24-cell
In four-dimensional geometry, the 24-cell is the convex regular 4-polytope with Schläfli symbol. It is also called C24, or the icositetrachoron, octaplex, icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.
The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual. It and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.
The 24-cell does not have a regular analogue in 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.
Geometry
The 24-cell is the symmetric union of the geometries of every convex regular polytope in the first four dimensions, except those with a 5 or [|above] in their Schlӓfli symbol. It is especially useful to explore the 24-cell, because one can see all the geometric relationships among all of these polytopes in a single 24-cell or its honeycomb.The 24-cell is the fourth in the sequence of 6 convex regular 4-polytopes. It can be deconstructed into 3 overlapping instances of its predecessor the tesseract, as the 8-cell can be deconstructed into 2 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 different edge length.
Coordinates
Squares
The 24-cell is the convex hull of its vertices which can be described as the 24 coordinate permutations of:Those coordinates can be constructed as, rectifying the 16-cell,, with 8 vertices permutations of. The vertex figure of a 16-cell is the octahedron; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.
In this form the 24-cell has edges of length and is inscribed in a 3-sphere of radius. Remarkably, the edge length equals the circumradius, as in the hexagon, or the cuboctahedron. Such polytopes are radially equilateral.
The 24 vertices can be seen as the vertices of 6 orthogonal equatorial squares which intersect only at their common center.
Hexagons
The 24-cell is self-dual, having the same number of vertices as cells and the same number of edges as faces.If the dual of the above 24-cell of edge length is taken by reciprocating it about its inscribed sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this form the vertices of the 24-cell can be given as follows:
8 vertices obtained by permuting the integer coordinates:
and 16 vertices with half-integer coordinates of the form:
all 24 of which lie at distance 1 from the origin.
Viewed as quaternions, these are the unit Hurwitz quaternions.
The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as unit radius coordinates to distinguish it from others, such as the radius coordinates used above.
The 24 vertices can be seen as the vertices of 4 orthogonal equatorial hexagons which intersect only at their common center.
Triangles
The 24 vertices can be seen as the vertices of 8 equilateral triangles lying in 4 orthogonal equatorial planes which intersect only at their common center.Hypercubic chords
The 24 vertices of the 24-cell are distributed at four different chord lengths from each other:,, and.Each vertex is joined to 8 others by an edge of length 1, spanning 60° = of arc. Next nearest are 6 vertices located 90° = away, along an interior chord of length. Another 8 vertices lie 120° = away, along an interior chord of length. The opposite vertex is 180° = away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center can be treated as a 25th canonical apex vertex, which is 1 edge length away from all the others.
To visualize how the interior polytopes of the 24-cell fit together, keep in mind that the four chord lengths are the long diameters of the hypercubes of dimensions 1 through 4: the long diameter of the square is ; the long diameter of the cube is ; and the long diameter of the tesseract is. Moreover, the long diameter of the octahedron is like the square; and the long diameter of the 24-cell itself is like the tesseract.
Geodesics
The vertex chords of the 24-cell are arranged in geodesic great circles which lie in sets of orthogonal planes. The geodesic distance between two 24-cell vertices along a path of edges is always 1, 2, or 3, and it is 3 only for opposite vertices.The edges occur in 16 hexagonal great circles, 4 of which cross at each vertex. The 96 distinct edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell.
The chords occur in 18 square great circles, 3 of which cross at each vertex. The 72 distinct chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its cell centers below one of its mid-edges.
The chords occur in 32 triangular great circles in 16 planes, 4 of which cross at each vertex. The 96 distinct chords run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.
The chords occur as 12 vertex-to-vertex diameters, the 24 radii around the 25th central vertex.
The edges occur in 48 parallel pairs, apart. The chords occur in 36 parallel pairs, apart. The chords occur in 48 parallel pairs, apart.
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.
Constructions
Triangles and squares come together uniquely in the 24-cell to generate, as interior features, all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions. Consequently, there are numerous ways to construct or deconstruct the 24-cell.Reciprocal constructions from 8-cell and 16-cell
The 8 integer vertices are the vertices of a regular 16-cell, and the 16 half-integer vertices are the vertices of its dual, the tesseract. The tesseract gives Gosset's construction of the 24-cell, equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction, equivalent to rectifying a 16-cell. The analogous construction in 3-space gives the cuboctahedron which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.We can further divide the last 16 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.
Truncations
We can truncate the 24-cell by cutting 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 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.8-cell
Starting with a complete 24-cell, remove 8 orthogonal vertices, and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by edges to remove 8 cubic pyramids whose apexes are the vertices to be removed. This removes 4 edges from each hexagonal great circle, 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, 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, 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.16-cell
Starting with a complete 24-cell, remove the 16 vertices of a tesseract, by cutting through 16 tetrahedral cells bounded by chords to remove 16 tetrahedral pyramids whose apexes are the vertices to be removed. This removes 12 square great circles and all the edges, exposing chords as the new edges. Now the remaining 6 square great circles cross perpendicularly, 3 at each of 8 remaining vertices, 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, 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.Tetrahedral constructions
The 24-cell can be constructed radially from 96 equilateral triangles of edge length which meet at the center of the polytope, each contributing two radii and an edge. They form 96 tetrahedra, all sharing the 25th central apex vertex. These form 24 octahedral pyramids with their apexes at the center.The 24-cell can be constructed from 96 equilateral triangles of edge length, where the three vertices of each triangle are located 90° = away from each other. They form 48 tetrahedra, centered at the 24 mid-radii of the 24-cell.
