24cell  

Type  Convex regular 4polytope 
Schläfli symbol  {3,4,3} r{3,3,4} = {3^{1,1,1}} = 
Coxeter diagram  or or 
Cells  24 {3,4} 
Faces  96 {3} 
Edges  96 
Vertices  24 
Vertex figure  Cube 
Petrie polygon  dodecagon 
Coxeter group  F_{4}, [3,4,3], order 1152 B_{4}, [4,3,3], order 384 D_{4}, [3^{1,1,1}], order 192 
Dual  Selfdual 
Properties  convex, isogonal, isotoxal, isohedral 
Uniform index  22 
In fourdimensional geometry, the 24cell is the convex regular 4polytope^{[1]} (fourdimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C_{24}, or the icositetrachoron,^{[2]} octaplex (short for "octahedral complex"), icosatetrahedroid,^{[3]} octacube, hyperdiamond or polyoctahedron, being constructed of octahedral cells.
The boundary of the 24cell 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 24cell is selfdual.^{[a]} The 24cell and the tesseract are the only convex regular 4polytopes in which the edge length equals the radius.^{[b]}
The 24cell does not have a regular analogue in 3 dimensions. It is the only one of the six convex regular 4polytopes which is not the fourdimensional analogue of one of the five regular Platonic solids. It is the unique regular polytope, in any number of dimensions, which has no regular analogue in the adjacent dimension, either below or above.^{[4]} However, it can be seen as the analogue of a pair of irregular solids: the cuboctahedron and its dual the rhombic dodecahedron.^{[5]}
Translated copies of the 24cell can tile fourdimensional space facetoface, forming the 24cell honeycomb. As a polytope that can tile by translation, the 24cell is an example of a parallelotope, the simplest one that is not also a zonotope.^{[6]}
The 24cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5cell, those with a 5 in their Schlӓfli symbol,^{[c]} and the regular polygons with 7 or more sides. It is especially useful to explore the 24cell, because one can see the geometric relationships among all of these regular polytopes in a single 24cell or its honeycomb.
The 24cell is the fourth in the sequence of 6 convex regular 4polytopes (in order of size and complexity).^{[d]} It can be deconstructed into 3 overlapping instances of its predecessor the tesseract (8cell), as the 8cell can be deconstructed into 2 overlapping instances of its predecessor the 16cell.^{[8]} 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.^{[e]}
The 24cell is the convex hull of its vertices which can be described as the 24 coordinate permutations of:
Those coordinates^{[9]} can be constructed as , rectifying the 16cell with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16cell is the octahedron; thus, cutting the vertices of the 16cell at the midpoint of its incident edges produces 8 octahedral cells. This process^{[10]} also rectifies the tetrahedral cells of the 16cell which become 16 octahedra, giving the 24cell 24 octahedral cells.
In this frame of reference the 24cell has edges of length √2 and is inscribed in a 3sphere of radius √2. Remarkably, the edge length equals the circumradius, as in the hexagon, or the cuboctahedron. Such polytopes are radially equilateral.^{[b]}
Regular convex 4polytopes of radius √2  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors  
Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 squares x 3  4 rectangles x 4  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon x 2  1 octagon x 3  2 octagons x 4  2 dodecagons x 4  4 30gons x 6  20 30gons x 4  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
The 24 vertices form 18 great squares^{[f]} (3 sets of 6 orthogonal^{[g]} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24cell can be seen as the vertices of 3 pairs of completely orthogonal great squares which intersect^{[i]} at no vertices.^{[j]}
The 24cell is selfdual, having the same number of vertices (24) as cells and the same number of edges (96) as faces.
If the dual of the above 24cell of edge length √2 is taken by reciprocating it about its inscribed sphere, another 24cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24cell lies vertexup, and its vertices can be given as follows:
8 vertices obtained by permuting the integer coordinates:
and 16 vertices with halfinteger coordinates of the form:
all 24 of which lie at distance 1 from the origin.
Viewed as quaternions,^{[k]} these are the unit Hurwitz quaternions.
The 24cell has unit radius and unit edge length^{[b]} 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.^{[l]}
Regular convex 4polytopes of radius 1  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors  
Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 squares x 3  4 rectangles x 4  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon x 2  1 octagon x 3  2 octagons x 4  2 dodecagons x 4  4 30gons x 6  20 30gons x 4  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
The 24 vertices and 96 edges form 16 nonorthogonal great hexagons,^{[o]} four of which intersect^{[i]} at each vertex.^{[q]} By viewing just one hexagon at each vertex, the 24cell can be seen as the 24 vertices of 4 nonintersecting hexagonal great circles which are Clifford parallel to each other.^{[r]}
The 12 axes and 16 hexagons of the 24cell constitute a Reye configuration, which in the language of configurations is written as 12_{4}16_{3} to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.^{[11]}
The 24 vertices form 32 equilateral great triangles, of edge length √3 in the unitradius 24cell,^{[u]} inscribed in the 16 great hexagons.^{[v]} Each great triangle is a ring linking three completely disjoint^{[w]} great squares.^{[aa]}
The 24 vertices of the 24cell are distributed^{[12]} at four different chord lengths from each other: √1, √2, √3 and √4.
Each vertex is joined to 8 others^{[ab]} by an edge of length 1, spanning 60° = π/3 of arc. Next nearest are 6 vertices^{[ac]} 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.^{[ad]} The opposite vertex is 180° = π away along a diameter of length 2. Finally, as the 24cell is radially equilateral, its center is 1 edge length away from all vertices.
To visualize how the interior polytopes of the 24cell 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.^{[ae]} Moreover, the long diameter of the octahedron is √2 like the square; and the long diameter of the 24cell itself is √4 like the tesseract. In the 24cell, the √2 chords are the edges of central squares, and the √4 chords are the diagonals of central squares.
The vertex chords of the 24cell are arranged in geodesic great circle polygons.^{[ag]} The geodesic distance between two 24cell vertices along a path of √1 edges is always 1, 2, or 3, and it is 3 only for opposite vertices.^{[ah]}
The √1 edges occur in 16 hexagonal great circles (in planes inclined at 60 degrees to each other), 4 of which cross^{[q]} at each vertex.^{[p]} The 96 distinct √1 edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24cell. The 16 hexagonal great circles can be divided into 4 sets of 4 nonintersecting 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.^{[ak]}
Coxeter plane  F_{4}  

