24-cell

In geometry, the 24-cell is the convex regular 4-polytope[1] (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, icositetrachoron,[2], octaplex (short for "octahedral complex"), icosatetrahedroid,[3] octacube,[citation needed] hyper-diamond or polyoctahedron, being constructed of octahedral cells.

24-cell
Schlegel diagram
(vertices and edges)
TypeConvex regular 4-polytope
Schläfli symbol{3,4,3}
r{3,3,4} = ${\displaystyle \left\{{\begin{array}{l}3\\3,4\end{array}}\right\}}$
{31,1,1} = ${\displaystyle \left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}}$
Coxeter diagram
or
or
Cells24 {3,4}
Faces96 {3}
Edges96
Vertices24
Vertex figureCube
Petrie polygondodecagon
Coxeter groupF4, [3,4,3], order 1152
B4, [4,3,3], order 384
D4, [31,1,1], order 192
DualSelf-dual
Propertiesconvex, isogonal, isotoxal, isohedral
Uniform index22

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. In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex. Due to this singular property, it 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 analogue of one of the five Platonic solids. Instead, it is 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 greater in their Schlafli symbol.[a] It is especially useful to explore the 24-cell, because one can see all the geometric relationships among all of those polytopes in a single 24-cell or its honeycomb.

Coordinates

Orthogonal disjoint squares

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

${\displaystyle (\pm 1,\pm 1,0,0)\in \mathbb {R} ^{4}}$ .

Those coordinates[4] can be constructed as        , rectifying the 16-cell,        , with 8 vertices permutations of (±2,0,0,0). The vertex figure[5] of a 16-cell is the octahedron; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produce 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 2 and is inscribed in a 3-sphere of radius 2. Remarkably, the edge length equals the radius, as in the hexagon, or the cuboctahedron. Such polytopes are radially equilateral.[b]

The 24 vertices are the vertices of 6 perpendicular[c] squares which intersect only at their common center.[d]

Orthogonal disjoint hexagons

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 is taken instead, and its edge length set to 1, the coordinates of another 24-cell can be found which reveal more structure. In this form the vertices of the 24-cell can be given as follows:

8 vertices obtained by permuting

(±1, 0, 0, 0)

and 16 vertices of the form

1/2, ±1/2, ±1/2, ±1/2).

all 24 of which lie at distance 1 from the origin.[b]

Viewed as quaternions, these are the unit Hurwitz quaternions.

The 24 vertices are the vertices of 4 perpendicular hexagons[e] which intersect[d] only at their common center.[f]

Hypercubic chords

Vertex geometry of the 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.

The 24 vertices of the 24-cell are distributed[6] at four different chord lengths from each other: 1, 2, 3, or 4.

Each vertex is joined to 8 others by an edge of length 1, spanning 60 degrees of arc. Next nearest are 6 vertices located 90 degrees away, along an interior chord of length 2. Another 8 lie 120 degrees away, along an interior chord of length 3. The opposite vertex is 180 degrees away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center can be treated[g] as a 25th canonical apex vertex[h], 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. 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.

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 1 edges is always 1, 2, or 3, and it is 3 only for opposite vertices.[i]

The 1 edges occur in 16 hexagonal great circles (4 sets of 4 orthogonal[f] planes), 4 of which cross at each vertex[j]. 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[c] planes), 3 of which cross at each vertex[k]. 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 cell centers below one of its mid-edges.[l]

The 3 chords occur in 32 triangular great circles in 16 planes (4 sets of 4 orthogonal planes), 6 of which cross at each vertex[m]. The 96 distinct 3 chords run vertex-to-every-other-vertex in the same planes as the hexagonal great circles[n]; but they do not follow the 24-cell's edges, they pass below two of its face centers.

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

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.

Each great circle plane intersects[d] 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.[p]

Constructions

Triangles and squares come together uniquely in the 24-cell to generate, as interior features[g], 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).[q] Consequently, there are numerous ways to construct or deconstruct the 24-cell.

