanes orthogonal to the F4 roots. This is a solvable group of order 1152. The rotational symmetry group of the 24-cell is of order 576.

Quaternionic interpretation

The 24 quaternion 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 quaternions, the F4 root lattice (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a ring. This is the ring of Hurwitz integral quaternions. 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 D4 root lattice is the dual of the F4 and is given by the subring of Hurwitz quaternions with even norm squared.

Viewed as the 24 unit Hurwitz quaternions, the unit radius coordinates of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.[29]

Vertices of other convex regular 4-polytopes also form multiplicative groups of quaternions, but few of them generate a root lattice.

Voronoi cells

The Voronoi cells of the D4 root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the tessellation of 4-dimensional Euclidean space by regular 24-cells, the 24-cell honeycomb. The 24-cells are centered at the D4 lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F4 lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The Schläfli symbol for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of R4.

The unit balls inscribed in the 24-cells of this tessellation give rise to the densest known lattice packing of hyperspheres in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the highest possible kissing number in 4 dimensions.

Radially equilateral honeycomb

The dual tessellation of the 24-cell honeycomb {3,4,3,3} is the 16-cell honeycomb {3,3,4,3}. The third regular tessellation of four dimensional space is the tesseractic honeycomb {4,3,3,4}, whose vertices can be described by 4-integer Cartesian coordinates. The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.[lower-alpha 2]

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.[30] The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.[31] Of the 24 center-to-vertex radii[lower-alpha 64] of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,[17] but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).[lower-alpha 65]

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.[lower-alpha 3]


The regular convex 4-polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations about a fixed point in 4-dimensional Euclidean space.[lower-alpha 68]

The 3 Cartesian bases of the 24-cell

There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell honeycomb, depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell) was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.[lower-alpha 33] The distance from one of these orientations to another is an isoclinic rotation through 60 degrees (a double rotation of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).[lower-alpha 69] This rotation can be seen most clearly in the hexagonal central planes, where the hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.[lower-alpha 13]

Planes of rotation

Rotations in 4-dimensional Euclidean space can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.[33] Thus the general rotation in 4-space is a double rotation. There are two important special cases, called a simple rotation and an isoclinic rotation.[lower-alpha 72]

Simple rotations
A 3D projection of a 24-cell performing a simple rotation

In 3 dimensions a spinning polyhedron has a single invariant central plane of rotation. The plane is called invariant because each point in the plane moves in a circle but stays within the plane. Only one of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed axis of rotation perpendicular to the invariant plane), but the circles do not lie within a central plane.

When a 4-polytope is rotating with only one invariant central plane, the same kind of simple rotation is happening that occurs in 3 dimensions. The only difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is completely orthogonal[lower-alpha 7] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex directly to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great digon, and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. [lower-alpha 43]

Double rotations
A 3D projection of a 24-cell performing a double rotation

The points in the completely orthogonal central plane are not constrained to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a double rotation in two perpendicular non-intersecting planes of rotation at once.[lower-alpha 71] In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane as the whole plane tilts sideways in the completely orthogonal rotation. A rotation in 4-space always has (at least) two completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two chiral forms: left and right rotations. In a double rotation each vertex moves in a spiral along two completely orthogonal great circles at once.[lower-alpha 70] Either the path is right-hand threaded (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the right hand rule by which we conventionally say which way is "up" on each of the 4 coordinate axes).

Isoclinic rotations

When the angles of rotation in the two invariant planes are exactly the same, a remarkably symmetric transformation occurs: all the great circle planes Clifford parallel[lower-alpha 24] to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates isoclinically in many directions at once.[35] Each vertex moves an equal distance in four orthogonal directions at the same time.[lower-alpha 55] In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a neighboring vertex, rotates all 16 hexagons by 60 degrees, and takes every great circle polygon (square,[lower-alpha 39] hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 60 degrees away. An isoclinic rotation is also called a Clifford displacement, after its discoverer.[lower-alpha 69]

The 24-cell in the double rotation animation appears to turn itself inside out.[lower-alpha 73] It appears to, because it actually does, reversing the chirality of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).[36]

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in two completely orthogonal planes one of which is a great hexagon,[lower-alpha 43] each vertex rotates first to a vertex two edge lengths away (3 and 120° distant).[lower-alpha 74] The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.[lower-alpha 75] Each 3 chord of the helical geodesic crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both. The 3 chords meet at a 60° angle, but since they lie in different planes they form a helix not a triangle. The helix of 3 chords closes into a loop only after six 3 chords: a 720° rotation twice around the 24-cell on a skew hexagon with 3 edges. Even though all the vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation hits only half the vertices in the 24-cell. After 360 degrees each helix has passed through 3 vertices, but has not arrived back at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's orientation in the 4-space in which it is embedded is now different. Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same direction through another 360 degrees, the moving vertices will pass through the other half of the vertices they missed on the first revolution (the 12 antipodal vertices of the 12 they hit the first time around), and each isoclinic geodesic will arrive back at the vertex it departed from, forming a closed hexagonal helix.[lower-alpha 76] It takes a 720 degree isoclinic rotation for each hexagram2 isoclinic geodesic to complete a circuit through its six vertices by winding around the 24-cell twice, returning the 24-cell to its original chiral orientation.[37]