Graph  
Dihedral symmetry  [12]  
Coxeter plane  B_{3} / A_{2} (a)  B_{3} / A_{2} (b) 
Graph  
Dihedral symmetry  [6]  [6] 
Coxeter plane  B_{4}  B_{2} / A_{3} 
Graph  
Dihedral symmetry  [8]  [4] 
The √2 chords occur in 18 square great circles (3 sets of 6 orthogonal planes^{[x]}), 3 of which cross at each vertex.^{[al]} The 72 distinct √2 chords do not run in the same planes as the hexagonal great circles; they do not follow the 24cell's edges, they pass through its octagonal cell centers.^{[am]} 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 nonintersecting Clifford parallel geodesics,^{[af]} 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.^{[aq]}
The √3 chords occur in 32 triangular great circles in 16 planes, 4 of which cross at each vertex.^{[ad]} The 96 distinct √3 chords^{[u]} run vertextoeveryothervertex in the same planes as the hexagonal great circles.^{[v]} 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.^{[t]}
The √4 chords occur as 12 vertextovertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.
The sum of the squared lengths^{[ar]} of all these distinct chords of the 24cell is 576 = 24^{2}.^{[as]} These are all the central polygons through vertices, but in 4space there are geodesics on the 3sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24cell vertices that are helical rather than simply circular; they correspond to diagonal isoclinic rotations rather than simple rotations.^{[at]}
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.^{[au]}
The central planes of the 24cell can be divided into 4 orthogonal central hyperplanes (3spaces) 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.^{[aw]} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of nonintersecting Clifford parallel polygons (of 6 squares or 4 hexagons).^{[ax]} Each set of Clifford parallel great circles is a parallel fiber bundle which visits all 24 vertices just once.
Each great circle intersects^{[i]} with the other great circles to which it is not Clifford parallel at one √4 diameter of the 24cell.^{[ay]} Great circles which are completely orthogonal or otherwise Clifford parallel^{[af]} do not intersect at all: they pass through disjoint sets of vertices.^{[az]}
Triangles and squares come together uniquely in the 24cell to generate, as interior features,^{[ba]} all of the trianglefaced and squarefaced regular convex polytopes in the first four dimensions (with caveats for the 5cell and the 600cell).^{[bb]} Consequently, there are numerous ways to construct or deconstruct the 24cell.
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular 16cell, and the 16 halfinteger vertices (±1/2, ±1/2, ±1/2, ±1/2) are the vertices of its dual, the tesseract (8cell).^{[21]} The tesseract gives Gosset's construction^{[22]} of the 24cell, equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. The analogous construction in 3space gives the rhombic dodecahedron which, however, is not regular.^{[bc]} The 16cell gives the reciprocal construction of the 24cell, Cesaro's construction,^{[23]} equivalent to rectifying a 16cell (truncating its corners at the midedges, as described above). The analogous construction in 3space gives the cuboctahedron (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16cell are the only regular 4polytopes in the 24cell.^{[24]}
We can further divide the 16 halfinteger 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 16cell. This shows that the vertices of the 24cell can be grouped into three disjoint sets of eight with each set defining a regular 16cell, and with the complement defining the dual tesseract.^{[25]} This also shows that the symmetries of the 16cell form a subgroup of index 3 of the symmetry group of the 24cell.^{[z]}
We can facet the 24cell by cutting^{[bd]} through interior cells bounded by vertex chords to remove vertices, exposing the facets of interior 4polytopes inscribed in the 24cell. One can cut a 24cell 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^{[be]} of interior polytopes.^{[bf]}
Starting with a complete 24cell, 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,^{[bg]} 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 24cell.^{[t]} 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.^{[bh]} They do share 4content, their common core.^{[bi]}
Starting with a complete 24cell, 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,^{[bj]} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a 16cell. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16cells inscribed in the 24cell.^{[y]} They overlap with each other, but all of their element sets are disjoint:^{[w]} they do not share any vertex count, edge length,^{[bk]} or face area, but they do share cell volume. They also share 4content, their common core.^{[bi]}
The 24cell 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.^{[b]} They form 96 √1 tetrahedra (each contributing one 24cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half16cells) with their apexes at the center.
The 24cell 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 3sphere. They form 48 √2edge tetrahedra (the cells of the three 16cells), centered at the 24 midedgeradii of the 24cell.^{[bk]}
The 24cell can be constructed directly from its characteristic simplex , the irregular 5cell which is the fundamental region of its symmetry group F_{4}, by reflection of that 4orthoscheme in its own cells (which are 3orthoschemes).^{[bl]}
The 24cell is not only the 24octahedralcell, it is also the 24cubicalcell, although the cubes are cells of the three 8cells, not cells of the 24cell, in which they are not volumetrically disjoint.
The 24cell can be constructed from 24 cubes of its own edge length (three 8cells).^{[t]} Each of the cubes is shared by 2 8cells, each of the cubes' square faces is shared by 4 cubes (in 2 8cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8cells).
The 24cell, three tesseracts, and three 16cells are deeply entwined around their common center, and intersect in a common core.^{[bi]} The tesseracts and the 16cells are rotated 60° isoclinically^{[m]} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16cells are √3 (120°) apart.^{[t]}
The tesseracts are inscribed in the 24cell^{[bm]} such that their vertices and edges are exterior elements of the 24cell, but their square faces and cubical cells lie inside the 24cell (they are not elements of the 24cell). The 16cells are inscribed in the 24cell^{[bn]} such that only their vertices are exterior elements of the 24cell: their edges, triangular faces, and tetrahedral cells lie inside the 24cell. The interior^{[bo]} 16cell edges have length √2.^{[aa]}
The 16cells 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 16cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16cell is inscribed in two of the three 8cells.^{[29]}^{[z]} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.^{[28]} 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.^{[30]}^{[bk]}
The 24cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16cells, leaving 4dimensional space in some places between its envelope and each 16cell's envelope of tetrahedra. Thus there are measurable^{[7]} 4dimensional interstices^{[bp]} between the 24cell, 8cell and 16cell envelopes. The shapes filling these gaps are 4pyramids, alluded to above.^{[bq]}
Despite the 4dimensional interstices between 24cell, 8cell and 16cell envelopes, their 3dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3membrane in those places, not two separate but adjacent 3dimensional layers.^{[bs]} 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 3space of the (outer) boundary envelope of the 24cell 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 16cell edges (one from each 16cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.^{[br]}
As we saw above, 16cell √2 tetrahedral cells are inscribed in tesseract √1 cubic cells, sharing the same volume. 24cell √1 octahedral cells overlap their volume with √1 cubic cells: they are bisected by a square face into two square pyramids,^{[32]} the apexes of which also lie at a vertex of a cube.^{[bt]} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24cell, tesseracts, and 16cells all share some boundary volume.^{[bs]}
This configuration matrix^{[33]} represents the 24cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Since the 24cell is selfdual, its matrix is identical to its 180 degree rotation.
The 24 root vectors of the D_{4} root system of the simple Lie group SO(8) form the vertices of a 24cell. The vertices can be seen in 3 hyperplanes,^{[av]} 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 16cell, represent the 32 root vectors of the B_{4} and C_{4} simple Lie groups.
The 48 vertices (or strictly speaking their radius vectors) of the union of the 24cell and its dual form the root system of type F_{4}.^{[35]} The 24 vertices of the original 24cell form a root system of type D_{4}; its size has the ratio √2:1. This is likewise true for the 24 vertices of its dual. The full symmetry group of the 24cell is the Weyl group of F_{4}, which is generated by reflections through the hyperplanes orthogonal to the F_{4} roots. This is a solvable group of order 1152. The rotational symmetry group of the 24cell is of order 576.