Reciprocal constructions from 8-cell and 16-cell

The first 8 vertices (±1, 0, 0, 0) are the vertices of a regular 16-cell, and the other 16 (±1/2, ±1/2, ±1/2, ±1/2) are the vertices of its dual, the tesseract (8-cell). The tesseract gives Gosset's construction[7] 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,[8] 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.[9]

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.[10] 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 slicing through planes 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 square of 4 vertices, or any planar 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.[r]

Tesseract

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 slicing through 24 square face planes 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[s], 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.

16-cell

Starting with a complete 24-cell, remove 16 tesseract vertices (retaining the 8 vertices you removed above), by slicing through 32 triangular face planes 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[t], 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 do not overlap with each other, and all of their element sets are disjoint: they do not share any vertex count, edge length, face area, or cell volume.

5-cell

Starting with a complete 24-cell, remove 20 vertices, by slicing through 4 mutually non-orthogonal triangular face planes bounded by 3 chords to remove 5 vertices above each plane.[u] This removes 20 triangular great circles, and all the 1 and 2 chords, exposing 3 chords as the new edges. Now the remaining 4 triangular great circles meet but do not cross, 3 at each of the 4 remaining vertices[v], and their 6 edges divide the surface into 4 non-orthogonal triangle faces[w] comprising a single regular tetrahedral cell: an irregular 5-cell.[y] There are 24 ways you can do this (find a tetrahedron made of 4 face planes out of 96), so there are 24 such 5-cells inscribed in the 24-cell. They overlap with each other, but some of their element sets are disjoint: they share vertex count and edge length, but not face area.

Relationships among interior polytopes

The 24-cell, three tesseracts, three 16-cells and twenty-four 5-cells are deeply entwined around their common center. The tesseracts are inscribed in the 24-cell such that their vertices and edges lie on the surface of the 24-cell (they are 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 such that only their vertices lie on the surface: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior[z]. 16-cell edges have length 2. The 5-cells are inscribed in the 24-cell such that only four of their five vertices lie on the surface: their fifth vertex is the center of the 24-cell, and all their edges, triangular faces and tetrahedral cells lie inside the 24-cell.

Kepler's drawing of tetrahedra inscribed in the cube.[11]

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.[11] In fact it is the exact analogy (the demihypercubes), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.[12]

The long 3 edges of the irregular 5-cells are each also a long diagonal of a cube (in two different tesseracts). Each 3 tetrahedral face triangle has one edge in a different pair of the three tesseracts. The short 1 edges of the irregular 5-cells are each also a long radius of the 24-cell (and therefore also a long radius of two different tesseracts).

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[13] 4-dimensional interstices[aa] between the 24-cell, 8-cell and 16-cell envelopes.[ab] The shapes filling these gaps are 4-pyramids[ac], alluded to above.

Boundary cells

Despite the 4-dimensional interstices between 24-cell, 8-cell, 16-cell and 5-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. Because there are a total of 31 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 inside 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,[14] the apexes of which also lie at a vertex of a cube.[ad] 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.[ae]

As a configuration

This configuration matrix[15] 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.

${\displaystyle {\begin{bmatrix}{\begin{matrix}24&8&12&6\\2&96&3&3\\3&3&96&2\\6&12&8&24\end{matrix}}\end{bmatrix}}}$

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

Symmetries, root systems, and tessellations

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

The 24 root vectors of the D4 root system of the simple Lie group SO(8) form the vertices of a 24-cell. The vertices can be seen in 3 hyperplanes,[x] 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.

Quaternionic interpretation

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.

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

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

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

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.[16] Of the 24 center-to-vertex radii[af] 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,[7] 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).

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 (4 perpendicular axes) was chosen to align it. (Three tesseracts can be inscribed in a 24-cell, rotated with respect to each other.) The distance from one of these orientations to another is an isoclinic rotation through 45 degrees (a double rotation of 45 degrees in each of two orthogonal axes planes, around a single fixed point).

Projections

Parallel projections

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

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

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

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

Perspective projections

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

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

Cell-first perspective projection

In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.

In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).

Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.

Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells.

 Stereographic projection Animated cross-section of 24-cell A 3D projection of a 24-cell performing a simple rotation. A stereoscopic 3D projection of an icositetrachoron (24-cell). Isometric Orthogonal Projection of:8 Cell(Tesseract) + 16 Cell = 24 Cell

Orthogonal projections

orthographic projections
Coxeter plane F4
Graph
Dihedral symmetry [12]
Coxeter plane B3 / A2 (a) B3 / A2 (b)
Graph
Dihedral symmetry [6] [6]
Coxeter plane B4 B2 / A3
Graph
Dihedral symmetry [8] [4]

Visualization

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. One can easily follow this path in a rendering of the equatorial cuboctahedron cross-section.