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a Möbius ring, so that the two strands of the double helix form a continuous single strand in a closed loop. In the first revolution the vertex traverses one 3-vertex strand of the double helix; in the second revolution it traverses the second strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic ring is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.[lower-alpha 36]

Clifford parallel polytopes

Two planes are also called isoclinic if an isoclinic rotation will bring them together.[lower-alpha 40] The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.[39] Clifford parallel great circles do not intersect, so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others.[lower-alpha 31] We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).[lower-alpha 16] We can pick out 6 mutually isoclinic (Clifford parallel) great squares (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.[40] Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are completely disjoint polytopes.[lower-alpha 52] A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all double rotations, isoclinic rotations come in two chiral forms: there is a disjoint 16-cell to the left of each 16-cell, and another to its right.

All Clifford parallel 4-polytopes are related by an isoclinic rotation,[lower-alpha 69] but not all isoclinic polytopes are Clifford parallels (completely disjoint).[lower-alpha 79] The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).[lower-alpha 33]

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate[lower-alpha 80] a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by isoclinic rotations of the 16-cell, generating isoclinic copies of itself.) The convex regular 4-polytopes nest inside each other, and hide next to each other in the Clifford parallel spaces that comprise the 3-sphere. For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation.[lower-alpha 81]


In the 24-cell there are sets of rings of five different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are intertwined.

The 24-cell contains three kinds of geodesic fibers (polygonal rings running through vertices): great circle squares, great circle hexagons, and isoclinic helix hexagrams. It also contains two kinds of cell rings (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 42. The axis can then be bent into a square of edge length 2. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the 2 axes of the four octahedra occupy the same plane, forming one of the 18 2 great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,[lower-alpha 82] and all four dimensions are utilized. The 24-cell can be partitioned into 6 such rings (three different ways), mutually interlinked like adjacent links in a chain (but these links all have a common center). A simple rotation in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

Six unit-edge-length octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.[lower-alpha 84] However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.[lower-alpha 85] The ring has two sets of three great hexagons, each on three Clifford parallel great circles.[lower-alpha 86] The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A simple rotation in any of the great hexagon planes by a multiple of 60° rotates all three parallel great hexagon planes similarly, and takes each octahedron in the ring to an octahedron in the ring.

The third kind of geodesic fiber, the isoclinic helix hexagrams, can also be found within a ring of six octahedral cells. Each of these geodesics runs through the six vertices of a skew hexagon of six 3 chords: a hexagon that does not lie in a single central plane, but is composed of six linked chords of six different hexagonal great circles. This geodesic fiber is the path of an isoclinic rotation, a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.[lower-alpha 36] Rather than a flat hexagon, it forms a skew hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.[lower-alpha 76] An isoclinic rotation in any of the great hexagon planes by 60° rotates all six great hexagon planes by 60°, and takes each octahedron in the ring to a non-adjacent octahedron in the ring.[lower-alpha 87]

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of 3 chords from octahedron to octahedron. In the 24-cell the 1 edges are great hexagon edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The 3 chords are great hexagon diagonals, joining great hexagon vertices two 1 edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each 3 chord is a chord of just one great hexagon (an edge of a great triangle inscribed in that great hexagon), but successive 3 chords belong to different great hexagons. At each vertex the isoclinic path of 3 chords bends 60 degrees in two completely orthogonal directions at once: 60 degrees around the great hexagon the current chord is part of, and 60 degrees orthogonally (sideways) into the plane of a different great hexagon entirely (that the next chord is part of).[lower-alpha 88] Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between hexagonal central planes, but it is not: any isoclinic path we can pick out always bends either right or left, never changing its inherent chiral "direction", as it visits all six of the great hexagons in the 6-cell ring. When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagon and begins to repeat itself (still bending in the same direction).

Characteristic orthoscheme

Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell.[lower-alpha 54] The characteristic 5-cell of the regular 24-cell is represented by the Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, which can be read as a list of the dihedral angles between its mirror facets.[lower-alpha 89] It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular octahedron. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.