When interpreted as the quaternions,^{[k]} the F_{4} root lattice (which is the integral span of the vertices of the 24cell) is closed under multiplication and is therefore a ring. This is the ring of Hurwitz integral quaternions. The vertices of the 24cell 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 24cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24cell are those with norm squared 2. The D_{4} root lattice is the dual of the F_{4} and is given by the subring of Hurwitz quaternions with even norm squared.^{[37]}
Viewed as the 24 unit Hurwitz quaternions, the unit radius coordinates of the 24cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.^{[38]}
Vertices of other convex regular 4polytopes also form multiplicative groups of quaternions, but few of them generate a root lattice.^{[39]}
The Voronoi cells of the D_{4} root lattice are regular 24cells. The corresponding Voronoi tessellation gives the tessellation of 4dimensional Euclidean space by regular 24cells, the 24cell honeycomb. The 24cells are centered at the D_{4} lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F_{4} lattice points with odd norm squared. Each 24cell 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 24cells 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 R^{4}.
The unit balls inscribed in the 24cells of this tessellation give rise to the densest known lattice packing of hyperspheres in 4 dimensions. The vertex configuration of the 24cell has also been shown to give the highest possible kissing number in 4 dimensions.
The dual tessellation of the 24cell honeycomb {3,4,3,3} is the 16cell 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 4integer Cartesian coordinates.^{[k]} The congruent relationships among these three tessellations can be helpful in visualizing the 24cell, in particular the radial equilateral symmetry which it shares with the tesseract.^{[b]}
A honeycomb of unit edge length 24cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4integer coordinate) is also the vertex of a 24cell (and tesseract edges are also 24cell edges), and every center of a 24cell is also the center of a tesseract.^{[40]} The 24cells are twice as large as the tesseracts by 4dimensional content (hypervolume), so overall there are two tesseracts for every 24cell, only half of which are inscribed in a 24cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4dimensional checkerboard results.^{[41]} Of the 24 centertovertex radii^{[bu]} of each 24cell, 16 are also the radii of a black tesseract inscribed in the 24cell. 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 24cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24cell 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 24cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,^{[22]} but instead of being removed the pyramids are simply colored red and left in place). Eight 24cells 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 24cells and tesseracts, plus the centers of the red tesseracts. Adding the 24cell centers (which are also the black tesseract centers) to this honeycomb yields a 16cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24cells and tesseracts. The formerly empty centers of adjacent 24cells become the opposite vertices of a unit edge length 16cell. 24 half16cells (octahedral pyramids) meet at each formerly empty center to fill each 24cell, and their octahedral bases are the 6vertex octahedral facets of the 24cell (shared with an adjacent 24cell).^{[bv]}
Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24cell, 4dimensional 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 5cell), but that complex does not require (or permit) any of the pentagonal polytopes.^{[c]}
The regular convex 4polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations^{[42]} about a fixed point in 4dimensional Euclidean space.^{[by]}
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24cell honeycomb, depending on which of the 24cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16cell)^{[n]} was chosen to align it, just as three tesseracts can be inscribed in the 24cell, rotated with respect to each other.^{[t]} 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).^{[bz]} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.^{[o]}
Rotations in 4dimensional Euclidean space can be seen as the composition of two 2dimensional rotations in completely orthogonal planes.^{[44]} Thus the general rotation in 4space is a double rotation.^{[45]} There are two important special cases, called a simple rotation and an isoclinic rotation.^{[ce]}
In 3 dimensions a spinning polyhedron has a single invariant central plane of rotation. The plane is an invariant set 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 and direction 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 4polytope is rotating with only one invariant central plane, the same kind of simple rotation is happening that occurs in 3 dimensions. One 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 to the invariant plane of rotation. In the 24cell, 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. ^{[az]}
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 nonintersecting planes^{[h]} of rotation at once.^{[cc]} 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 4space 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.^{[cf]} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.^{[ca]} Either the path is righthand threaded (like most screws and bolts), moving along the circles in the "same" directions, or it is lefthand threaded (like a reversethreaded 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).^{[47]}
In double rotations of the 24cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24cell there are several noteworthy kinds of double rotation permitted by these parameters.^{[48]}
When the angles of rotation in the two invariant planes are exactly the same, a remarkably symmetric transformation occurs:^{[49]} all the great circle planes Clifford parallel^{[af]} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4polytope rotates isoclinically in many directions at once.^{[50]} Each vertex moves an equal distance in four orthogonal directions at the same time.^{[m]} In the 24cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates all 16 hexagons by 60 degrees, and takes every great circle polygon (square,^{[an]} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a Clifford displacement, after its discoverer.^{[bz]}
The 24cell in the double rotation animation appears to turn itself inside out.^{[ci]} It appears to, because it actually does, reversing the chirality of the whole 4polytope 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 24cell surface had been stripped off like a glove and turned inside out, making a righthand glove into a lefthand glove (or vice versa).^{[51]}
In a simple rotation of the 24cell 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,^{[az]} each vertex rotates first to a vertex two edge lengths away (√3 and 120° distant). The double 60degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.^{[s]} Each √3 chord of the helical geodesic^{[cp]} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.^{[cu]} The √3 chords meet at a 60° angle, but since they lie in different planes they form a helix not a triangle. Three √3 chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of √3 chords closes into a loop only after six √3 chords: a 720° rotation twice around the 24cell^{[cb]} on a skew hexagram with √3 edges.^{[ct]} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has not arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24cell) 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 24cell's orientation in the 4space in which it is embedded is now different.^{[53]} Because the 24cell is now insideout, if the isoclinic rotation is continued in the same direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic will arrive back at the vertex it departed from, forming a closed sixchord helical loop. It takes a 720 degree isoclinic rotation for each hexagram_{2} geodesic to complete a circuit through every second vertex of its six vertices by winding around the 24cell twice, returning the 24cell to its original chiral orientation.^{[dc]}
The hexagonal winding path that each vertex takes as it loops twice around the 24cell 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.^{[cw]} In the first revolution the vertex traverses one 3chord strand of the double helix; in the second revolution it traverses the second 3chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius ring is a closed spiral not a 2dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.^{[at]}
Two planes are also called isoclinic if an isoclinic rotation will bring them together.^{[aw]} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.^{[55]} Clifford parallel great circles do not intersect,^{[af]} so isoclinic great circle polygons have disjoint vertices. In the 24cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24cell just once (a hexagonal fibration).^{[ak]} We can pick out 6 mutually isoclinic (Clifford parallel) great squares^{[ck]} (three different ways) covering all 24 vertices of the 24cell just once (a square fibration).^{[aq]} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.^{[dg]}
