24-cell | |
---|---|

Type | Convex regular 4-polytope |

Schläfli symbol | {3,4,3} r{3,3,4} = {3 ^{1,1,1}} = |

Coxeter diagram | or or |

Cells | 24 {3,4} |

Faces | 96 {3} |

Edges | 96 |

Vertices | 24 |

Vertex figure | Cube |

Petrie polygon | dodecagon |

Coxeter group | F_{4}, [3,4,3], order 1152B _{4}, [4,3,3], order 384D _{4}, [3^{1,1,1}], order 192 |

Dual | Self-dual |

Properties | convex, isogonal, isotoxal, isohedral |

Uniform index | 22 |

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 **C _{24}**, or the

- Geometry
- Coordinates
- Constructions
- As a configuration
- Symmetries, root systems, and tessellations
- Quaternionic interpretation
- Voronoi cells
- Radially equilateral honeycomb
- Rotations
- The 3 Cartesian bases of the 24-cell
- Planes of rotation
- Clifford parallel polytopes
- Rings
- Characteristic orthoscheme
- Reflections
- Visualization
- Cell rings
- Parallel projections
- Perspective projections
- Related polytopes
- Three Coxeter group constructions
- Related complex polygons
- Related 4-polytopes
- Related uniform polytopes
- See also
- Notes
- Citations
- References
- External links

The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual.^{ [lower-alpha 1] } It and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.^{ [lower-alpha 2] }

The 24-cell does not have a regular analogue in 3 dimensions. It is the only one of the six convex regular 4-polytopes which is not the four-dimensional analogue of one of the five regular Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the cuboctahedron and its dual the rhombic dodecahedron.

Translated copies of the 24-cell can tile four-dimensional space face-to-face, forming the 24-cell honeycomb. As a polytope that can tile by translation, the 24-cell is an example of a parallelotope, the simplest one that is not also a zonotope.

The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,^{ [lower-alpha 3] } and the polygons {7} and above. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or its honeycomb.

The 24-cell is the fourth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).^{ [lower-alpha 4] } It can be deconstructed into 3 overlapping instances of its predecessor the tesseract (8-cell), as the 8-cell can be deconstructed into 2 overlapping instances of its predecessor the 16-cell.^{ [5] } The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.^{ [lower-alpha 5] }

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

- .

Those coordinates^{ [6] } 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^{ [7] } also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length √2 and is inscribed in a 3-sphere of radius √2. Remarkably, the edge length equals the circumradius, as in the hexagon, or the cuboctahedron. Such polytopes are *radially equilateral*.^{ [lower-alpha 2] }

Regular convex 4-polytopes of radius √2 | |||||||
---|---|---|---|---|---|---|---|

Symmetry group | A_{4} | B_{4} | F_{4} | H_{4} | |||

Name | 5-cell Hyper-tetrahedron | 16-cell Hyper-octahedron | 8-cell Hyper-cube | 24-cell
| 600-cell Hyper-icosahedron | 120-cell Hyper-dodecahedron | |

Schläfli symbol | {3, 3, 3} | {3, 3, 4} | {4, 3, 3} | {3, 4, 3} | {3, 3, 5} | {5, 3, 3} | |

Coxeter mirrors | |||||||

Mirror dihedrals | 𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2 | 𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2 | 𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | |

Graph | |||||||

Vertices | 5 tetrahedral | 8 octahedral | 16 tetrahedral | 24 cubical | 120 icosahedral | 600 tetrahedral | |

Edges | 10 triangular | 24 square | 32 triangular | 96 triangular | 720 pentagonal | 1200 triangular | |

Faces | 10 triangles | 32 triangles | 24 squares | 96 triangles | 1200 triangles | 720 pentagons | |

Cells | 5 tetrahedra | 16 tetrahedra | 8 cubes | 24 octahedra | 600 tetrahedra | 120 dodecahedra | |

Tori | 1 5-tetrahedron | 2 8-tetrahedron | 2 4-cube | 4 6-octahedron | 20 30-tetrahedron | 12 10-dodecahedron | |

Inscribed | 120 in 120-cell | 675 in 120-cell | 2 16-cells | 3 8-cells | 25 24-cells | 10 600-cells | |

Great polygons | 2 𝝅/2 squares x 3 | 4 𝝅/2 rectangles x 3 | 4 𝝅/3 hexagons x 4 | 12 𝝅/5 decagons x 6 | 50 𝝅/15 dodecagons x 4 | ||

Petrie polygons | 1 pentagon | 1 octagon | 2 octagons | 2 dodecagons | 4 30-gons | 20 30-gons | |

Isocline polygrams | 1 octagram _{3} √8 | 2 octagram _{3}√8 | 4 hexagram _{2} √6 | 4 30-gram _{2} √2 | 20 30-gram _{2}√2 | ||

Long radius | |||||||

Edge length | |||||||

Short radius | |||||||

Area | |||||||

Volume | |||||||

4-Content |

The 24 vertices form 18 great squares^{ [lower-alpha 6] } (3 sets of 6 orthogonal^{ [lower-alpha 8] } central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of completely orthogonal^{ [lower-alpha 7] } great squares which intersect^{ [lower-alpha 10] }</ref> at no vertices.^{ [lower-alpha 11] }

The 24-cell is self-dual, having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length √2 is taken by reciprocating it about its *inscribed* sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the *integer* coordinates:

- (±1, 0, 0, 0)

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

- (±1/2, ±1/2, ±1/2, ±1/2)

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

Viewed as quaternions,^{ [lower-alpha 12] } these are the unit Hurwitz quaternions.

The 24-cell has unit radius and unit edge length^{ [lower-alpha 2] } in this coordinate system. We refer to the system as *unit radius coordinates* to distinguish it from others, such as the √2 radius coordinates used above.^{ [lower-alpha 13] }

Regular convex 4-polytopes of radius 1 | |||||||
---|---|---|---|---|---|---|---|

Symmetry group | A_{4} | B_{4} | F_{4} | H_{4} | |||

Name | 5-cell Hyper-tetrahedron | 16-cell Hyper-octahedron | 8-cell Hyper-cube | 24-cell
| 600-cell Hyper-icosahedron | 120-cell Hyper-dodecahedron | |

Schläfli symbol | {3, 3, 3} | {3, 3, 4} | {4, 3, 3} | {3, 4, 3} | {3, 3, 5} | {5, 3, 3} | |

Coxeter mirrors | |||||||

Mirror dihedrals | 𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2 | 𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2 | 𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | |

Graph | |||||||

Vertices | 5 tetrahedral | 8 octahedral | 16 tetrahedral | 24 cubical | 120 icosahedral | 600 tetrahedral | |

Edges | 10 triangular | 24 square | 32 triangular | 96 triangular | 720 pentagonal | 1200 triangular | |

Faces | 10 triangles | 32 triangles | 24 squares | 96 triangles | 1200 triangles | 720 pentagons | |

Cells | 5 tetrahedra | 16 tetrahedra | 8 cubes | 24 octahedra | 600 tetrahedra | 120 dodecahedra | |

Tori | 1 5-tetrahedron | 2 8-tetrahedron | 2 4-cube | 4 6-octahedron | 20 30-tetrahedron | 12 10-dodecahedron | |

Inscribed | 120 in 120-cell | 675 in 120-cell | 2 16-cells | 3 8-cells | 25 24-cells | 10 600-cells | |