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

Layer # Number of Cells Description Colatitude Region
1 1 cell North Pole Northern Hemisphere
2 8 cells First layer of meridian cells 60°
3 6 cells Non-meridian / interstitial 90° Equator
4 8 cells Second layer of meridian cells 120° Southern Hemisphere
5 1 cell South Pole 180°
Total 24 cells

An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator

The 24-cell can be partitioned into 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. This corresponds to traversing diagonally through the squares in the cuboctahedron cross-section. The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell. This great circle is self dual. This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells. The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract. Each of these subsets can be further equipartitioned into two interlocking great circle chains, four cells long. Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

Three Coxeter group constructions

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

Related complex polygons

The regular complex polygon 4{3}4,     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 4[3]4, order 96.[17]

The regular complex polytope 3{4}3,     or    , in ${\displaystyle \mathbb {C} ^{2}}$  has a real representation as a 24-cell in 4-dimensional space. 3{4}3 has 24 vertices, and 24 3-edges. Its symmetry is 3[4]3, order 72.

The regular complex polygon 3{6}2,     or     contains the 24 vertices of the 24-cell. It has 16 3-edges. Its symmetry is 3[6]2, order 48.

Related figures in orthogonal projections
Name {3,4,3},         4{3}4,     3{4}3,     3{6}2,
Symmetry [3,4,3],        , order 1152 4[3]4,    , order 96 3[4]3,    , order 72 3[6]2,    , order 48
Vertices 24 24 24 24
Edges 96 2-edges 24 4-edge 24 3-edges 16 3-edges
Image
24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.

4{3}4,     has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.

3{4}3 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.

3{6}2,     has 24 vertices and 16 3-edges, shown here with 8 red and 8 blue 3-edges.