Two dimensional great circle polygons are not the only polytopes in the 24cell which are parallel in the Clifford sense.^{[57]} 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 16cells inscribed in the 24cell are Clifford parallels. Clifford parallel polytopes are completely disjoint polytopes.^{[w]} A 60 degree isoclinic rotation in hexagonal planes takes each 16cell to a disjoint 16cell. Like all double rotations, isoclinic rotations come in two chiral forms: there is a disjoint 16cell to the left of each 16cell, and another to its right.^{[y]}
All Clifford parallel 4polytopes are related by an isoclinic rotation,^{[bz]} but not all isoclinic polytopes are Clifford parallels (completely disjoint).^{[dh]} The three 8cells in the 24cell are isoclinic but not Clifford parallel. Like the 16cells, 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 8cells (as each 16cell occurs in two of the three 8cells).^{[t]}
Isoclinic rotations relate the convex regular 4polytopes to each other. An isoclinic rotation of a single 16cell will generate^{[di]} a 24cell. A simple rotation of a single 16cell will not, because its vertices will not reach either of the other two 16cells' vertices in the course of the rotation. An isoclinic rotation of the 24cell will generate the 600cell, and an isoclinic rotation of the 600cell will generate the 120cell. (Or they can all be generated directly by an isoclinic rotation of the 16cell, generating isoclinic copies of itself.) The convex regular 4polytopes nest inside each other, and hide next to each other in the Clifford parallel spaces that comprise the 3sphere.^{[58]} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation.^{[dj]}
In the 24cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are intertwined.
The 24cell contains four kinds of geodesic fibers (polygonal rings running through vertices): great circle squares and their isoclinic helix octagrams,^{[aq]} and great circle hexagons and their isoclinic helix hexagrams.^{[ak]} It also contains two kinds of cell rings (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertextovertex and bent into a square, and six octahedra connected facetoface and bent into a hexagon.
Four unitedgelength octahedra can be connected vertextovertex along a common axis of length 4√2. 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 24cell. Although the √2 axes of the four octahedra occupy the same plane, forming one of the 18 √2 great squares of the 24cell, each octahedron occupies a different 3dimensional hyperplane,^{[dk]} and all four dimensions are utilized. The 24cell can be partitioned into 6 such 4cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these links all have a common center). An isoclinic rotation in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.
Six regular octahedra can be connected facetoface 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 24cell 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 24cell.^{[dn]} 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.^{[dl]} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.^{[dp]} 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 only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An isoclinic rotation by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a nonadjacent octahedron in the ring.^{[dr]}
Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original orientation.
Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a Hopf fibration. Four celldisjoint 6cell rings comprise the same discrete fibration. The 24cell has four such discrete hexagonal fibrations, and each is the domain (container) of a unique leftright pair of isoclinic rotations (left and right Hopf fiber bundles). Each great hexagon belongs to just one fibration,^{[60]} but each 6cell ring belongs to three fibrations. The 24cell contains 16 great hexagons, divided among four fibrations, each of which is a set of four 6cell rings, but the 24cell has only four distinct 6cell rings. Each 6cell ring contains 3 of the great hexagons in each of three fibrations: only 3 of the 4 Clifford parallel hexagons of each of the three fibrations, and only 18 of the 24 vertices.^{[dg]}
Another kind of geodesic fiber, the helical hexagram isoclines, can be found within a 6cell ring of octahedra. Each of these geodesics runs through every second vertex of a skew hexagram_{2}, which in the unitradius, unitedgelength 24cell has six √3 edges. The hexagram does not lie in a single central plane, but is composed of six linked √3 chords from the six different hexagon great circles in the 6cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,^{[at]} a helical rather than simply circular path around the 24cell which links vertices two edge lengths apart and consequently must wrap twice around the 24cell before completing its sixvertex loop.^{[cl]} Rather than a flat hexagon, it forms a skew hexagram out of two threesided 360 degree halfloops: open triangles joined endtoend to each other in a sixsided Möbius loop.^{[cw]}
Each 6cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.^{[do]} Each of the three blackwhite pairs of isoclines belongs to one of the three fibrations in which the 6cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.^{[s]}
Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of √3 chords of an isocline from octahedron to octahedron. In the 24cell 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.^{[cu]} At each vertex the isoclinic path of √3 chords bends 60 degrees in two central planes^{[ds]} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.^{[dv]} 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_{2} 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 zigzagging between three adjacent parallel hexagonal central planes (like a Petrie polygon), but it is not: any isoclinic path we can pick out always zigzags between two sets of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6cell ring in rotation.^{[cg]} 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 hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.
At each vertex, there are four great hexagons^{[dx]} and four hexagram isoclines (all black or all white) that cross at the vertex.^{[dy]} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.^{[dz]} There are 16 distinct hexagram isoclines in the 24cell (8 black and 8 white).^{[ea]} Each isocline is a skew Clifford polygon of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.^{[cl]}
The 24cell contains 18 helical octagram isoclines (9 black and 9 white). Three pairs of octagram edgehelices are found in each of the three inscribed 16cells, described elsewhere as the helical construction of the 16cell. In summary, each 16cell can be decomposed (three different ways) into a leftright pair of 8cell rings of √2edged tetrahedral cells. Each 8cell ring twists either left or right around an axial octagram helix of eight chords. In each 16cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are √2 = 90° apart, so the circumference of the isocline is 4𝝅. An isoclinic rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and √4 = 180° distant across the diameter of the isocline.
Each of the 3 fibrations of the 24cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16cell) to great squares in a neighboring 16cell, on 8chord helical isoclines characteristic of the 16cell.^{[eb]}
In the 24cell, these 18 helical octagram isoclines can be found within the six orthogonal 4cell rings of octahedra. Each 4cell ring has cells bonded vertextovertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A √4 chord (the diameter of the great square and of the isocline) connects them. Boundary cells describes how the √2 axes of the 24cell's octahedral cells are the edges of the 16cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.^{[br]} The vertexbonded octahedra of the 4cell ring also lie in different tesseracts.^{[bh]} The isocline's four √4 diameter chords form an octagram_{8{4}=4{2}} with √4 edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24cell on one of the 12 √4 axes.
The octahedra in the 4cell rings are vertexbonded to more than two other octahedra, because three 4cell rings (and their three axial great squares, which belong to different 16cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two rightangled turns at once: 90° around the great square, and 90° orthogonally into a different 4cell ring entirely. The 180° fouredge arc joining two ends of each √4 diameter chord of the octagram runs through the volumes and opposite vertices of two facebonded √2 tetrahedra (in the same 16cell), which are also the opposite vertices of two vertexbonded octahedra in different 4cell rings (and different tesseracts). The 720° octagram isocline runs through 8 vertices of the fourcell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three blackwhite pairs) that cross at the vertex.^{[ck]}
This is the characteristic rotation of the 16cell, not the 24cell's characteristic rotation, and it does not take whole 16cells of the 24cell to each other the way the 24cell's rotation in great hexagon planes does.^{[ec]}
Five ways of looking at a skew 24gram  