Great polygons | 2 𝝅/2 squares x 3 | 4 𝝅/2 rectangles x 3 | 4 𝝅/3 hexagons x 4 | 12 𝝅/5 decagons x 6 | 50 𝝅/15 dodecagons x 4 | ||

Petrie polygons | 1 pentagon | 1 octagon | 2 octagons | 2 dodecagons | 4 30-gons | 20 30-gons | |

Isocline polygrams | 1 octagram _{3} √4 | 2 octagram _{3}√4 | 4 hexagram _{2} √3 | 4 30-gram _{2} √1 | 20 30-gram _{2}√1 | ||

Long radius | |||||||

Edge length | |||||||

Short radius | |||||||

Area | |||||||

Volume | |||||||

4-Content |

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,^{ [lower-alpha 15] } The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of *integer* coordinate vertices (one of the four coordinate axes), and two opposite pairs of *half-integer* coordinate vertices (not coordinate axes). For example:

( 0, 0, 1, 0)

( 1/2,–1/2, 1/2,–1/2) ( 1/2, 1/2, 1/2, 1/2)

(–1/2,–1/2,–1/2,–1/2) (–1/2, 1/2,–1/2, 1/2)

( 0, 0,–1, 0)

is a hexagon on the *y* axis. Unlike the √2 squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.</ref> four of which intersect^{ [lower-alpha 10] } at each vertex.^{ [lower-alpha 17] } By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are Clifford parallel to each other.^{ [lower-alpha 18] }

The 12 axes and 16 hexagons of the 24-cell constitute a Reye configuration, which in the language of configurations is written as 12_{4}16_{3} to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.^{ [8] }

The 24 vertices form 32 equilateral great triangles^{ [lower-alpha 19] } inscribed in the 16 great hexagons.^{ [lower-alpha 20] }

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

Each vertex is joined to 8 others^{ [lower-alpha 21] } by an edge of length 1, spanning 60° = π/3 of arc. Next nearest are 6 vertices^{ [lower-alpha 22] } located 90° = π/2 away, along an interior chord of length √2. Another 8 vertices lie 120° = 2π/3 away, along an interior chord of length √3. The opposite vertex is 180° = π away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center can be treated^{ [lower-alpha 23] } as a 25th canonical apex vertex,^{ [lower-alpha 24] } which is 1 edge length away from all the others.

To visualize how the interior polytopes of the 24-cell fit together (as described below), keep in mind that the four chord lengths (√1, √2, √3, √4) are the long diameters of the hypercubes of dimensions 1 through 4: the long diameter of the square is √2; the long diameter of the cube is √3; and the long diameter of the tesseract is √4.^{ [lower-alpha 25] } Moreover, the long diameter of the octahedron is √2 like the square; and the long diameter of the 24-cell itself is √4 like the tesseract. In the 24-cell, the √2 chords are the edges of central squares, and the √4 chords are the diagonals of central squares.

The vertex chords of the 24-cell are arranged in geodesic great circle polygons.^{ [lower-alpha 27] } The geodesic distance between two 24-cell vertices along a path of √1 edges is always 1, 2, or 3, and it is 3 only for opposite vertices.^{ [lower-alpha 28] }

The √1 edges occur in 16 hexagonal great circles (in planes inclined at 60 degrees to each other), 4 of which cross^{ [lower-alpha 17] } at each vertex.^{ [lower-alpha 16] } and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two √1-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a cubic pyramid . Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its "radius" equals its edge length.^{ [lower-alpha 30] }</ref> The 96 distinct √1 edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.^{ [lower-alpha 18] }

Coxeter plane | F_{4} | |
---|---|---|

Graph | ||

Dihedral symmetry | [12] | |

Coxeter plane | B_{3} / A_{2} (a) | B_{3} / A_{2} (b) |

Graph | ||

Dihedral symmetry | [6] | [6] |

Coxeter plane | B_{4} | B_{2} / A_{3} |

Graph | ||

Dihedral symmetry | [8] | [4] |

The √2 chords occur in 18 square great circles (3 sets of 6 orthogonal planes^{ [lower-alpha 31] }), 3 of which cross at each vertex.^{ [lower-alpha 32] } The 72 distinct √2 chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.^{ [lower-alpha 33] } The 72 √2 chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 √1 edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,^{ [lower-alpha 26] } Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a Hopf link .</ref> such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.^{ [lower-alpha 11] }

The √3 chords occur in 32 triangular great circles in 16 planes, 4 of which cross at each vertex.^{ [lower-alpha 35] } The 96 distinct √3 chords^{ [lower-alpha 19] } run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.^{ [lower-alpha 20] } They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 √1 edges apart on a great circle.^{ [lower-alpha 36] }

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

The sum of the squared lengths^{ [lower-alpha 37] } of all these distinct chords of the 24-cell is 576 = 24^{2}.^{ [lower-alpha 38] } These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they corresponding to diagonal isoclinic rotations rather than simple rotations.^{ [lower-alpha 39] }

The √1 edges occur in 48 parallel pairs, √3 apart. The √2 chords occur in 36 parallel pairs, √2 apart. The √3 chords occur in 48 parallel pairs, √1 apart.^{ [lower-alpha 40] }

The central planes of the 24-cell can be divided into 4 central hyperplanes (3-spaces) each forming a cuboctahedron. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees *and* 60 degrees apart.^{ [lower-alpha 43] } Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).^{ [lower-alpha 44] } Each set of Clifford parallel great circles is a parallel fiber bundle which visits all 24 vertices just once.

Each great circle intersects^{ [lower-alpha 10] } with the other great circles to which it is not Clifford parallel at one √4 diameter of the 24-cell.^{ [lower-alpha 45] } Great circles which are completely orthogonal^{ [lower-alpha 7] } or otherwise Clifford parallel^{ [lower-alpha 26] } do not intersect at all: they pass through disjoint sets of vertices.^{ [lower-alpha 46] }

Triangles and squares come together uniquely in the 24-cell to generate, as interior features,^{ [lower-alpha 23] } all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the 5-cell and the 600-cell).^{ [lower-alpha 47] } 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).^{ [16] } The tesseract gives Gosset's construction^{ [17] } of the 24-cell, equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular.^{ [lower-alpha 48] } The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,^{ [18] } equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described above). The analogous construction in 3-space gives the cuboctahedron (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.^{ [19] }

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.^{ [20] } This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.^{ [lower-alpha 49] }

We can facet the 24-cell by cutting^{ [lower-alpha 50] } through interior cells bounded by vertex chords to remove vertices, exposing the facets of interior 4-polytopes inscribed in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes (above) are only some of those planes. Here we shall expose some of the others: the face planes^{ [lower-alpha 51] } of interior polytopes.^{ [lower-alpha 52] }

Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by √1 edges to remove 8 cubic pyramids whose apexes are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,^{ [lower-alpha 53] } and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a tesseract. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.^{ [lower-alpha 36] } 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.^{ [lower-alpha 54] } They do share 4-content, their common core.^{ [lower-alpha 55] }

Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by √2 chords to remove 16 tetrahedral pyramids whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the √1 edges, exposing √2 chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,^{ [lower-alpha 56] } 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.^{ [lower-alpha 57] } They overlap with each other, but all of their element sets are disjoint:^{ [lower-alpha 58] } they do not share any vertex count, edge length,^{ [lower-alpha 59] } or face area, but they do share cell volume. They also share 4-content, their common core.^{ [lower-alpha 55] }