The characteristic 4-orthoscheme has four more edges than its base 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).[lower-alpha 90] If the regular 24-cell has radius and edge length 1, its characteristic 5-cell's ten edges have lengths [math]\displaystyle{ \sqrt{\tfrac{1}{12}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{4}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{3}} }[/math] (the exterior right triangle face, the characteristic triangle), plus [math]\displaystyle{ \sqrt{\tfrac{1}{2}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{4}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{6}} }[/math] (the other three edges of the exterior 3-orthoscheme facet, the characteristic tetrahedron of the octahedron), plus [math]\displaystyle{ \sqrt{\tfrac{1}{2}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{3}{4}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{2}{3}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{2}} }[/math] (edges that are the characteristic radii of the regular 24-cell).[43] The 4-edge path along orthogonal edges of the orthoscheme is [math]\displaystyle{ \sqrt{\tfrac{1}{4}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{12}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{6}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{2}} }[/math], first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.


Parallel projections

Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)

The vertex-first parallel projection of the 24-cell into 3-dimensional space has a rhombic dodecahedral envelope. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The cell-first parallel projection of the 24-cell into 3-dimensional space has a cuboctahedral envelope. Two of the octahedral cells, the nearest and farther from the viewer along the w-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The edge-first parallel projection has an elongated hexagonal dipyramidal envelope, and the face-first parallel projection has a nonuniform hexagonal bi-antiprismic envelope.

Perspective projections

The vertex-first perspective projection of the 24-cell into 3-dimensional space has a tetrakis hexahedral envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

Cell-first perspective projection
In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells.
24cell section anim.gif
Animated cross-section of 24-cell
3D stereoscopic projection icositetrachoron.PNG
A stereoscopic 3D projection of an icositetrachoron (24-cell).
Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell

Orthogonal projections


Octacube steel sculpture at Pennsylvania State University

The 24-cell is bounded by 24 octahedral cells. For visualization purposes, it is convenient that the octahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 120-cell). One can stack octahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 6 cells. The cell locations lend themselves to a hyperspherical description. Pick an arbitrary cell and label it the "North Pole". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "South Pole" cell. This skeleton accounts for 18 of the 24 cells (2 + 8×2). See the table below.

There is another related great circle in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the hexagonal geodesics described above.[lower-alpha 16] One can easily follow this path in a rendering of the equatorial cuboctahedron cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices. This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a tesseract (8-cell), although they touch at their vertices instead of their faces.

Layer # Number of Cells Description Colatitude Region
1 1 cell North Pole Northern Hemisphere
2 8 cells First layer of meridian cells 60°
3 6 cells Non-meridian / interstitial 90° Equator
4 8 cells Second layer of meridian cells 120° Southern Hemisphere
5 1 cell South Pole 180°
Total 24 cells
An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator

The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete Hopf fibration of four interlocking rings.[44] One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees. This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a great circle route, through the octahedrons' opposing vertices, that is four cells long. These are the square geodesics along four 2 chords described above. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two interlocking great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

Three Coxeter group constructions

There are two lower symmetry forms of the 24-cell, derived as a rectified 16-cell, with B4 or [3,3,4] symmetry drawn bicolored with 8 and 16 octahedral cells. Lastly it can be constructed from D4 or [31,1,1] symmetry, and drawn tricolored with 8 octahedra each.

Related complex polygons

The regular complex polygon 4{3}4, CDel 4node 1.pngCDel 3.pngCDel 4node.png or CDel node h.pngCDel 6.pngCDel 4node.png contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is 4[3]4, order 96.[45]

The regular complex polytope 3{4}3, CDel 3node 1.pngCDel 4.pngCDel 3node.png or CDel node h.pngCDel 8.pngCDel 3node.png, in [math]\displaystyle{ \mathbb{C}^2 }[/math] has a real representation as a 24-cell in 4-dimensional space. 3{4}3 has 24 vertices, and 24 3-edges. Its symmetry is 3[4]3, order 72.

Related figures in orthogonal projections
Name {3,4,3}, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 4{3}4, CDel 4node 1.pngCDel 3.pngCDel 4node.png 3{4}3, CDel 3node 1.pngCDel 4.pngCDel 3node.png
Symmetry [3,4,3], CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, order 1152 4[3]4, CDel 4node.pngCDel 3.pngCDel 4node.png, order 96 3[4]3, CDel 3node.pngCDel 4.pngCDel 3node.png, order 72
Vertices 24 24 24
Edges 96 2-edges 24 4-edge 24 3-edges
Image 24-cell t0 F4.svg
24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
Complex polygon 4-3-4.png
4{3}4, CDel 4node 1.pngCDel 3.pngCDel 4node.png has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
Complex polygon 3-4-3-fill1.png
3{4}3 or CDel 3node 1.pngCDel 4.pngCDel 3node.png has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.