Edge path  Petrie polygons  In a 600cell  Discrete fibration  Diameter chords 
16cells_{3{3/8}}  Dodecagons_{2{12}}  24gram_{{24/5}}  Squares_{6{4}}  _{{24/12}={12/2}} 
The 24cell's three inscribed Clifford parallel 16cells revealed as disjoint 8point 4polytopes with √2 edges.^{[eb]}  2 skew polygons of 12 √1 edges each. The 24cell can be decomposed into 2 disjoint zigzag dodecagons (4 different ways).^{[64]}  In compounds of 5 24cells, isoclines with golden chords of length φ = √2.𝚽 connect all 24cells in 24chord circuits.^{[65]}  Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with √2 edges to each other.  Two vertices four √2 chords apart on the circular isocline are antipodal vertices joined by a √4 axis. 
Characteristics of the 24cell^{[66]}  

edge^{[67]}  arc  dihedral^{[68]}  
𝒍  60°  120°  
𝟀  45°  45°  
𝝉^{[ed]}  30°  60°  
𝟁  30°  60°  
45°  90°  
30°  90°  
30°  90°  
^{[ch]}  
Every regular 4polytope has its characteristic 4orthoscheme, an irregular 5cell.^{[bl]} The characteristic 5cell of the regular 24cell is represented by the CoxeterDynkin diagram , which can be read as a list of the dihedral angles between its mirror facets.^{[ee]} It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular octahedron. The regular 24cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5cell that all meet at its center.^{[70]}
The characteristic 5cell (4orthoscheme) has four more edges than its base characteristic tetrahedron (3orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4orthoscheme, at the center of the regular 24cell).^{[ef]} If the regular 24cell has radius and edge length 𝒍 = 1, its characteristic 5cell's ten edges have lengths , , around its exterior righttriangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),^{[ed]} plus , , (the other three edges of the exterior 3orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the octahedron), plus , , , (edges which are the characteristic radii of the 24cell). The 4edge path along orthogonal edges of the orthoscheme is , , , , first from a 24cell vertex to a 24cell edge center, then turning 90° to a 24cell face center, then turning 90° to a 24cell octahedral cell center, then turning 90° to the 24cell center.
The 24cell can be constructed by the reflections of its characteristic 5cell in its own facets (its tetrahedral mirror walls).^{[eg]} Reflections and rotations are related: a reflection in an even number of intersecting mirrors is a rotation.^{[71]} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any 720° isoclinic rotation of the 24cell in a hexagonal invariant plane takes each of the 24 vertices to and through 5 other vertices and back to itself, on a skew hexagram_{2} geodesic isocline that winds twice around the 3sphere on every second vertex of the hexagram. Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the three inscribed 16cells) performing half such an orbit visits 3 * 8 = 24 distinct vertices and generates the 24cell sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.
Tracing the orbit of one such 16cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.^{[ej]} The vertex follows an isocline (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each √3 chord of the rotation.^{[cu]} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.^{[cp]} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24cell it stays within a 6cell ring of spherical^{[73]} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.
A symmetry operation is a rotation or reflection which leaves the object in the same orientation, indistinguishable from itself before the transformation. The 24cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two reflections, in a distinct pair of nonparallel mirror planes.^{[ej]}
Pictured are sets of disjoint great circle polygons, each in a distinct central plane of the 24cell. For example, {24/4}=4{6} is an orthogonal projection of the 24cell picturing 4 of its [16] great hexagon planes.^{[r]} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete Hopf fibration of 4 nonintersecting great circles which visit all 24 vertices just once.
Each row of the table describes a class of distinct rotations. Each rotation class takes the left planes pictured to the corresponding right planes pictured.^{[ek]} The vertices of the moving planes move in parallel along the polygonal isocline paths pictured. For example, the rotation class consists of [32] distinct rotational displacements by an arcdistance of 2𝝅/3 = 120° between 16 great hexagon planes represented by quaternion group and a corresponding set of 16 great hexagon planes represented by quaternion group .^{[em]} One of the [32] distinct rotations of this class moves the representative vertex coordinate to the vertex coordinate .^{[en]}
Proper rotations of the 24cell symmetry group F_{4} ^{[74]}  