The 24-cell can be constructed radially from 96 equilateral triangles of edge length √1 which meet at the center of the polytope, each contributing two radii and an edge.^{ [lower-alpha 2] } They form 96 √1 tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length √2, where the three vertices of each triangle are located 90° = π/2 away from each other on the 3-sphere. They form 48 √2 tetrahedra (the cells of the three 16-cells), centered at the 24 mid-edge-radii of the 24-cell.^{ [lower-alpha 59] }

The 24-cell can be constructed directly from its characteristic simplex , the irregular 5-cell which is the fundamental region of its symmetry group F_{4}, by reflection of that 4-orthoscheme in its own cells (which are 3-orthoschemes).^{ [lower-alpha 60] }

The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.^{ [lower-alpha 55] } The tesseracts and the 16-cells are rotated 60° isoclinically^{ [lower-alpha 61] } with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are √3 (120°) apart.^{ [lower-alpha 36] }

The tesseracts are inscribed in the 24-cell^{ [lower-alpha 62] } such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell^{ [lower-alpha 63] } such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior^{ [lower-alpha 64] } 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.^{ [22] }^{ [lower-alpha 49] } This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.^{ [21] } In fact it is the exact dimensional analogy (the demihypercubes), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.^{ [23] }^{ [lower-alpha 59] }

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable^{ [4] } 4-dimensional interstices^{ [lower-alpha 65] } between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are 4-pyramids,^{ [lower-alpha 66] } 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.^{ [lower-alpha 68] } Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.^{ [lower-alpha 67] }

As we saw above, 16-cell √2 tetrahedral cells are inscribed in tesseract √1 cubic cells, sharing the same volume. 24-cell √1 octahedral cells overlap their volume with √1 cubic cells: they are bisected by a square face into two square pyramids,^{ [25] } the apexes of which also lie at a vertex of a cube.^{ [lower-alpha 69] } The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.^{ [lower-alpha 68] }

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

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

The 24 root vectors of the D_{4} root system of the simple Lie group SO(8) form the vertices of a 24-cell. The vertices can be seen in 3 hyperplanes,^{ [lower-alpha 41] } 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 B_{4} and C_{4} 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 F_{4}.^{ [28] } The 24 vertices of the original 24-cell form a root system of type D_{4}; its size has the ratio √2:1. This is likewise true for the 24 vertices of its dual. The full symmetry group of the 24-cell is the Weyl group of F_{4}, which is generated by reflections through the hyperplanes orthogonal to the F_{4} roots. This is a solvable group of order 1152. The rotational symmetry group of the 24-cell is of order 576.

When interpreted as the quaternions,^{ [lower-alpha 12] } the F_{4} 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 D_{4} root lattice is the dual of the F_{4} and is given by the subring of Hurwitz quaternions with even norm squared.^{ [30] }

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.^{ [31] }

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

The Voronoi cells of the D_{4} root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the tessellation of 4-dimensional Euclidean space by regular 24-cells, the 24-cell honeycomb. The 24-cells are centered at the D_{4} lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F_{4} lattice points with odd norm squared. Each 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 **R**^{4}.

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.^{ [lower-alpha 2] }

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.^{ [32] } 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.^{ [33] } Of the 24 center-to-vertex radii^{ [lower-alpha 70] } of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,^{ [17] } but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).^{ [lower-alpha 71] }

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

The regular convex 4-polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations^{ [34] } about a fixed point in 4-dimensional Euclidean space.^{ [lower-alpha 74] }</ref>

There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell honeycomb, depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell)^{ [lower-alpha 14] } was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.^{ [lower-alpha 36] } The distance from one of these orientations to another is an isoclinic rotation through 60 degrees (a double rotation of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).^{ [lower-alpha 75] } This rotation can be seen most clearly in the hexagonal central planes, where the hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.^{ [lower-alpha 15] }

Rotations in 4-dimensional Euclidean space can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.^{ [35] } Thus the general rotation in 4-space is a *double rotation*. There are two important special cases, called a *simple rotation* and an *isoclinic rotation*.^{ [lower-alpha 77] } An **isoclinic rotation** is a different special case, similar but not identical to two simple rotations through the *same* angle.^{ [lower-alpha 75] }</ref>

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

When a 4-polytope is rotating with only one invariant central plane, the same kind of simple rotation is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. (Another difference is that there is more than one plane of rotation, because in four dimensions there is a set of central planes Clifford parallel^{ [lower-alpha 26] } to the invariant plane of rotation.) The fixed plane is the one central plane that is completely orthogonal^{ [lower-alpha 7] } to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex *directly* to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation (and in its Clifford parallel planes) between adjacent vertices of a great hexagon, a great square or a great digon, and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. ^{ [lower-alpha 46] }

The points in the completely orthogonal central plane are not *constrained* to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a double rotation in two perpendicular non-intersecting planes^{ [lower-alpha 9] } of rotation at once.^{ [lower-alpha 76] } In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane *as the whole plane tilts sideways* in the completely orthogonal rotation. A rotation in 4-space always has (at least) *two* completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two chiral forms: *left* and *right* rotations. In a double rotation each vertex moves in a spiral along two completely orthogonal great circles at once.^{ [lower-alpha 78] } Either the path is right-hand threaded (like most screws and bolts), moving along the circles in the "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the right hand rule by which we conventionally say which way is "up" on each of the 4 coordinate axes).

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.^{ [36] }

When the angles of rotation in the two invariant planes are exactly the same, a remarkably symmetric transformation occurs: all the great circle planes Clifford parallel^{ [lower-alpha 26] } to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates isoclinically in many directions at once.^{ [37] } Each vertex moves an equal distance in four orthogonal directions at the same time.^{ [lower-alpha 61] } In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates *all 16* hexagons by 60 degrees, and takes *every* great circle polygon (square,^{ [lower-alpha 42] } hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a *Clifford displacement*, after its discoverer.^{ [lower-alpha 75] }

The 24-cell in the *double* rotation animation appears to turn itself inside out.^{ [lower-alpha 79] } It appears to, because it actually does, reversing the chirality of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).^{ [38] }