Related 4-polytopes

Several uniform 4-polytopes can be derived from the 24-cell via truncation:

  • truncating at 1/3 of the edge length yields the truncated 24-cell;
  • truncating at 1/2 of the edge length yields the rectified 24-cell;
  • and truncating at half the depth to the dual 24-cell yields the bitruncated 24-cell, which is cell-transitive.

The 96 edges of the 24-cell can be partitioned into the golden ratio to produce the 96 vertices of the snub 24-cell. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an octahedron produces an icosahedron, or "snub octahedron."

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a polygon nor a simplex. Relaxing the condition of convexity admits two further figures: the great 120-cell and grand stellated 120-cell. With itself, it can form a polytope compound: the compound of two 24-cells.

Related uniform polytopes

The 24-cell can also be derived as a rectified 16-cell:

See also


  1. The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a polygon nor a simplex. The other two are also 4-polytopes, but not convex: the grand stellated 120-cell and the great 120-cell. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.
  2. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and tesseract, the three-dimensional cuboctahedron, and the two-dimensional hexagon. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
  3. 3.0 3.1 The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the pentagon {5}, the dodecahedron {5, 3}, the 600-cell {3,3,5} and the 120-cell {5,3,3}. In other words, the 24-cell possesses all of the triangular and square features that exist in four dimensions except the regular 5-cell, but none of the pentagonal features. (The 5-cell is also pentagonal in the sense that its Petrie polygon is the pentagon.)
  4. The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content[4] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.
  5. The edge length will always be different unless predecessor and successor are both radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes[lower-alpha 2] are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.
  6. The edges of six of the squares are aligned with the grid lines of this coordinate system. For example:
         (  0,–1,  1,  0)   (  0,  1,  1,  0)
         (  0,–1,–1,  0)   (  0,  1,–1,  0)
    is the square in the xy plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90o distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.
  7. 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Two flat planes A and B of a Euclidean space of four dimensions are called completely orthogonal if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.[lower-alpha 10]
  8. Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time. Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is completely orthogonal[lower-alpha 7] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, just as two edges of the tetrahedron are perpendicular and opposite.
  9. To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w=0, z=0) shares no axis with the wz central plane (where x=0, y=0). The xy plane exists at only a single instant in time (w=0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).
  10. 10.0 10.1 10.2 In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.
  11. 11.0 11.1 11.2 11.3 Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) they can intersect in a single point[lower-alpha 9] (and they must, if they are completely orthogonal).[lower-alpha 7]
  12. 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 the coordinate system, but their edges are the 2 diagonals of unit edge length squares of the coordinate lattice. For example:
                     (  0,  0,  1,  0)
         (  0,–1,  0,  0)   (  0,  1,  0,  0)
                     (  0,  0,–1,  0)
    is the square in the xy plane. Notice that the 8 integer coordinates comprise the vertices of the 6 orthogonal squares.
  13. 13.0 13.1 13.2 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. 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:
                     (  0,  0,  1,  0)
         (  1/2,–1/2,  1/2,–1/2)   (  1/2,  1/2,  1/2,  1/2)
         (–1/2,–1/2,–1/2,–1/2)   (–1/2,  1/2,–1/2,  1/2)
                     (  0,  0,–1,  0)
    is a hexagon on the y axis. Unlike the 2 squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.
  14. 14.0 14.1 Eight 1 edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure[lower-alpha 27] and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two 1-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a cubic pyramid. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.[lower-alpha 28]
  15. 15.0 15.1 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.[lower-alpha 14]
  16. 16.0 16.1 16.2 16.3 The 24-cell has 4 sets of 4 non-intersecting Clifford parallel[lower-alpha 24] great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices. Each set constitutes a discrete Hopf fibration of interlocking great circles. The 24-cell can also be divided into 4 disjoint subsets of 6 vertices that do not lie in a hexagonal central plane, each skew hexagon forming an isoclinic geodesic or isocline that is the path followed by those 6 vertices in one particular isoclinic rotation.
  17. 17.0 17.1 These triangles' edges of length 3 are the diagonals of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract) cells are not cells of the unit radius coordinate lattice.
  18. 18.0 18.1 These triangles lie in the same planes containing the hexagons;[lower-alpha 13] two triangles of edge length 3 are inscribed in each hexagon. For example, in unit radius coordinates:
                     (  0,  0,  1,  0)
         (  1/2,–1/2,  1/2,–1/2)   (  1/2,  1/2,  1/2,  1/2)
         (–1/2,–1/2,–1/2,–1/2)   (–1/2,  1/2,–1/2,  1/2)
                     (  0,  0,–1,  0)
    are two opposing central triangles on the y axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the 3 triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the 2 squares.
  19. They surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The vertex figure of the 24-cell is a cube.)
  20. They surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.
  21. 21.0 21.1 Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in its configuration matrix, which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.
  22. 22.0 22.1 The central vertex is a canonical apex because it is one edge length equidistant from the ordinary vertices in the 4th dimension, as the apex of a canonical pyramid is one edge length equidistant from its other vertices.
