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.[a] It and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.[b]
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 is the fourth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[d] 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. 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]
Those coordinates can be constructed as , rectifying the 16-cell, , with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the octahedron; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.
In this frame of reference the 24-cell has edges of length √2 and is inscribed in a 3-sphere of radius √2. Remarkably, the edge length equals the circumradius, as in the hexagon, or the cuboctahedron. Such polytopes are radially equilateral.[b]
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:
The 24-cell 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.[i]
Each vertex is joined to 8 others[n] by an edge of length 1, spanning 60° = π/3 of arc. Next nearest are 6 vertices[o] 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 as a 25th canonical apex vertex,[q] 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.[r] 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.
The vertex chords of the 24-cell are arranged in geodesic great circles which lie in sets of orthogonal planes. 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.[s]
The √1 edges occur in 16 hexagonal great circles (4 sets of 4 orthogonal planes[t]), 4 of which cross at each vertex.[v] The 96 distinct √1 edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell.
The √2 chords occur in 18 square great circles (3 sets of 6 orthogonal planes[t]), 3 of which cross at each vertex.[w] 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.[x]
The √3 chords occur in 32 triangular great circles in 16 planes (4 sets of 4 orthogonal planes),[m] 4 of which cross at each vertex.[y] The 96 distinct √3 chords run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.[l]
The √4 chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.[q]
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.[z]
Each great circle plane intersects with each of the other great circle planes or face planes to which it is orthogonal at the center point only, and with each of the others to which it is not orthogonal at a single edge of some kind. In every case that edge is one of the vertex chords of the 24-cell.
Triangles and squares come together uniquely in the 24-cell to generate, as interior features, all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the 5-cell and the 600-cell).[ac] Consequently, there are numerous ways to construct or deconstruct the 24-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 of the 24-cell, equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction, 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.
We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract. This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.
We can facet the 24-cell by cutting[ad] 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 into two parts through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle planes (above) are only some of those planes. Here we shall expose some of the others: the face planes of interior polytopes, which divide the 24-cell into two unequal parts.[ag]
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,[ah] 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. 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.[ai]
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 square great circles (retaining just one orthogonal set) and all the √1 edges, exposing √2 chords as the new edges. Now the remaining 6 square great circles cross perpendicularly, 3 at each of 8 remaining vertices,[aj] 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 most of their element sets are disjoint: they do not share any vertex count, edge length, or face area, but they do share cell volume. They also share 4-content, their common core.[ai]
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.[b] 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. They form 48 √2 tetrahedra (the cells of the three 16-cells), centered at the 24 mid-radii of the 24-cell.
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.[ai] The tesseracts are inscribed in the 24-cell[ak] 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[al] 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[am] 16-cell edges have length √2.
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. This is reminiscent of the way, in 3 dimensions, two tetrahedra can be inscribed in a cube, as discovered by Kepler. 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.
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-dimensional interstices[an] between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are 4-pyramids,[ao] alluded to above.
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.[aq] 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.
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, the apexes of which also lie at a vertex of a cube.[ar] 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.[ap]
This configuration matrix 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.
Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.
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,[ae] 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. 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.
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.
Vertices of other convex regular 4-polytopes also form multiplicative groups of quaternions, but few of them generate a root lattice.
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.
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. 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. Of the 24 center-to-vertex radii[as] 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, 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).[at]
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell honeycomb in this manner, depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes) was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other. 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 axes planes, around a single fixed point).[aw] 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.[j]
The vertex set of the 24-cell is the binary tetrahedral group. 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.
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 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.
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. 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.
The 24-cell can be partitioned into disjoint sets of four of these 6-cell great circle rings, forming a discrete Hopf fibration of four interlocking rings. 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.
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.
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.