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in *two* completely orthogonal planes one of which is a great hexagon,^{ [lower-alpha 46] } each vertex rotates first to a vertex *two* edge lengths away (√3 and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.^{ [lower-alpha 81] } But in an *isoclinic* rotation between two vertices √3 apart (perhaps the same two vertices as in the simple rotation) the vertex moves along a helical arc (not a great circle), which does *not* pass through an intervening vertex: it misses the vertex nearest to its midpoint.</ref> Each √3 chord of the helical geodesic crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.^{ [lower-alpha 83] } The second vertex reached V_{2} is 120 degrees beyond V_{1} along a second √3 chord lying in another hexagonal plane P_{2} that is Clifford parallel to P_{0}.^{ [lower-alpha 85] }, P_{0} and P_{2} are just one √1 edge apart (at every pair of *nearest* vertices).</ref> (Notice that V_{1} lies in both intersecting planes P_{1} and P_{2}, as V_{0} lies in both P_{0} and P_{1}. But P_{0} and P_{2} have *no* vertices in common; they do not intersect.) The third vertex reached V_{3} is 120 degrees beyond V_{2} along a third √3 chord lying in another hexagonal plane P_{3} that is Clifford parallel to P_{1}. V_{0} and V_{3} are adjacent vertices, √1 apart.^{ [lower-alpha 86] } The three √3 chords lie in different 8-cells.^{ [lower-alpha 36] } V_{0} to V_{3} is a 360° isoclinic rotation.</ref> The √3 chords meet at a 60° angle, but since they lie in different planes they form a helix not a triangle. Three √3 chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of √3 chords closes into a loop only after six √3 chords: a 720° rotation twice around the 24-cell on a skew hexagram with √3 edges.^{ [lower-alpha 86] } Even though all the vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation hits only half the vertices in the 24-cell. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has *not* arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees *and* been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's orientation in the 4-space in which it is embedded is now different. Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the *same* direction through another 360 degrees, the moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic *will* arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each hexagram_{2} geodesic to complete a circuit through every *second* vertex of its six vertices by winding around the 24-cell twice, returning the 24-cell to its original chiral orientation.^{ [lower-alpha 87] }

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a Möbius ring, so that the two strands of the double helix form a continuous single strand in a closed loop.^{ [lower-alpha 88] } In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius ring is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex in some rotation.^{ [lower-alpha 39] } They are truly circles, and even form fibrations like ordinary 2-dimensional great circles.^{ [lower-alpha 18] }^{ [lower-alpha 11] } These **isoclines** are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere^{ [lower-alpha 90] } they always occur in chiral pairs which form Villarceau circles on the Clifford torus ,^{ [lower-alpha 91] } the paths of the left and the right isoclinic rotation . They are helices bent into a Möbius loop in the fourth dimension, taking a diagonal winding route twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's skew polygon .</ref>

Two planes are also called *isoclinic* if an isoclinic rotation will bring them together.^{ [lower-alpha 43] } The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.^{ [40] } Clifford parallel great circles do not intersect, so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).^{ [lower-alpha 18] } We can pick out 6 mutually isoclinic (Clifford parallel) great squares^{ [lower-alpha 93] } (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).^{ [lower-alpha 11] }</ref>

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.^{ [42] } Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are *completely disjoint* polytopes.^{ [lower-alpha 58] } A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all double rotations, isoclinic rotations come in two chiral forms: there is a disjoint 16-cell to the *left* of each 16-cell, and another to its *right*.^{ [lower-alpha 57] } Two 16-cells have vertex-pairs which are one √1 edge (one hexagon edge) apart. But a *simple* rotation of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell *can* be taken to another 16-cell by a 60° *isoclinic* rotation , because an isoclinic rotation is 3-sphere symmetric: four Clifford parallel hexagonal planes rotate together, but in four different rotational directions,^{ [lower-alpha 75] } taking each 16-cell to another 16-cell. But since an isoclinic 60° rotation is a *diagonal* rotation by 60° in *two* completely orthogonal directions at once,^{ [lower-alpha 39] } the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: *two*√1 hexagon edges (or one √3 hexagon chord) apart, not one √1 edge (60°) apart.^{ [lower-alpha 61] } By the chiral diagonal nature of isoclinic rotations, the 16-cell *cannot* reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell *beyond* it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic rotation *will* take every 16-cell to another 16-cell: a 60° *right* isoclinic rotation will take the middle 16-cell to the 16-cell we originally visualized as the *left* 16-cell, and a 60° *left* isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the *right* 16-cell. (If so, that was our error in visualization; the 16-cell to the "left" is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)</ref>

All Clifford parallel 4-polytopes are related by an isoclinic rotation,^{ [lower-alpha 75] } but not all isoclinic polytopes are Clifford parallels (completely disjoint).^{ [lower-alpha 94] } The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).^{ [lower-alpha 36] }

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate^{ [lower-alpha 95] } a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The convex regular 4-polytopes nest inside each other, and hide next to each other in the Clifford parallel spaces that comprise the 3-sphere. For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation.^{ [lower-alpha 96] }

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

The 24-cell contains four kinds of geodesic fibers (polygonal rings running through vertices): great circle squares and their isoclinic helix octagrams,^{ [lower-alpha 11] } and great circle hexagons and their isoclinic helix hexagrams.^{ [lower-alpha 18] } It also contains two kinds of cell rings (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.^{ [43] }

Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4√2. The axis can then be bent into a square of edge length √2. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the √2 axes of the four octahedra occupy the same plane, forming one of the 18 √2 great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,^{ [lower-alpha 98] } and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these links all have a common center). A simple rotation in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.^{ [lower-alpha 101] } Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.^{ [lower-alpha 55] }</ref> However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.^{ [lower-alpha 99] } The ring has two sets of three great hexagons, each on three Clifford parallel great circles.^{ [lower-alpha 102] } The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices. A simple rotation in any of the great hexagon planes by a multiple of 60° rotates all three parallel great hexagon planes similarly, and takes each octahedron in the ring to an octahedron in the ring.

Another kind of geodesic fiber, the isoclinic helix hexagrams, can be found within a 6-cell ring of octahedra.^{ [lower-alpha 97] } Each of these geodesics runs through every *second* vertex of a skew hexagram _{2}, which in the unit-radius, unit-edge-length 24-cell has six √3 edges. It does not lie in a single central plane, but is composed of six linked √3 chords from six different hexagonal great circles. This geodesic fiber is the path of an isoclinic rotation, a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.^{ [lower-alpha 39] } Rather than a flat hexagon, it forms a skew hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.^{ [lower-alpha 88] } An isoclinic rotation in any of the great hexagon planes by 60° rotates all *six* great hexagon planes by 60°, and takes each octahedron in the ring to a *non-adjacent* octahedron in the ring.^{ [lower-alpha 103] } Each isoclinically displaced octahedron is itself rotated isoclinically. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original orientation.

Each 6-cell ring contains six such isoclinic skew hexagrams, three left-handed and three right-handed, each left-right pair belonging to a different fibration of hexagrams.^{ [lower-alpha 92] } Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of √3 chords from octahedron to octahedron. In the 24-cell the √1 edges are great hexagon edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The √3 chords are great hexagon diagonals, joining great hexagon vertices two √1 edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each √3 chord is a chord of just one great hexagon (an edge of a great triangle inscribed in that great hexagon), but successive √3 chords belong to different great hexagons.^{ [lower-alpha 83] } At each vertex the isoclinic path of √3 chords bends 60 degrees in two completely orthogonal central planes^{ [lower-alpha 105] } at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.^{ [lower-alpha 106] }</ref> Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram_{2} path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between two hexagonal central planes (like a Petrie polygon), but it is not: any isoclinic path we can pick out always bends either right or left, never changing its inherent chiral "direction", as it visits all six of the great hexagons in the 6-cell ring. When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself (still bending in the same direction).