  23. Thus (1, 2, 3, 4) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.
  24. 24.0 24.1 24.2 24.3 24.4 24.5 24.6 24.7 Clifford parallels are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.[11] A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the 3-sphere.[12] Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.[lower-alpha 10] Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.[lower-alpha 31] Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a Hopf link.
  25. A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does not bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.[lower-alpha 24]
  26. If the Pythagorean distance between any two vertices is 1, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is 2, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90o bend in it as the path through the center). If their Pythagorean distance is 3, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60o bend, or as a straight line with one 60o bend in it through the center). Finally, if their Pythagorean distance is 4, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).
  27. 27.0 27.1 27.2 27.3 27.4 The vertex figure is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a full size vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".[10] That is what serves the illustrative purpose here.
  28. The vertex cubic pyramid is not actually radially equilateral,[lower-alpha 2] because the edges radiating from its apex are not actually its radii: the apex of the cubic pyramid is not actually its center, just one of its vertices.
  29. Six 2 chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure[lower-alpha 27] and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight 1 edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six 2 chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six 2-distant vertices that surrounds the first shell of eight 1-distant vertices. The face-center through which the 2 chord passes is the mid-point of the 2 chord, so it lies inside the 24-cell.
  30. One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the cuboctahedron (the central hyperplane of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).
  31. 31.0 31.1 31.2 Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal[lower-alpha 7] to only one of them.[lower-alpha 39] Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).
  32. Eight 3 chords converge from the corners of the 24-cell's cubical vertex figure[lower-alpha 27] and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight 3 chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube, which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight 3-distant vertices surrounding the second shell of six 2-distant vertices that surrounds the first shell of eight 1-distant vertices.
  33. 33.0 33.1 33.2 33.3 33.4 33.5 The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are 3 (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four 3 chords (its long diagonals). The 8-cells are not completely disjoint[lower-alpha 52] (they share vertices), but each cube and each 3 chord belongs to just one 8-cell. The 3 chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.
  34. The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.
  35. The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.[13]
  36. 36.0 36.1 36.2 36.3 A point under isoclinic rotation traverses the diagonal[lower-alpha 55] straight line of a single isoclinic geodesic, reaching its destination directly, instead of the bent line of two successive simple geodesics. A geodesic is the shortest path through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do not lie in a single plane; they are 4-dimensional spirals rather than simple 2-dimensional circles.[lower-alpha 70] But they are not like 3-dimensional screw threads either, because they form a closed loop like any circle (after two revolutions).[lower-alpha 76] Isoclinic geodesics are 4-dimensional great circles, and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.[lower-alpha 77] These isoclines are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere[lower-alpha 78] they always occur in chiral pairs and form a pair of Villarceau circles on the Clifford torus, the paths of the left and the right isoclinic rotation. They are helices bent into a Möbius loop in the fourth dimension, taking a diagonal winding route twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's skew polygon.
  37. Each pair of parallel 1 edges joins a pair of parallel 3 chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel 2 chords joins another pair of parallel 2 chords to form one of the 18 central squares.
  38. 38.0 38.1 38.2 38.3 One way to visualize the n-dimensional hyperplanes is as the n-spaces which can be defined by n + 1 points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These simplex figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the universe (the enclosing space) into two parts (above and below the hyperplane). The n points bound a finite simplex figure (from the outside), and they define an infinite hyperplane (from the inside).[27] These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.
  39. 39.0 39.1 39.2 In the 16-cell the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically[lower-alpha 55] with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.[lower-alpha 31]) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.
  40. 40.0 40.1 Two angles are required to fix the relative positions of two planes in 4-space.[14] Since all planes in the same hyperplane[lower-alpha 38] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in both angles. Great squares in different hyperplanes are 90 degrees apart in both angles (completely orthogonal)[lower-alpha 7] or 60 degrees apart in both angles.[lower-alpha 39] Planes which are separated by two equal angles are called isoclinic. Planes which are isoclinic have Clifford parallel great circles.[lower-alpha 24] A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle and a 60 degree angle.
  41. Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.
  42. Two intersecting great squares or great hexagons share two opposing vertices, but squares and hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.
  43. 43.0 43.1 43.2 43.3 In the 24-cell each great square plane is completely orthogonal[lower-alpha 7] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great digon plane.
  44. The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.[15] The regular 5-cell is not found in the interior of any convex regular 4-polytope except the 120-cell,[16] though every convex 4-polytope can be deconstructed into irregular 5-cells.