Related 4-polytopes

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

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

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

Related uniform polytopes

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

Notes

1. ^ The polygons {5} and above, the dodecahedron {5, 3}, the 600-cell {3,3,5} and the 120-cell {5,3,3}. In other words, the 24-cell only lacks the pentagonal features; it possesses all of the triangular and square features that exist in four dimensions.
2. ^ a b c The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few polytopes have this property, including the four-dimensional 24-cell and tesseract, the three-dimensional cuboctahedron, and the two-dimensional hexagon. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
3. ^ a b Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time. Three such pairs (perpendicular planes) meet at each vertex (for the same reason that three edges of the tetrahedron meet at each vertex).
4. ^ a b c Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) they can intersect in a single point: and they must, if and only if they are perpendicular; this is the surprising, counterintuitive thing about how planes intersect in 4-space.
5. ^ The hexagons are tilted with respect to the coordinate system's 6 perpendicular planes (the 6 perpendicular squares). Each hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair from the first 8 vertices (one of the four coordinate axes) and two opposite pairs from the second 16 vertices (not coordinate axes). Specifying any two of these diameter-pairs will determine the hexagonal plane (and its third diameter-pair). The hexagonal plane is also determined by specifying one vertex from each of three different diameter-pairs.
6. ^ a b It is of course difficult to visualize four hexagonal planes that are all perpendicular to each other. One can see them in the cuboctahedron (a projection of the 24-cell into 3-dimensions), where they appear to be at 60 degrees to each other. In 3 dimensions two hexagons appear to intersect at each of 12 vertices, but in 4 dimensions only one hexagon passes through each disjoint set of 6 of the 24 vertices: the four perpendicular planes intersect only at their common center.
7. ^ a b Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in its configuration matrix, which counts only surface features. Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from surface edges.
8. ^ a b The central vertex is a canonical apex because it is one edge length equidistant from the ordinary vertices in the 4th dimension, as the apex of a canonical pyramid is one edge length equidistant from its other vertices.
9. ^ If the Pythagorean distance between any two vertices is 1, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is 2, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90o bend in it as the path through the center). If their Pythagorean distance is 3, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60o bend, or as a straight line with one 60o bend in it through the center). Finally, if their Pythagorean distance is 4, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).
10. ^ Eight 1 edges converge in 3-dimensional space, enter the 24-cell's cubical vertex figure through its corners and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the 8 edges connects this center to the center of a diagonally adjacent vertex cube (just where it would be in an ordinary 3-dimensional cubic honeycomb), which is one of the eight nearest other vertices of the 24-cell. Where the edge passes diagonally out of one cube into another (where their adjacent corners touch) is the mid-point of the edge (not a vertex of the 24-cell). The straight lines are geodesics: six 1-edge-length segments of a straight line (in the 3-space of the 24-cell's curved surface, here tessellated by cubes) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Viewed from inside this cubical honeycomb 3-space, the bends in the hexagons are invisible (they appear to be hidden in the corners of the cubes). From outside (viewing the 24-cell in 4-space), the straight lines can be seen to bend at the vertices (not the mid-edges), because the vertices are displaced outwards in the 4th dimension (farther from the center of the 24-cell than the mid-edges).
11. ^ Six 2 chords enter the cubical 3-space of the 24-cell's vertex figure through its face centers and converge at its center (the vertex), where they form 3 straight lines which cross there. (This is the same vertex cube in which eight 1 edges enter through the corners, but let us ignore those now, since 7 straight lines crossing at the center would be too confusing to visualize all at once.) Each of the six 2 chords runs from this cube's center (the vertex) straight through the center of an adjacent (face-bonded) cube and on out through its opposite face center into a third cube, where the 2 chord terminates at that cube center, which is another vertex of the 24-cell. That vertex-center is not a nearest vertex (1-distant from the original vertex), but one in a second concentric shell of six 2-distant vertices that surrounds the first shell of eight vertices. (The center of the intermediate cube through which the 2 chord passes is not a vertex of the 24-cell at all: it is the mid-point of the 2 chord, so it lies inside the 24-cell, not on its surface.)
12. ^ One can cut the 24-cell into two equal parts through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the cuboctahedron (the central hyperplane of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).
13. ^ Twelve 3 chords enter the 24-cell's cubical vertex figure through its mid-edges, and meet at its center (the vertex), where they form six straight lines which cross there. (This is the same vertex cube in which eight 1 edges enter through the corners and six 2 chords enter through the face centers, but let us ignore those now, since 13 straight lines crossing at the center would be much too confusing to visualize all at once.) Each of the twelve 3 chords runs from this cube's center straight through the center of a diagonally-adjacent (edge-bonded) cube and on out through its opposite mid-edge into a third cube, where it terminates at that cube center, which is another vertex of the 24-cell: not a nearest vertex (1-distant from the original vertex), nor a next-nearest vertex (2-distant), but one in a third concentric shell of eight 3-distant vertices surrounding the second shell of six vertices. (The center of the intermediate cube through which the 3 chord passes is not a vertex of the 24-cell at all: it is the mid-point of the 3 chord, so it lies inside the 24-cell, not on its surface.)
14. ^ There are two 3 triangles in each great circle plane, inscribed in a 1 hexagon.
15. ^ a b Three dimensional rotations occur around an axis line. Four dimensional rotations occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when folding a flat net of 8 cubes up into a tesseract).
16. ^ Each great circle plane intersects with the other great circle planes to which it is not orthogonal at one 4 diameter of the 24-cell. Thus two non-orthogonal great circles share two vertices (unlike two orthogonal great circles which share no points except their common center). Each face plane intersects with the other face planes of its kind to which it is not orthogonal at their characteristic edge. It may seem paradoxical that the face planes of orthogonally-faced cells (such as cubes) intersect at an edge (as they obviously do), since planes are not supposed to be able to intersect in 4-space (except at a single point) if they are perpendicular. The resolution of this apparent paradox is that the face planes of 4-polytope cells are folded in the 4th dimension around their edges of intersection (as the boundary 3-manifold of the 4-polytope is folded around its face planes[o]), so if their dihedral angle is 90 degrees in the boundary 3-space, it is some other angle in 4-space, and they are not orthogonal in 4-space.