At each vertex, there are four great hexagons^{ [lower-alpha 107] } and four hexagram isoclines (all four either left-handed right-handed) that cross at the vertex and share a 24-cell axis chord. Each left-right pair at the axis^{ [lower-alpha 108] } is Clifford parallel^{ [lower-alpha 109] } and belongs to one of four different fibrations of the hexagram isoclines. Four left-right pairs of isoclines together (one pair at each axis) visit all 24 vertices and comprise a discrete fibration of the 24-cell's 60° isoclines (just as four great hexagons together, one at each axis, comprise a discrete fibration of the 24-cell's 60° great circles).^{ [lower-alpha 18] }

Eighteen isoclinic helix octagrams (nine left-handed and nine right-handed) can be found within the six orthogonal 4-cell rings of octahedra. (Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis). Three left-right pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the helical construction of the 16-cell. In summary, each 16-cell consists of a left-right pair of 8-cell rings of √2-edged tetrahedral cells. Each 8-cell ring is bounded by three intersecting octagram edge-helices which twist around a common isocline axis that circles through all eight vertices. The chords of the isoclines connect opposite vertices of face-bonded tetrahedral cells, which are also opposite vertices (antipodal vertices) of the 16-cell, so they are √4 chords.

In the 24-cell's 4-cell ring of octahedra with a great square axis, we find antipodal vertices at opposite vertices of the great square. A √4 chord (the diagonal of the great square) connects them; this is a chord of two octagram isoclines (one left-handed and one right-handed). Boundary cells describes how the √2 axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.^{ [lower-alpha 67] } The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.^{ [lower-alpha 54] } In the 24-cell, the 16-cells' isoclines' chords describe an octagram_{4{2}} with √4 edges that run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the twelve √4 axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° completely orthogonally into a different 4-cell ring entirely. The 180° arc of each √4 chord of the octagram runs through the volumes and opposite vertices of two face-bonded √2 tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The arc does not hit any vertices of those two octahedra except the chord endpoints; in particular, it misses the vertex near the chord midpoint where the two octahedra are vertex-bonded. The 720° octagram isocline runs through *one* vertex of *one* octahedron in eight different 4-cell rings (of the 18 4-cell rings in the 24-cell), and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and two octagram isoclines (a left and a right)^{ [lower-alpha 93] } that cross at the vertex and share a 24-cell axis chord.

Characteristics of the 24-cell^{ [44] } | |||||
---|---|---|---|---|---|

edge^{ [45] } | arc | dihedral^{ [46] } | |||

𝒍 | 60° | 120° | |||

𝟀 | 45° | 45° | |||

𝝓 | 30° | 60° | |||

𝟁 | 30° | 60° | |||

45° | 90° | ||||

30° | 90° | ||||

30° | 90° | ||||

Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell.^{ [lower-alpha 60] } The **characteristic 5-cell of the regular 24-cell** is represented by the Coxeter-Dynkin diagram , which can be read as a list of the dihedral angles between its mirror facets.^{ [lower-alpha 110] } It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular octahedron. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.^{ [48] }

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell^{ [lower-alpha 24] }).^{ [lower-alpha 111] } If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths , , (the exterior right triangle face, the *characteristic triangle* 𝟀, 𝝓, 𝟁), plus , , (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the *characteristic radii* of the octahedron), plus , , , (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is , , , , first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

The 24-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).^{ [lower-alpha 112] } Reflections and rotations are related: a reflection in an *even* number of *intersecting* mirrors is a rotation.^{ [49] } Consequently, regular polytopes can be generated by reflections or by rotations. For example, any 720° isoclinic rotation of the 24-cell in a hexagonal invariant plane takes *each* of the 24 vertices to and through 5 other vertices and back to itself, on a skew hexagram_{2} geodesic isocline that winds twice around the 3-sphere on every *second* vertex of the hexagram. Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the three inscribed 16-cells) performing *half* such an orbit visits 3 * 8 = 24 distinct vertices and generates the 24-cell sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of *one* such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.^{ [lower-alpha 83] } The vertex follows an isocline (a doubly curved geodesic circle) rather than any one of the singly curved geodesic circles that are the great circle segments over each √3 chord of the rotation. The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.^{ [lower-alpha 81] } Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a 6-cell ring of spherical^{ [50] } octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

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.^{ [51] } The cell locations lend themselves to a hyperspherical description. Pick an arbitrary cell and label it the "North Pole". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "South Pole" cell. This skeleton accounts for 18 of the 24 cells (2 + 8×2). See the table below.

There is another related great circle in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the hexagonal geodesics described above.^{ [lower-alpha 18] } 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 | 0° | 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 |

The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete Hopf fibration of four interlocking rings.^{ [52] } 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.^{ [lower-alpha 97] }

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.

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.

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.

Animated cross-section of 24-cell | ||

A stereoscopic 3D projection of an icositetrachoron (24-cell). | ||

Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |

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

Three nets of the 24-cell with cells colored by D_{4}, B_{4}, and F_{4} symmetry | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Rectified demitesseract | Rectified 16-cell | Regular 24-cell | |||||||||

D_{4}, [3^{1,1,1}], order 192 | B_{4}, [3,3,4], order 384 | F_{4}, [3,4,3], order 1152 | |||||||||

Three sets of 8 rectified tetrahedral cells | One set of 16 rectified tetrahedral cells and one set of 8 octahedral cells. | One set of 24 octahedral cells | |||||||||

Vertex figure (Each edge corresponds to one triangular face, colored by symmetry arrangement) | |||||||||||

The regular complex polygon _{4}{3}_{4}, or contains the 24 vertices of the 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.^{ [53] }

The regular complex polytope _{3}{4}_{3}, or , in 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.

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

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

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

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

D_{4} uniform polychora | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

{3,3^{1,1}} h{4,3,3} | 2r{3,3^{1,1}} h _{3}{4,3,3} | t{3,3^{1,1}} h _{2}{4,3,3} | 2t{3,3^{1,1}} h _{2,3}{4,3,3} | r{3,3^{1,1}} {3 ^{1,1,1}}={3,4,3} | rr{3,3^{1,1}} r{3 ^{1,1,1}}=r{3,4,3} | tr{3,3^{1,1}} t{3 ^{1,1,1}}=t{3,4,3} | sr{3,3^{1,1}} s{3 ^{1,1,1}}=s{3,4,3} |

24-cell family polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Name | 24-cell | truncated 24-cell | snub 24-cell | rectified 24-cell | cantellated 24-cell | bitruncated 24-cell | cantitruncated 24-cell | runcinated 24-cell | runcitruncated 24-cell | omnitruncated 24-cell | |

Schläfli symbol | {3,4,3} | t_{0,1}{3,4,3}t{3,4,3} | s{3,4,3} | t_{1}{3,4,3}r{3,4,3} | t_{0,2}{3,4,3}rr{3,4,3} | t_{1,2}{3,4,3}2t{3,4,3} | t_{0,1,2}{3,4,3}tr{3,4,3} | t_{0,3}{3,4,3} | t_{0,1,3}{3,4,3} | t_{0,1,2,3}{3,4,3} | |

Coxeter diagram | |||||||||||

Schlegel diagram | |||||||||||

F_{4} | |||||||||||

B_{4} | |||||||||||

B_{3}(a) | |||||||||||

B_{3}(b) | |||||||||||

B_{2} |

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

B4 symmetry polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Name | tesseract | rectified tesseract | truncated tesseract | cantellated tesseract | runcinated tesseract | bitruncated tesseract | cantitruncated tesseract | runcitruncated tesseract | omnitruncated tesseract | ||