  45. R1-cube.gif
    This animation shows the construction of a rhombic dodecahedron from a cube, by inverting the center-face-pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process.[17]
  46. We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.
  47. Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.[lower-alpha 11] Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane[lower-alpha 38] (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.
  48. The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the 16 hexagonal great circles. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 2 square great circles, the 72 1 square (tesseract) faces, and 144 1 by 2 rectangles. The planes through exactly 3 vertices are the 96 2 equilateral triangle (16-cell) faces, and the 96 1 equilateral triangle (24-cell) faces. There are an infinite number of central planes through exactly two vertices (great circle digons); 16 are distinguished, as each is completely orthogonal[lower-alpha 7] to one of the 16 hexagonal great circles.
  49. The 24-cell's cubical vertex figure[lower-alpha 27] has been truncated to a tetrahedral vertex figure (see Kepler's drawing). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).
  50. 50.0 50.1 50.2 50.3 The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2. Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.
  51. The 24-cell's cubical vertex figure[lower-alpha 27] has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 2 chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.
  52. 52.0 52.1 52.2 52.3 Polytopes are completely disjoint if all their element sets are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.
  53. 53.0 53.1 53.2 53.3 Each of the 72 2 chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of just one tetrahedral cell (of one 16-cell).
  54. 54.0 54.1 An orthoscheme is a chiral irregular simplex with right triangle faces that is characteristic of some polytope because it will exactly fill that polytope with the reflections of itself in its own facets (its mirror walls). Every regular polytope can be dissected radially into instances of its 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.
  55. 55.0 55.1 55.2 55.3 55.4 In an isoclinic rotation, each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a 4-dimensional diagonal. The point is displaced a total Pythagorean distance equal to the square root of four times that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,[lower-alpha 43] all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex 3 (120°) away, moving 3/4 in four orthogonal coordinate directions.
  56. The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.
  57. The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.
  58. The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.
  59. The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length 2) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.
  60. Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right tetrahedral pyramids, with their apexes filling the corners of the tesseract.
  61. 61.0 61.1 Consider the three perpendicular 2 long diameters of the octahedral cell.[24] Two of them are the face diagonals of the square face between two cubes; each is a 2 chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).[lower-alpha 53] The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face (but a different pair of cubes, from one of the other tesseracts in the 24-cell).
  62. 62.0 62.1 62.2 Because there are three overlapping tesseracts inscribed in the 24-cell, each octahedral cell lies on a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and in two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).[lower-alpha 61]
  63. This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do not lie at the corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices do lie at the corner of a cube: but a cube in another (overlapping) tesseract.[lower-alpha 62]
  64. It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).
  65. Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius 2/2. The three 16-cells inscribed in each 24-cell have edge length 2, and unit radius.
  66. Three dimensional rotations occur around an axis line. Four dimensional rotations may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when folding a flat net of 8 cubes up into a tesseract). Folding around a square face is just folding around two of its orthogonal edges at the same time; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).
  67. There are (at least) two kinds of correct dimensional analogies: the usual kind between dimension n and dimension n + 1, and the much rarer and less obvious kind between dimension n and dimension n + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the n-sphere rule that the surface area of the sphere embedded in n+2 dimensions is exactly 2π r times the volume enclosed by the sphere embedded in n dimensions, the most well-known examples being that the circumference of a circle is 2π r times 1, and the surface area of the ordinary sphere is 2π r times 2r. Coxeter cites[32] this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.
  68. Rotations in 4-dimensional Euclidean space may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).[lower-alpha 66] But in four dimensions there is yet another way in which rotations can occur, called a double rotation. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of simple rotations, the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves (as in a 2-dimensional rotation!).[lower-alpha 67]
  69. 69.0 69.1 69.2 69.3 In an isoclinic rotation, also known as a Clifford displacement, all the Clifford parallel[lower-alpha 24] invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted sideways by that same angle. A Clifford displacement is 4-dimensionally diagonal.[lower-alpha 55] Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane moves sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.
  70. 70.0 70.1 70.2 In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be invariant because the points in each stay in the plane as the plane moves, tilting sideways by the same angle that the other plane rotates.