17. ^ The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature. The regular 5-cell is not found in the interior any 4-polytope except the 120-cell, but the 24-cell contains irregular 5-cells.
18. ^ The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the 16 hexagonal great circles. There are no planes through exactly 5 vertices. There are two kinds of planes through exactly 4 vertices: the 18 2 square great circles, and 72 1 square (tesseract) faces. There is always a plane through any 3 vertices; some of them are actually through exactly 6 (the 32 3 equilateral triangles in the central hexagons), some are actually through exactly 4 (isosceles right triangles with a 3 hypotenuse in the central 2 squares, and isosceles right triangles with a 2 hypotenuse in the 1 squares), but some are through exactly 3: 96 3 equilateral triangle (5-cell) faces that do not lie in a central plane, and 96 2 equilateral triangle (16-cell) faces; 1 2 isosceles triangles and 1 2 3 scalene right triangles; and 96 1 equilateral triangle (24-cell) faces.
19. ^ The 24-cell's cubical vertex figure has been truncated to a tetrahedral vertex figure (see Kepler's drawing). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges enter at the corners and converge at the center, where they meet but do not cross (since the tetrahedron does not have opposing vertices).
20. ^ The 24-cell's cubical vertex figure has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 2 chords which formerly entered at cube face centers are now 16-cell edges which enter at octahedron corners; but just as before, they converge at the center where 3 straight lines cross perpendicularly. The octahedron corners are located outside the vanished cube, at the mid-points of the 2 chords; recall that before truncation those mid-points occurred at the centers of 6 adjacent cubes face-bonded to the (now-vanished) vertex cube.
21. ^ Starting at any vertex, choose any one of the six 1 square face planes nearest it (any square face of the vertex cube), and slice through those 4 vertices, removing the chosen vertex and exposing a tesseract square face. Now find the 3 triangular face plane just below and parallel to the exposed square face and slice through those 3 vertices, removing the 4 vertices of the square face and exposing a 3 triangular face. Repeat for the other three sides of the tetrahedron.
22. ^ The 24-cell's cubical vertex figure has been truncated to a triangular vertex figure. The vertex cube has vanished, and now there are only 3 corners of the vertex figure where before there were 8. Three 3 chords enter through the vertex triangle's corners and meet at its center. Those corners are located outside the vanished cube, at the mid-points of the 3 chords; recall that before truncation those mid-points occurred at the centers of diagonally adjacent cubes edge-bonded to the (now-vanished) vertex cube. Only three of the 12 3 chords which formerly entered at cube mid-edges remain; they are now three of the 5-cell's 6 3 edges. The 5-cell has 10 edges, but being irregular, it has only 6 3 edges, three of which meet at this vertex. Its other 4 edges are 1 24-cell radii (because the 5-cell's 5th vertex is the 24-cell center). Since those radii are perpendicular to the tetrahedral bounding surface of the irregular 5-cell, each enters a (different) vertex figure perpendicularly from the 4th dimension directly at its center; the 1 line is foreshortened into a point in the perspective view from this vertex.
23. ^ These 3 tetrahedron faces are not the same triangles as the great circle 3 triangles: they are both the same size because they are made from the same 96 3 chords, so they are easily confused, but they lie in different planes, and there are different quantities of them. The 32 great circle triangles lie in 16 hexagonal planes (each with two inscribed 3 triangles) in 4 sets of 4 mutually orthogonal planes, all with their centers at the center of the 24-cell. In contrast, 96 3 triangular tetrahedron faces lie in 24 sets of 4 mutually non-orthogonal planes, none of which pass through the center of the 24-cell: the center of each face plane is 1/2 unit length above the center of the 24-cell. There is only one triangle in each face plane (not two in opposite orientations as in the hexagonal great circle planes). The three vertices that each face triangle connects do not lie in the same great circle of the 24-cell. Each face plane divides the 24-cell into two unequal parts: there are 5 vertices above the plane, 3 and only 3 vertices in the face plane, and 16 below it (not counting the 25th central vertex, 1/2 unit length below the face center).
24. ^ a b One way to visualize the n-dimensional hyperplanes is as the spaces which can be defined by n points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These simplex figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the universe (the enclosing space) into two parts (above and below the hyperplane). The n points bound a finite simplex figure (from the outside), and they define an infinite hyperplane (from the inside).
25. ^ The tetrahedron is one cell of a canonical apex 5-cell because it is the base of a canonical tetrahedral pyramid with its apex at the center of the 24-cell. The base is bent in the fourth dimension completely around the central apex so its vertices are equidistant from it (and slightly closer to it than they would be to the center of an ordinary tetrahedron in 3-dimensional space). Tetrahedral pyramid is another name for a 5-cell. Even though the tetrahedron is regular, the 5-cell is irregular, because its edge length is 3 but its height is 1. Its other four cells are irregular tetrahedra (the sides of the pyramid) which meet at the center. Although irregular, the 5-cell is not degenerate, because its 5 points are not cocellular (the central apex is below the hyperplane[x] defined by the other 4 vertices); the 5-cell defines a 4-space and has non-zero 4-dimensional content.
26. ^ The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell (they lie on its surface).
27. ^ The 4-dimensional content of the unit-edge-length tesseract is 1 (by definition). The content of the unit-edge-length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length 2) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.
28. ^ Each irregular 5-cell is not wholly enclosed within any one tesseract or 16-cell.
29. ^ Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right tetrahedral pyramids, with their apexes filling the corners of the tesseract.
30. ^ This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, however, the boundary 3-spaces of 4-polytopes are bent. The tesseract's boundary 3-manifold (a tessellation of the 3-sphere by 8 cubes) is folded around its square face planes,[o] so that the adjacent face-bonded cubes are oriented with respect to each other differently than they would be in an ordinary 3-dimensional room, and all 6 of the octahedron's vertices lie at the vertex of a cube. This is only possible because the 24-cell's boundary 3-manifold (a tessellation of the 3-sphere by 24 octahedra) is folded differently, around its triangular face planes, i.e. it is folded in different places than the tesseract's 3-manifold. The individual octahedra are not bent at the square cube faces which are their central sections; the individual cubes are not bent either; each 4-polytope's 3-manifold folds only at its own characteristic face planes, independently of the way the other 4-polytope is folded.
31. ^ Consider the three perpendicular 2 long diameters of the octahedron. Two of them are the face diagonals of the square face between the two cubes; each is a 2 chord that connects two vertices of a tesseract cube across its square face, connects two vertices of 16-cell tetrahedron (inscribed in the cube), and connects two vertices of a 24-cell octahedron across its square central section. The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a cube across its square face (but a face of a different pair of cubes, from one of the other tesseracts in the 24-cell).
32. ^ It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).