Coxeter diagram | = | = | |||||||||

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

Schlegel diagram | |||||||||||

B_{4} | |||||||||||

Name | 16-cell | rectified 16-cell | truncated 16-cell | cantellated 16-cell | runcinated 16-cell | bitruncated 16-cell | cantitruncated 16-cell | runcitruncated 16-cell | omnitruncated 16-cell | ||

Coxeter diagram | = | = | = | = | = | = | |||||

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

Schlegel diagram | |||||||||||

B_{4} |

{3,p,3} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | S^{3} | H^{3} | |||||||||

Form | Finite | Compact | Paracompact | Noncompact | |||||||

{3,p,3} | {3,3,3} | {3,4,3} | {3,5,3} | {3,6,3} | {3,7,3} | {3,8,3} | ... {3,∞,3} | ||||

Image | |||||||||||

Cells | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} | ||||

Vertex figure | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} |

- ↑ The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a polygon nor a simplex. The other two are also 4-polytopes, but not convex: the grand stellated 120-cell and the great 120-cell. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.
- 1 2 3 4 5 6 7 8 The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and tesseract, the three-dimensional cuboctahedron, and the two-dimensional hexagon. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.)
**Radially equilateral**polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. - 1 2 The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the pentagon {5}, the dodecahedron {5, 3}, the 600-cell {3,3,5} and the 120-cell {5,3,3}. In other words, the 24-cell possesses
*all*of the triangular and square features that exist in four dimensions except the regular 5-cell, but*none*of the pentagonal features. (The 5-cell is also pentagonal in the sense that its Petrie polygon is the pentagon.) - ↑ The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is
*rounder*than its predecessor, enclosing more content^{ [4] }within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope. - ↑ The edge length will always be different unless predecessor and successor are
*both*radially equilateral, i.e. their edge length is the*same*as their radius (so both are preserved). Since radially equilateral polytopes^{ [lower-alpha 2] }are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius. - ↑ The edges of six of the squares are aligned with the grid lines of the
*√2 radius coordinate system*. For example:

( 0,–1, 1, 0) ( 0, 1, 1, 0)

( 0,–1,–1, 0) ( 0, 1,–1, 0)

is the square in the*xy*plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90^{o}distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features. - 1 2 3 4 5 6 7 8 9 Two flat planes A and B of a Euclidean space of four dimensions are called
*completely orthogonal*if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.<ref name='six orthogonal planes of the Cartesian basis' group='lower-alpha'> - ↑ Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time. Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is completely orthogonal
^{ [lower-alpha 7] }to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, just as two edges of the tetrahedron are perpendicular and opposite. - 1 2 To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w=0, z=0) shares no axis with the wz central plane (where x=0, y=0). The xy plane exists at only a single instant in time (w=0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).
- 1 2 3 4 Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4)
**they can intersect in a single point**^{ [lower-alpha 9] }(and they*must*, if they are completely orthogonal).^{ [lower-alpha 7] } - 1 2 3 4 5 The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices. Each set constitutes a discrete Hopf fibration of interlocking great circles. The 24-cell can also be divided (three different ways) into 3 disjoint subsets of 8 vertices that do
*not*lie in a square central plane, each skew octagram_{3}forming an isoclinic geodesic or*isocline*that is the path followed by those 8 vertices in one particular isoclinic rotation. Each of these sets of 3 Clifford parallel isoclines corresponds to one of the three discrete Hopf fibrations of great circles.^{ [lower-alpha 92] }polygrams, but the two fiber bundles together constitute the*same*discrete Hopf fibration, because they enumerate the 24 vertices by their intersection. They are the warp and woof of the same woven fabric that is the fibration. - 1 2 3 In four-dimensional Euclidean geometry, a quaternion is simply a (w, x, y, z) Cartesian coordinate. Hamilton did not see them as such when he discovered the quaternions. Schläfli would be the first to consider four-dimensional Euclidean space, publishing his discovery of the regular polyschemes in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.
^{ [29] }Although he described a quaternion as an*ordered four-element multiple of real numbers*, the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions. - ↑ The edges of the orthogonal great squares are
*not*aligned with the grid lines of the*unit radius coordinate system*. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the √2*diagonals*of unit edge length squares of the coordinate lattice. For example:

( 0, 0, 1, 0)

( 0,–1, 0, 0) ( 0, 1, 0, 0)

( 0, 0,–1, 0)

is the square in the*xy*plane. Notice that the 8*integer*coordinates comprise the vertices of the 6 orthogonal squares. - 1 2 3 Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically<ref name='isoclinic 4-dimensional diagonal' group='lower-alpha'>
- 1 2 3 4 The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only
*one*of the 4 coordinate system axes.^{ [lower-alpha 14] }with respect to each other (so their corresponding vertices are 120° = √3 apart). A 16-cell is an orthonormal*basis*for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only*one*axis which is a coordinate system axis. - 1 2 3 Eight √1 edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure
^{ [lower-alpha 29] }That is what serves the illustrative purpose here. - 1 2 3 4 It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the cuboctahedron. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the
*edges*around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical cubic pyramid.^{ [lower-alpha 16] } - 1 2 3 4 5 6 7 8 9 The 24-cell has four sets of 4 non-intersecting Clifford parallel
^{ [lower-alpha 26] }great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices. Each set constitutes a discrete Hopf fibration of interlocking great circles. The 24-cell can also be divided (four different ways) into 4 disjoint subsets of 6 vertices that do*not*lie in a hexagonal central plane, each skew hexagram forming an isoclinic geodesic or*isocline*that is the path followed by those 6 vertices in one particular isoclinic rotation. Each of these sets of four Clifford parallel isoclines corresponds to one of the four discrete Hopf fibrations of great circles.^{ [lower-alpha 92] } - 1 2 These triangles' edges of length √3 are the diagonals of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract) cells are not cells of the unit radius coordinate lattice.
- 1 2 These triangles lie in the same planes containing the hexagons;
^{ [lower-alpha 15] }two triangles of edge length √3 are inscribed in each hexagon. For example, in unit radius coordinates:

( 0, 0, 1, 0)

( 1/2,–1/2, 1/2,–1/2) ( 1/2, 1/2, 1/2, 1/2)

(–1/2,–1/2,–1/2,–1/2) (–1/2, 1/2,–1/2, 1/2)