  71. 71.0 71.1 71.2 Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations a and b: the left double rotation as a then b, and the right double rotation as b then a. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: it reaches its destination directly without passing through the intermediate point touched by a then b, or the other intermediate point touched by b then a, by rotating on a single helical geodesic (so it is the shortest path).[lower-alpha 70] Conversely, any simple rotation can be seen as the composition of two equal-angled double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by Cayley; perhaps surprisingly, this composition is commutative, and is possible for any double rotation as well.[34]
  72. A rotation in 4-space is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing). Thus the general rotation in 4-space is a double rotation, characterized by two angles. A simple rotation is a special case in which one rotational angle is 0.[lower-alpha 71] An isoclinic rotation is a different special case, similar but not identical to two simple rotations through the same angle.[lower-alpha 69]
  73. That a double rotation can turn a 4-polytope inside out is even more noticeable in the tesseract double rotation.
  74. The first vertex reached is 120 degrees away along a 3 chord (lying in a different hexagonal plane than the original great hexagon plane of rotation), in a different 8-cell than the original vertex.[lower-alpha 33] The second vertex reached will be another 120 degrees away along another 3 chord (lying in a different hexagonal plane than the first 3 chord), in the third 8-cell.
  75. Although adjacent vertices on the geodesic are a 3 chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a simple rotation between two vertices 3 apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex halfway. But in an isoclinic rotation between two vertices 3 apart (perhaps the same two vertices as in the simple rotation) the vertex moves along a helical arc (not a great circle), which does not pass through an intervening vertex: it misses the vertices nearest to its midpoint.
  76. 76.0 76.1 76.2 Because the 24-cell's helical hexagram2 geodesic is bent into a twisted ring in the fourth dimension like a Möbius strip, its screw thread doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right). The 6-vertex isoclinic path forms a Möbius double helix (a 3-dimensional double helix with the ends of its two 3-vertex helices joined), a regular compound polygon denoted {6/2}=2{3} or hexagram2. It passes through non-adjacent (alternate) vertices of a regular {12/4}=4{3} skew dodecagon4, as the 24-cell's Petrie dodecagon passes through adjacent vertices, but unlike the Petrie polygon it does not zig-zag: it always bends either right or left, along a chiral helical geodesic "straight line" or isocline.[lower-alpha 36] The Petrie dodecagon has 1 edges which zig-zag back and forth between the same two Clifford parallel great hexagon planes; the isoclinic hexagram2 has 3 edges which either zig or zag (along a left or right handed geodesic spiral) visiting six Clifford parallel great hexagon planes in rotation, and connecting skew dodecagram4 vertices which are 2 vertices apart. Each 3 edge belongs to a different great hexagon and successive 3 edges belong to different 8-cells, as the 720° isoclinic rotation takes each hexagon through all the Clifford parallel hexagons and passes through successive isoclinic 8-cells in rotation.
  77. Isoclinic geodesics are 4-dimensional great circles in the sense that they are 1-dimensional geodesic lines that curve in 4-space in two completely orthogonal planes at once. They should not be confused with great 2-spheres,[38] which are the 4-dimensional analogues of 3-dimensional great circles.
  78. All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.
  79. All isoclinic planes are Clifford parallels (completely disjoint).[lower-alpha 52] Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and not disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).
  80. By generate we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.
  81. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders in order to move the short distance between Clifford parallel subspaces.
  82. Just as each face of a polyhedron occupies a different (2-dimensional) face plane, each cell of a polychoron occupies a different (3-dimensional) cell hyperplane.[lower-alpha 38]
  83. When unit-edge octahedra are placed face-to-face the distance between their centers of volume is 2/3 ≈ 0.816.[41] When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still 2/3 apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.
  84. The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.[lower-alpha 83] Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.[lower-alpha 50]
  85. There is a choice of 16 planes in which to fold the column into a ring, but they are all equivalent in that they produce congruent rings. Whichever folding plane is chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are not helices: they lie on ordinary flat great circles. Only one kind of 6-cell ring exists, not two different chiral kinds (left-handed and right-handed). The 6-cell ring has no torsion, either clockwise or counterclockwise.
  86. The three great hexagons are Clifford parallel, which is different than ordinary parallelism.[lower-alpha 24] Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a Hopf link. Unlike links in a 3-dimensional chain, they share the same center point.
  87. An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra. It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by an isoclinic rotation.[lower-alpha 71] Isoclinic rotation partitions the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), which shift places among themselves.
  88. Notice that at each vertex there is only one adjacent great hexagon plane that the path can bend 60 degrees into: the isoclinic path is deterministic in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges of a different color, we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, because two great hexagons intersect at antipodal vertices. Each isoclinic skew hexagon contains one 3 edge of each color, and visits 6 of the 9 different color-pairs of vertex. Each 6-cell ring contains six such isoclinic skew hexagons, three left-handed and three right-handed. Each vertex is visited by one left-handed and one right-handed skew hexagon.
  89. For a regular k-polytope, the Coxeter-Dynkin diagram of the characteristic k-orthoscheme is the k-polytope's diagram without the generating point ring. The regular k-polytope is subdivided by its symmetry (k-1)-elements into g instances of its characteristic k-orthoscheme that surround its center, where g is the order of the k-polytope's symmetry group.[42]
  90. The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.