Isocline^{[db]}  Rotation class^{[ep]}  Left planes ^{[eo]}  Right planes  
{24/8}=4{6/2}^{[er]} [16] 4𝝅 {6/2} 
^{[es]}  {24/4}=4{6}^{[r]} [16] 2𝝅 {6} 
^{[en]}  {24/8}=4{6/2}^{[et]} [16] 2𝝅 {6} 

2𝝅/3  120°  √3  1.732~  𝝅/3  60°  √1  1  2𝝅/3  120°  √3  1.732~  
{24/2}=2{12}^{[ev]} [16] 4𝝅 {12} 
^{[ex]}  {24/4}=4{6} [16] 2𝝅 {6} 
{24/4}=4{6} [16] 2𝝅 {6} 

𝝅/3  60°  √1  1  𝝅/3  60°  √1  1  𝝅/3  60°  √1  1  
{24/1}={24} [16] 4𝝅 {1} 
^{[ey]}  {24/4}=4{6} [16] 2𝝅 {6} 
{24/4}=4{6} [16] 2𝝅 {6} 

2𝝅  360°  √0  0  𝝅/3  60°  √1  1  𝝅/3  60°  √1  1  
{24/12}=12{2} [16] 4𝝅 {2} 
^{[ez]}  {24/4}=4{6} [16] 2𝝅 {6} 
{24/8}=4{6/2}^{[et]} [16] 2𝝅 {6} 

𝝅  180°  √4  2  𝝅/3  60°  √1  1  2𝝅/3  120°  √3  1.732~  
{24/2}=2{12}^{[ev]} [8] 4𝝅 {12} 
^{[fd]}  {24/4}=4{6} [8] 2𝝅 {6} 
{24/6}=6{4}^{[j]} [8] 2𝝅 {4} 