Citations

1. ^ Coxeter 1973, p. 118, Chapter VII: Ordinary Polytopes in Higher Space.
2. ^ Johnson 2018, p. 249, 11.5.
3. ^ Matila Ghyka, The Geometry of Art and Life (1977), p.68
4. ^ Coxeter 1973, p. 156, §8.7. Cartesian Coordinates.
5. ^ Coxeter 1973, pp. 145-146, §8.1 The simple truncations of the general regular polytope.
6. ^ Coxeter 1973, p. 298, Table V(i), column a: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections.
7. ^ a b Coxeter 1973, p. 150, Gosset.
8. ^ Coxeter 1973, p. 148, §8.2. Cesaro's construction for {3, 4, 3}..
9. ^ Coxeter 1973, p. 302, Table VI (ii), Result column.
10. ^ Coxeter 1973, pp. 149-150, §8.22. see illustrations Fig. 8.2A and Fig 8.2B
11. ^ a b Kepler, Johannes (1619). Harmonicis Mundi, Book V. p. 181.
12. ^ Coxeter 1973, p. 269, §14.32. "For instance, in the case of ${\displaystyle \gamma _{4}[2\beta _{4}]}$ ...."
13. ^ Coxeter 1973, pp. 292-293, Table I (ii) The sixteen regular polytopes {p, q, r} in four dimensions. [An invaluable table providing all 20 metrics of each 4-polytope.]
14. ^ Coxeter 1973, p. 150: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the ${\displaystyle \gamma _{4}}$ . (Their centres are the mid-points of the 24 edges of the ${\displaystyle \beta _{4}}$ .)"
15. ^ Coxeter 1973, p. 12, §1.8. Configurations.
16. ^ Coxeter 1973, p. 156: "...the chess-board has an n-dimensional analogue."
17. ^