Relationships among interior polytopes
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core. The tesseracts are inscribed in the 24-cell such that their vertices and edges lie on the surface of the 24-cell, but their square faces and cubical cells lie inside the 24-cell. The 16-cells are inscribed in the 24-cell such that only their vertices lie on the surface: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior 16-cell edges have length.The 16-cells are also inscribed in the tesseracts: their 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, 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, 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 between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are 4-pyramids, alluded to above.
Boundary cells
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places. Where 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. 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.Some interior features lie inside the boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares, and the diagonals of those squares are 16-cell edges. 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 tetrahedral cells are inscribed in tesseract cubic cells, sharing the same volume. 24-cell octahedral cells overlap their volume with 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. 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.
As a configuration
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.
Symmetries, root systems, and tessellations
The 24 root vectors of the D4 root system of the simple Lie group SO form the vertices of a 24-cell. The vertices can be seen in 3 hyperplanes, 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 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 :1. This is likewise true for the 24 vertices of its dual. The full symmetry group of the 24-cell is the Weyl group of F4, which is generated by reflections through the hyperplanes orthogonal to the F4 roots. This is a solvable group of order 1152. The rotational symmetry group of the 24-cell is of order 576.
Quaternionic interpretation
When interpreted as the quaternions, the F4 root lattice 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 in the Hurwitz quaternion ring. 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.
Voronoi cells
The Voronoi cells of the D4 root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the tessellation of 4-dimensional Euclidean space by regular 24-cells, the icositetrachoric honeycomb. The 24-cells are centered at the D4 lattice points while the vertices are at the F4 lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The Schläfli symbol for this tessellation is. It is one of only three regular tessellations of R4.The unit balls inscribed in the 24-cells of this tessellation give rise to the densest known lattice packing of hyperspheres in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the highest possible kissing number in 4 dimensions.
Radially equilateral honeycomb
The dual tessellation of the 24-cell honeycomb is the 16-cell honeycomb. The third regular tessellation of four dimensional space is the tesseractic honeycomb, whose vertices can be described by 4-integer Cartesian coordinates. The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.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 is also the vertex of a 24-cell, 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, 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 are colored red, a 4-dimensional checkerboard results. Of the 24 center-to-vertex radii 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 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. Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids. 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 ; 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 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 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.
Rotations
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 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 45 degrees.Projections
Parallel projections
The vertex-first parallel projection of the 24-cell into 3-dimensional space has a rhombic dodecahedral envelope. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.The cell-first parallel projection of the 24-cell into 3-dimensional space has a cuboctahedral envelope. Two of the octahedral cells, the nearest and farther from the viewer along the w-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.
The edge-first parallel projection has an elongated hexagonal dipyramidal envelope, and the face-first parallel projection has a nonuniform hexagonal bi-antiprismic envelope.
Perspective projections
The vertex-first perspective projection of the 24-cell into 3-dimensional space has a tetrakis hexahedral envelope. The layout of cells in this image is similar to the image under parallel projection.The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.
Orthogonal projections
Visualization
The 24-cell is bounded by 24 octahedral cells. For visualization purposes, it is convenient that the octahedron has opposing parallel faces. 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 radiate out in 3 dimensions, converging at the 3rd "South Pole" cell. This skeleton accounts for 18 of the 24 cells. 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, 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 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 of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two interlocking great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration.
Three Coxeter group constructions
There are two lower symmetry forms of the 24-cell, derived as a rectified 16-cell, with B4 or symmetry drawn bicolored with 8 and 16 octahedral cells. Lastly it can be constructed from D4 or symmetry, and drawn tricolored with 8 octahedra each.Related complex polygons
The regular complex polygon 44, or contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is 44, order 96.The regular complex polytope 33, or, in has a real representation as a 24-cell in 4-dimensional space. 33 has 24 vertices, and 24 3-edges. Its symmetry is 33, order 72.
| Name | , | 44, | 33, |
| Symmetry | ,, order 1152 | 44,, order 96 | 33,, order 72 |
| Vertices | 24 | 24 | 24 |
| Edges | 96 2-edges | 24 4-edge | 24 3-edges |
| Image | 24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges. | 44, has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges. | 33 or has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled. |
Related 4-polytopes
Several uniform 4-polytopes can be derived from the 24-cell via truncation:- truncating at 1/3 of the edge length yields the truncated 24-cell;
- truncating at 1/2 of the edge length yields the rectified 24-cell;
- and truncating at half the depth to the dual 24-cell yields the bitruncated 24-cell, which is cell-transitive.
The 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.