𝝅/3  60°  √1  1  𝝅/3  60°  √1  1  𝝅/2  90°  √2  1.414~  
{24/8}=4{6/2}^{[er]} [8] 4𝝅 {6/2} 
^{[fg]}  {24/4}=4{6} [8] 2𝝅 {6} 
{24/6}=6{4} [8] 2𝝅 {4} 

2𝝅/3  120°  √3  1.732~  𝝅/3  60°  √1  1  𝝅/2  90°  √2  1.414~  
{24/1}={24} [18] 4𝝅 {1} 
^{[fh]}  {24/6}=6{4} [18] 2𝝅 {4} 
^{[fj]}  {24/6}=6{4} [18] 2𝝅 {4} 

2𝝅  360°  √0  0  𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  
{24/12}=12{2} [18] 4𝝅 {2} 
^{[fk]}  {24/6}=6{4} [18] 2𝝅 {4} 
{24/6}=6{4} [18] 2𝝅 {4} 

𝝅  180°  √4  2  𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  
{24/9}=3{8/3}^{[fm]} [72] 4𝝅 {8/3} 
^{[fo]}  {24/6}=6{4} [72] 2𝝅 {4} 
{24/6}=6{4} [72] 2𝝅 {4} 

𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  𝝅  180°  √4  2  
{24/1}={24} [36] 4𝝅 {1} 
^{[fp]}  {24/6}=6{4} [36] 2𝝅 {4} 
{24/6}=6{4} [36] 2𝝅 {4} 

2𝝅  360°  √0  0  𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  
{24/2}=2{12}^{[ev]} [48] 4𝝅 {12} 
^{[fq]}  {24/6}=6{4} [48] 2𝝅 {4} 
{24/4}=4{6} [48] 2𝝅 {6} 

𝝅/3  60°  √1  1  𝝅/2  90°  √2  1.414~  𝝅/3  60°  √1  1  
{24/12}=12{2} [9] 4𝝅 {2} 
^{[fr]}  {24/6}=6{4} [9] 2𝝅 {4} 
{24/6}=6{4} [9] 2𝝅 {4} 

𝝅  180°  √4  2  𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  
{24/12}=12{2} [12] 4𝝅 {2} 
^{[ft]}  {24/12}=12{2} [12] 2𝝅 {2} 
{24/12}=12{2} [12] 2𝝅 {2} 

𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  
{24/1}={24} [0] 0𝝅 {1} 
^{[fu]}  {24/12}=12{2} [0] 2𝝅 {2} 
{24/12}=12{2} [0] 2𝝅 {2} 

0  0°  √0  0  𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~  
{24/12}=12{2} [12] 2𝝅 {2} 
^{[fv]}  {24/12}=12{2} [12] 2𝝅 {2} 
{24/12}=12{2} [12] 2𝝅 {2} 

𝝅  180°  √4  2  𝝅/2  90°  √2  1.414~  𝝅/2  90°  √2  1.414~ 
In a rotation class each quaternion group may be representative not only of its own fibration of Clifford parallel planes^{[em]} but also of the other congruent fibrations.^{[r]} For example, rotation class takes the 4 hexagon planes of to the 4 hexagon planes of which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,^{[eo]} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes , and . The name is the conventional representation for all [16] congruent plane displacements.
These rotation classes are all subclasses of which has [32] distinct rotational displacements rather than [16] because there are two chiral ways to perform any class of rotations, designated its left rotations and its right rotations. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.^{[fw]} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.
Each rotation class (table row) describes a distinct left (and right) isoclinic rotation. The left (or right) rotations carry the left planes to the right planes simultaneously,^{[cd]} through a characteristic rotation angle.^{[aw]} For example, the rotation moves all [16] hexagonal planes at once by 2𝝅/3 = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same orientation, passing through all 4 planes of the left set and all 4 planes of the right set once each.^{[el]} The picture in the isocline column represents this union of the left and right plane sets. In the example it can be seen as a set of 4 Clifford parallel skew hexagrams, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.^{[cf]}
The 24cell 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 120cell). 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.^{[75]}^{[76]} 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 24cell, 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.^{[ak]} One can easily follow this path in a rendering of the equatorial cuboctahedron crosssection.
Starting at the North Pole, we can build up the 24cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2sphere, with the equator being a great 2sphere.^{[ap]} 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 nonmeridian and pole cells has the same relative position to each other as the cells in a tesseract (8cell), although they touch at their vertices instead of their faces.
Layer #  Number of Cells  Description  Colatitude  Region 

1  1 cell  North Pole  0°  Northern Hemisphere 
2  8 cells  First layer of meridian cells  60°  
3  6 cells  Nonmeridian / interstitial  90°  Equator 
4  8 cells  Second layer of meridian cells  120°  Southern Hemisphere 
5  1 cell  South Pole  180°  
Total  24 cells 
The 24cell can be partitioned into celldisjoint sets of four of these 6cell great circle rings, forming a discrete Hopf fibration of four interlocking rings.^{[dg]} 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.^{[77]}
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 24cells in a plane and form a 4D honeycomb of 24cells 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 crosssection. The 24cell 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 nonmeridian (equatorial) and pole cells.
The 24cell can be equipartitioned into three 8cell 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.
The vertexfirst parallel projection of the 24cell into 3dimensional 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 cellfirst parallel projection of the 24cell into 3dimensional space has a cuboctahedral envelope. Two of the octahedral cells, the nearest and farther from the viewer along the waxis, 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 edgefirst parallel projection has an elongated hexagonal dipyramidal envelope, and the facefirst parallel projection has a nonuniform hexagonal biantiprismic envelope.
The vertexfirst perspective projection of the 24cell into 3dimensional 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 cellfirst perspective projection of the 24cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertexcenter radius of the 24cell.
Animated crosssection of 24cell  
A stereoscopic 3D projection of an icositetrachoron (24cell).  
Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell 
There are two lower symmetry forms of the 24cell, derived as a rectified 16cell, with B_{4} or [3,3,4] symmetry drawn bicolored with 8 and 16 octahedral cells. Lastly it can be constructed from D_{4} or [3^{1,1,1}] symmetry, and drawn tricolored with 8 octahedra each.
Three nets of the 24cell with cells colored by D_{4}, B_{4}, and F_{4} symmetry  