( 0, 0,–1, 0)

are two opposing central triangles on the*y*axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the √3 triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the √2 squares. - ↑ They surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The vertex figure of the 24-cell is a cube.)
- ↑ They surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.
- 1 2 3 Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in its configuration matrix, which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.
- 1 2 3 The center of the regular 24-cell is a
**canonical apex**of the 24-cell because it is one edge length equidistant from the 24 ordinary vertices in the 4th dimension, as the apex of a canonical pyramid is one edge length equidistant from its other vertices. - ↑ Thus (√1, √2, √3, √4) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.
- 1 2 3 4 5 6 7 8 9 10 11 Clifford parallels are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.
^{ [10] }A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the 3-sphere.^{ [11] }Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.^{ [lower-alpha 31] }Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.^{ [lower-alpha 34] }Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). - ↑ A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does
*not*bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.^{ [lower-alpha 26] } - ↑ If the Pythagorean distance between any two vertices is √1, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is √2, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90
^{o}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 60^{o}bend, or as a straight line with one 60^{o}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). - 1 2 3 4 5 The vertex figure is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a
*full size*vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".<ref name='FOOTNOTEStillwell200117'>Stillwell 2001, p. 17. - ↑ The vertex cubic pyramid is not actually radially equilateral,
^{ [lower-alpha 2] }because the edges radiating from its apex are not actually its radii: the apex of the cubic pyramid is not actually its center, just one of its vertices. - 1 2 3 In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is
*completely orthogonal*to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin. - ↑ Six √2 chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure
^{ [lower-alpha 29] }and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight √1 edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six √2 chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six √2-distant vertices that surrounds the first shell of eight √1-distant vertices. The face-center through which the √2 chord passes is the mid-point of the √2 chord, so it lies inside the 24-cell. - ↑ One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the cuboctahedron (the central hyperplane of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).
- 1 2 Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal to only one of them.<ref name='Clifford parallel squares in the 16-cell and 24-cell' group='lower-alpha'>
- ↑ Eight √3 chords converge from the corners of the 24-cell's cubical vertex figure
^{ [lower-alpha 29] }and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight √3 chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube, which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight √3-distant vertices surrounding the second shell of six √2-distant vertices that surrounds the first shell of eight √1-distant vertices. - 1 2 3 4 5 6 7 The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are √3 (120°) apart (unless they are the same vertex). Each 8-cell contains 8 cubical cells, and each cube contains four √3 chords (its long diagonals). The 8-cells are not completely disjoint
^{ [lower-alpha 58] }(they share vertices), but each cube and each √3 chord belongs to just one 8-cell. The √3 chords joining the corresponding vertices of two 8-cells belong to the third 8-cell. - ↑ The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.
- ↑ The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.
^{ [12] } - 1 2 3 4 5 6 7 A point under isoclinic rotation traverses the diagonal
^{ [lower-alpha 61] }straight line of a single**isoclinic geodesic**, reaching its destination directly, instead of the bent line of two successive**simple geodesics**.^{ [lower-alpha 76] }A**geodesic**is the*shortest path*through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do*not*lie in a single plane; they are 4-dimensional spirals rather than simple 2-dimensional circles.^{ [lower-alpha 78] }But they are not like 3-dimensional screw threads either, because they form a closed loop like any circle (after*two*revolutions).^{ [lower-alpha 88] }Isoclinic geodesics are*4-dimensional great circles*, and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in*two*completely orthogonal directions at once.^{ [lower-alpha 89] }which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres). - ↑ Each pair of parallel √1 edges joins a pair of parallel √3 chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel √2 chords joins another pair of parallel √2 chords to form one of the 18 central squares.
- 1 2 3 4 One way to visualize the
*n*-dimensional hyperplanes is as the*n*-spaces which can be defined by*n + 1*points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These simplex figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the universe (the enclosing space) into two parts (above and below the hyperplane). The*n*points*bound*a finite simplex figure (from the outside), and they*define*an infinite hyperplane (from the inside).^{ [27] }These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane. - 1 2 3 In the 16-cell the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically
^{ [lower-alpha 61] }with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.^{ [lower-alpha 34] }) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell. - 1 2 3 Two angles are required to fix the relative positions of two planes in 4-space.
^{ [13] }Since all planes in the same hyperplane^{ [lower-alpha 41] }are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in*both*angles. Great squares in different hyperplanes are 90 degrees apart in*both*angles (completely orthogonal)^{ [lower-alpha 7] }or 60 degrees apart in*both*angles.^{ [lower-alpha 42] }Planes which are separated by two equal angles are called*isoclinic*. Planes which are isoclinic have Clifford parallel great circles.^{ [lower-alpha 26] }A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle*and*a 60 degree angle. - ↑ Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.
- ↑ Two intersecting great squares or great hexagons share two opposing vertices, but squares and hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.
- 1 2 3 4 5 In the 24-cell each great square plane is completely orthogonal
^{ [lower-alpha 7] }to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great digon plane. - ↑ The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.
^{ [14] }The regular 5-cell is not found in the interior of any convex regular 4-polytope except the 120-cell,^{ [15] }though every convex 4-polytope can be deconstructed into irregular 5-cells. - ↑ This animation shows the construction of a rhombic dodecahedron from a cube, by inverting the center-face-pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process.
^{ [17] } - 1 2 3 Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional hypercube (a tesseract or 8-cell), in dimensional analogy to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the 8-cells' cubic cells. The three pairs of 16-cells form three tesseracts. The tesseracts share vertices, but the 16-cells are completely disjoint.<ref name='completely disjoint' group='lower-alpha'>
- ↑ We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.
- ↑ Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.
^{ [lower-alpha 10] }Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane^{ [lower-alpha 41] }(they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space. - 1 2 The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the
**16 hexagonal great circles**. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 √2 square great circles, the**72 √1 square (tesseract) faces**, and 144 √1 by √2 rectangles. The planes through exactly 3 vertices are the 96 √2 equilateral triangle (16-cell) faces, and the**96 √1 equilateral triangle (24-cell) faces**. There are an infinite number of central planes through exactly two vertices (great circle digons); 16 are distinguished, as each is completely orthogonal^{ [lower-alpha 7] }to one of the 16 hexagonal great circles.**Only the polygons composed of 24-cell √1 edges are visible**in the projections and rotating animations illustrating this article; the others contain invisible interior chords.^{ [lower-alpha 23] } - ↑ The 24-cell's cubical vertex figure
^{ [lower-alpha 29] }has been truncated to a tetrahedral vertex figure (see Kepler's drawing). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices). - 1 2 3 Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.
- 1 2 3 4 The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2. Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.
- ↑ The 24-cell's cubical vertex figure
^{ [lower-alpha 29] }has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 √2 chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices. - 1 2 Visualize the three 16-cells inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes;
^{ [lower-alpha 31] }the other two are rotated 60° isoclinically to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's*surface*), the way the vertices of a cube surround its center.^{ [lower-alpha 16] }The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are √2, each vertex of the compound of three 16-cells is √1 away from its 8 surrounding vertices in other 16-cells. Now visualize those √1 distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The √1 edges form great hexagons of 6 vertices which run around the 24-cell in a central plane.*Four*hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.^{ [lower-alpha 17] }The hexagons are not perpendicular to each other, or to the 16-cells' perpendicular square central planes.^{ [lower-alpha 15] }The left and right 16-cells form a tesseract.^{ [lower-alpha 49] } - 1 2 3 4 Polytopes are