  1. Coxeter 1973, p. 118, Chapter VII: Ordinary Polytopes in Higher Space.
  2. Johnson 2018, p. 249, 11.5.
  3. Ghyka 1977, p. 68.
  4. 4.0 4.1 Coxeter 1973, pp. 292-293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
  5. Coxeter 1973, p. 302, Table VI (ii): 𝐈𝐈 = {3,4,3}: see Result column
  6. Coxeter 1973, p. 156, §8.7. Cartesian Coordinates.
  7. Coxeter 1973, pp. 145-146, §8.1 The simple truncations of the general regular polytope.
  8. Waegell & Aravind 2009, pp. 4-5, §3.4 The 24-cell: points, lines and Reye’s configuration; In the 24-cell Reye's "points" and "lines" are axes and hexagons, respectively.
  9. Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column a.
  10. Stillwell 2001, p. 17.
  11. Tyrrell & Semple 1971, pp. 5-6, §3. Clifford's original definition of parallelism.
  12. Kim & Rote 2016, pp. 8-10, Relations to Clifford Parallelism.
  13. Copher 2019, p. 6, §3.2 Theorem 3.4.
  14. Kim & Rote 2016, p. 7, §6 Angles between two Planes in 4-Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, k angles are defined between k-dimensional subspaces.)"
  15. Coxeter 1973, p. 153, 8.5. Gosset's construction for {3,3,5}: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."
  16. Coxeter 1973, p. 304, Table VI(iv) II={5,3,3}: Faceting {5,3,3}[120𝛼4]{3,3,5} of the 120-cell reveals 120 regular 5-cells.
  17. 17.0 17.1 17.2 Coxeter 1973, p. 150, Gosset.
  18. Coxeter 1973, p. 148, §8.2. Cesaro's construction for {3, 4, 3}..
  19. Coxeter 1973, p. 302, Table VI(ii) II={3,4,3}, Result column.
  20. Coxeter 1973, pp. 149-150, §8.22. see illustrations Fig. 8.2A and Fig 8.2B
  21. 21.0 21.1 Kepler 1619, p. 181.
  22. van Ittersum 2020, pp. 73-79, §4.2.
  23. Coxeter 1973, p. 269, §14.32. "For instance, in the case of [math]\displaystyle{ \gamma_4[2\beta_4] }[/math]...."
  24. van Ittersum 2020, p. 79.
  25. Coxeter 1973, p. 150: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the [math]\displaystyle{ \gamma_4 }[/math]. (Their centres are the mid-points of the 24 edges of the [math]\displaystyle{ \beta_4 }[/math].)"
  26. Coxeter 1973, p. 12, §1.8. Configurations.
  27. Coxeter 1973, p. 120, §7.2.: "... any n+1 points which do not lie in an (n-1)-space are the vertices of an n-dimensional simplex.... Thus the general simplex may alternatively be defined as a finite region of n-space enclosed by n+1 hyperplanes or (n-1)-spaces."
  28. van Ittersum 2020, p. 78, §4.2.5.
  29. Stillwell 2001, p. 22.
  30. Coxeter 1973, p. 163: Coxeter notes that Thorold Gosset was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.
  31. Coxeter 1973, p. 156: "...the chess-board has an n-dimensional analogue."
  32. Coxeter 1973, p. 119, §7.1. Dimensional Analogy: "For instance, seeing that the circumference of a circle is 2π r, while the surface of a sphere is 4π r 2, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2π 2r 3."
  33. Kim & Rote 2016, p. 6, §5. Four-Dimensional Rotations.
  34. Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications". Adv. Appl. Clifford Algebras 27: 523–538. doi:10.1007/s00006-016-0683-9. https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf. 
  35. Kim & Rote 2016, pp. 7-10, §6. Angles between two Planes in 4-Space
  36. Coxeter 1973, p. 141, §7.x. Historical remarks; "Möbius realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by H. G. Wells in The Plattner Story."
  37. Coxeter 1995, (Paper 3) Two aspects of the regular 24-cell in four dimensions.
  38. Stillwell 2001, p. 24.
  39. Kim & Rote 2016, pp. 8-9, Relations to Clifford parallelism.
  40. Tyrrell & Semple 1971, pp. 1-9, §1. Introduction.
  41. Coxeter 1973, pp. 292-293, Table I(i): Octahedron.
  42. Coxeter 1973, pp. 130-133, §7.6 The symmetry group of the general regular polytope.
  43. Coxeter 1973, pp. 292-293, Table I(ii); "24-cell".
  44. Banchoff 2013, pp. 265-266.
  45. Coxeter 1991.


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Fundamental convex regular and uniform polytopes in dimensions 2–10
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