Rectified demitesseract  Rectified 16cell  Regular 24cell  
D_{4}, [3^{1,1,1}], order 192  B_{4}, [3,3,4], order 384  F_{4}, [3,4,3], order 1152  
Three sets of 8 rectified tetrahedral cells  One set of 16 rectified tetrahedral cells and one set of 8 octahedral cells.  One set of 24 octahedral cells  
Vertex figure (Each edge corresponds to one triangular face, colored by symmetry arrangement)  
The regular complex polygon _{4}{3}_{4}, or contains the 24 vertices of the 24cell, and 24 4edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is _{4}[3]_{4}, order 96.^{[78]}
The regular complex polytope _{3}{4}_{3}, or , in has a real representation as a 24cell in 4dimensional space. _{3}{4}_{3} has 24 vertices, and 24 3edges. Its symmetry is _{3}[4]_{3}, order 72.
Several uniform 4polytopes can be derived from the 24cell via truncation:
The 96 edges of the 24cell can be partitioned into the golden ratio to produce the 96 vertices of the snub 24cell. This is done by first placing vectors along the 24cell's edges such that each twodimensional 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 24cell is the unique convex selfdual regular Euclidean polytope that is neither a polygon nor a simplex. Relaxing the condition of convexity admits two further figures: the great 120cell and grand stellated 120cell. With itself, it can form a polytope compound: the compound of two 24cells.
D_{4} uniform polychora  

{3,3^{1,1}} h{4,3,3} 
2r{3,3^{1,1}} h_{3}{4,3,3} 
t{3,3^{1,1}} h_{2}{4,3,3} 
2t{3,3^{1,1}} h_{2,3}{4,3,3} 
r{3,3^{1,1}} {3^{1,1,1}}={3,4,3} 
rr{3,3^{1,1}} r{3^{1,1,1}}=r{3,4,3} 
tr{3,3^{1,1}} t{3^{1,1,1}}=t{3,4,3} 
sr{3,3^{1,1}} s{3^{1,1,1}}=s{3,4,3} 
24cell family polytopes  

Name  24cell  truncated 24cell  snub 24cell  rectified 24cell  cantellated 24cell  bitruncated 24cell  cantitruncated 24cell  runcinated 24cell  runcitruncated 24cell  omnitruncated 24cell  
Schläfli symbol 
{3,4,3}  t_{0,1}{3,4,3} t{3,4,3} 
s{3,4,3}  t_{1}{3,4,3} r{3,4,3} 
t_{0,2}{3,4,3} rr{3,4,3} 
t_{1,2}{3,4,3} 2t{3,4,3} 
t_{0,1,2}{3,4,3} tr{3,4,3} 
t_{0,3}{3,4,3}  t_{0,1,3}{3,4,3}  t_{0,1,2,3}{3,4,3}  
Coxeter diagram 

Schlegel diagram 

F_{4}  
B_{4}  
B_{3}(a)  
B_{3}(b)  
B_{2} 
The 24cell can also be derived as a rectified 16cell:
B4 symmetry polytopes  

Name  tesseract  rectified tesseract 
truncated tesseract 
cantellated tesseract 
runcinated tesseract 
bitruncated tesseract 
cantitruncated tesseract 
runcitruncated tesseract 
omnitruncated tesseract  
Coxeter diagram 
= 
= 

Schläfli symbol 
{4,3,3}  t_{1}{4,3,3} r{4,3,3} 
t_{0,1}{4,3,3} t{4,3,3} 
t_{0,2}{4,3,3} rr{4,3,3} 
t_{0,3}{4,3,3}  t_{1,2}{4,3,3} 2t{4,3,3} 
t_{0,1,2}{4,3,3} tr{4,3,3} 
t_{0,1,3}{4,3,3}  t_{0,1,2,3}{4,3,3}  
Schlegel diagram 

B_{4}  
Name  16cell  rectified 16cell 
truncated 16cell 
cantellated 16cell 
runcinated 16cell 
bitruncated 16cell 
cantitruncated 16cell 
runcitruncated 16cell 
omnitruncated 16cell  
Coxeter diagram 
= 
= 
= 
= 
= 
= 

Schläfli symbol 
{3,3,4}  t_{1}{3,3,4} r{3,3,4} 
t_{0,1}{3,3,4} t{3,3,4} 
t_{0,2}{3,3,4} rr{3,3,4} 
t_{0,3}{3,3,4}  t_{1,2}{3,3,4} 2t{3,3,4} 
t_{0,1,2}{3,3,4} tr{3,3,4} 
t_{0,1,3}{3,3,4}  t_{0,1,2,3}{3,3,4}  
Schlegel diagram 

B_{4} 
{3,p,3} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Compact  Paracompact  Noncompact  
{3,p,3}  {3,3,3}  {3,4,3}  {3,5,3}  {3,6,3}  {3,7,3}  {3,8,3}  ... {3,∞,3}  
Image  
Cells  {3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞}  
Vertex figure 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 