**completely disjoint**if all their*element sets*are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage. - 1 2 3 4 Each of the 72 √2 chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).
- 1 2 An orthoscheme is a chiral irregular simplex with right triangle faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own facets (its
*mirror walls*). Every regular polytope can be dissected radially into instances of its characteristic orthoscheme surrounding its center. The characteristic orthoscheme has the shape described by the same Coxeter-Dynkin diagram as the regular polytope without the*generating point*ring. - 1 2 3 4 5 6 7 In an isoclinic rotation, each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a 4-dimensional diagonal. The point is displaced a total Pythagorean distance equal to the square root of four times the square of that distance. All vertices are displaced to a vertex two edge lengths away in an adjacent cell. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,
^{ [lower-alpha 46] }each vertex is displaced to another vertex √3 (120°) away, moving √3/4 ≈ 0.866 in four orthogonal directions. - ↑ The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.
- ↑ The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.
^{ [lower-alpha 14] } - ↑ 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.
- ↑ 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.
- ↑ 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.
- 1 2 3 Consider the three perpendicular √2 long diameters of the octahedral cell.
^{ [24] }Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a √2 chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).^{ [lower-alpha 59] }The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.^{ [lower-alpha 54] } - 1 2 3 Because there are three overlapping tesseracts inscribed in the 24-cell, each octahedral cell lies
*on*a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and*in*two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).^{ [lower-alpha 67] } - ↑ This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do
*not*lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices*do*lie at the corner of a cube: but a cube in another (overlapping) tesseract.^{ [lower-alpha 68] } - ↑ 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).
- ↑ Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius √2/2. The three 16-cells inscribed in each 24-cell have edge length √2, and unit radius.
- ↑ Three dimensional rotations occur around an axis line. Four dimensional rotations may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when folding a flat net of 8 cubes up into a tesseract). Folding around a square face is just folding around
*two*of its orthogonal edges*at the same time*; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point). - ↑ There are (at least) two kinds of correct dimensional analogies: the usual kind between dimension
*n*and dimension*n*+ 1, and the much rarer and less obvious kind between dimension*n*and dimension*n*+ 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the*n*-sphere rule that the*surface area*of the sphere embedded in*n*+2 dimensions is exactly 2*π r*times the*volume*enclosed by the sphere embedded in*n*dimensions, the most well-known examples being that the circumference of a circle is 2*π r*times 1, and the surface area of the ordinary sphere is 2*π r*times 2*r*. Coxeter cites<ref name='FOOTNOTECoxeter1973119§7.1. Dimensional Analogy'>Coxeter 1973, p. 119, §7.1. Dimensional Analogy: "For instance, seeing that the circumference of a circle is 2*π r*, while the surface of a sphere is 4*π r*^{2}, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2*π*^{2}*r*^{3}." - ↑ Rotations in 4-dimensional Euclidean space may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).
^{ [lower-alpha 72] }But in four dimensions there is yet another way in which rotations can occur, called a**double rotation**. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of**simple rotations**, the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves.*In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves*(as in a 2-dimensional rotation!).^{ [lower-alpha 73] }this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method. - 1 2 3 4 5 In a
*Clifford displacement*, also known as an isoclinic rotation, all the Clifford parallel^{ [lower-alpha 26] }invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted*sideways*by that same angle. A Clifford displacement is 4-dimensionally diagonal.^{ [lower-alpha 61] }Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away. - 1 2 3 4 Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations
*a*and*b*: the*left*double rotation as*a*then*b*, and the*right*double rotation as*b*then*a*. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: it reaches its destination*directly*without passing through the intermediate point touched by*a*then*b*, or the other intermediate point touched by*b*then*a*, by rotating on a single helical geodesic (so it is the shortest path). Conversely, any simple rotation can be seen as the composition of two*equal-angled*double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by Cayley; perhaps surprisingly, this composition*is*commutative, and is possible for any double rotation as well.<ref>Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF).*Adv. Appl. Clifford Algebras*.**27**: 523–538. doi:10.1007/s00006-016-0683-9. hdl: 2117/113067 . S2CID 12350382. - ↑ A rotation in 4-space is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing). Thus the general rotation in 4-space is a
**double rotation**, characterized by*two*angles. A**simple rotation**is a special case in which one rotational angle is 0.^{ [lower-alpha 76] } - 1 2 In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be
*invariant*because the points in each stay in their places in the plane*as the plane moves*, rotating*and*tilting sideways by the same angle that the other plane rotates. - ↑ That a double rotation can turn a 4-polytope inside out is even more noticeable in the tesseract double rotation.
- ↑ The √3 chord passes through the mid-edge of one of the 24-cell's √1 radii. Since the 24-cell can be constructed, with its long radii, from √1 triangles which meet at its center,<ref name='radially equilateral' group='lower-alpha'>
- 1 2 Although adjacent vertices on the geodesic are a √3 chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a
*simple*rotation between two vertices √3 apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway.^{ [lower-alpha 80] }this is a mid-edge of one of the six √1 triangles in a great hexagon, as seen in the chord diagram. - ↑ P
_{0}and P_{1}lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.<ref name='two angles between central planes' group='lower-alpha'> - 1 2 3 Departing from any vertex V
_{0}in the original great hexagon plane of isoclinic rotation P_{0}, the first vertex reached V_{1}is 120 degrees away along a √3 chord lying in a different hexagonal plane P_{1}. P_{1}is inclined to P_{0}at a 60° angle.^{ [lower-alpha 82] } - ↑ V
_{0}and V_{2}are two √3 chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than*one*√3 chord, unless they are antipodal vertices √4 apart.<ref name='Geodesic distance' group='lower-alpha'> - ↑ P
_{0}and P_{2}are 60° apart in*both*angles of separation.^{ [lower-alpha 43] }Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V_{0}and V_{2}are*two*√3 chords apart^{ [lower-alpha 84] }V_{0}and V_{2}are*one*√3 chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their*adjacent*vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics). - 1 2 3 4 Each half of a skew hexagram is an open triangle of three √3 chords, the two open ends of which are one √1 edge length apart. The two halves have the same chirality and are Clifford parallel:<ref name='Clifford parallels' group='lower-alpha'>
- ↑ In a 720° isoclinic rotation of a
*rigid*24-cell the 24 vertices rotate along four separate Clifford parallel hexagram_{2}geodesic loops (six vertices circling in each loop) and return to their original positions. - 1 2 3 Because the 24-cell's helical hexagram
_{2}geodesic is bent into a twisted ring in the fourth dimension like a Möbius strip, its screw thread doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right). The 6-vertex isoclinic path forms a Möbius double helix (like a 3-dimensional double helix with the ends of its two 3-vertex helices joined), a skewed instance of the regular compound polygon denoted {6/2}=2{3} or hexagram_{2}.^{ [lower-alpha 86] }(The isocline also passes through every 4th vertex of a skew regular {12/4}=4{3} dodecagram_{4}.) The 24-cell's Petrie dodecagon zig-zags between adjacent vertices, but unlike the Petrie polygon the hexagram_{2}isocline does not zig-zag, because it hits only every second vertex: it always bends*either*right or left, along a chiral helical geodesic "straight line" or*isocline*.^{ [lower-alpha 39] }The Petrie dodecagon has √1 edges which zig-zag back and forth between the same*two*Clifford parallel great hexagon planes; the isoclinic hexagram_{2}has √3 edges which either zig or zag through every second vertex (along a left or right handed geodesic spiral) visiting*six*great hexagon planes in rotation (switching at each vertex between two sets of three Clifford parallel hexagons), and connecting skew dodecagram_{4}vertices which are 4 vertices apart. Successive √3 edges belong to different great hexagons and different 8-cells, as the 720° isoclinic rotation takes each hexagon to all six hexagons, and each 8-cell to all three 8-cells^{ [lower-alpha 36] }twice in rotation. - ↑ Isoclinic geodesics or
*isoclines*are 4-dimensional great circles in the sense that they are 1-dimensional geodesic*lines*that curve in 4-space in two completely orthogonal planes at once. They should not be confused with*great 2-spheres*,<ref name='FOOTNOTEStillwell200124'>Stillwell 2001, p. 24.