Squares

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

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

Those coordinates^[9] 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^[10] 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	A4	B4		F4	H4
Name	5-cell Hyper-tetrahedron 5-point	16-cell Hyper-octahedron 8-point	8-cell Hyper-cube 16-point	24-cell 24-point	600-cell Hyper-icosahedron 120-point	120-cell Hyper-dodecahedron 600-point
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter mirrors
Mirror dihedrals	𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2	𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2	𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices	5 tetrahedral	8 octahedral	16 tetrahedral	24 cubical	120 icosahedral	600 tetrahedral
Edges	10 triangular	24 square	32 triangular	96 triangular	720 pentagonal	1200 triangular
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 tetrahedra	16 tetrahedra	8 cubes	24 octahedra	600 tetrahedra	120 dodecahedra
Tori	1 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 squares x 3	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	100 irregular hexagons x 4
Petrie polygons	1 pentagon x 3	1 octagon x 3	2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	20 30-gons x 4
Long radius	${\displaystyle {\sqrt {2}}}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle {\sqrt {2}}}$
Edge length	${\displaystyle {\sqrt {5}}\approx 2.236}$	${\displaystyle 2}$	${\displaystyle {\sqrt {2}}\approx 1.414}$	${\displaystyle {\sqrt {2}}\approx 1.414}$	${\displaystyle {\tfrac {\sqrt {2}}{\phi }}\approx 0.874}$	${\displaystyle 2-\phi \approx 0.382}$
Short radius	${\displaystyle {\tfrac {\sqrt {2}}{4}}\approx 0.354}$	${\displaystyle {\tfrac {\sqrt {2}}{2}}\approx 0.707}$	${\displaystyle {\tfrac {\sqrt {2}}{2}}\approx 0.707}$	${\displaystyle 1}$	${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309}$	${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{4}}}\approx 1.309}$
Area	${\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{4}}\right)\approx 21.651}$	${\displaystyle 32\left({\sqrt {3}}\right)\approx 55.425}$	${\displaystyle 48}$	${\displaystyle 96\left({\sqrt {\tfrac {3}{4}}}\right)\approx 83.138}$	${\displaystyle 1200\left({\tfrac {2{\sqrt {3}}}{4\phi ^{2}}}\right)\approx 396.95}$	${\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{4\phi ^{4}}}\right)\approx 180.73}$
Volume	${\displaystyle 5\left({\tfrac {5{\sqrt {10}}}{12}}\right)\approx 6.588}$	${\displaystyle 16\left({\tfrac {2{\sqrt {2}}}{3}}\right)\approx 15.085}$	${\displaystyle 8{\sqrt {8}}\approx 22.627}$	${\displaystyle 24\left({\tfrac {4}{3}}\right)=32}$	${\displaystyle 600\left({\tfrac {4}{12\phi ^{3}}}\right)\approx 47.214}$	${\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}}}\right)\approx 51.246}$
4-Content	${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\sqrt {5}}\right)^{4}\approx 2.329}$	${\displaystyle {\tfrac {8}{3}}\approx 2.666}$	${\displaystyle 4}$	${\displaystyle 8}$	${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 15.451}$	${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 16.770}$

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 11]} at no vertices.^{[lower-alpha 12]}

Hexagons

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

If the dual 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:

$${\displaystyle \left(\pm 1,0,0,0\right)}$$

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

$${\displaystyle \left(\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}\right)}$$

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

Viewed as quaternions,^{[lower-alpha 13]} 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 14]}

Regular convex 4-polytopes of radius 1
Symmetry group	A4	B4		F4	H4
Name	5-cell Hyper-tetrahedron 5-point	16-cell Hyper-octahedron 8-point	8-cell Hyper-cube 16-point	24-cell 24-point	600-cell Hyper-icosahedron 120-point	120-cell Hyper-dodecahedron 600-point
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter mirrors
Mirror dihedrals	𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2	𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2	𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2	𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices	5 tetrahedral	8 octahedral	16 tetrahedral	24 cubical	120 icosahedral	600 tetrahedral
Edges	10 triangular	24 square	32 triangular	96 triangular	720 pentagonal	1200 triangular
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 tetrahedra	16 tetrahedra	8 cubes	24 octahedra	600 tetrahedra	120 dodecahedra
Tori	1 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 squares x 3	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	100 irregular hexagons x 4
Petrie polygons	1 pentagon x 3	1 octagon x 3	2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	20 30-gons x 4
Long radius	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
Edge length	${\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}$	${\displaystyle {\sqrt {2}}\approx 1.414}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle {\tfrac {1}{\phi }}\approx 0.618}$	${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}$
Short radius	${\displaystyle {\tfrac {1}{4}}}$	${\displaystyle {\tfrac {1}{2}}}$	${\displaystyle {\tfrac {1}{2}}}$	${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$	${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$	${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$
Area	${\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}$	${\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}$	${\displaystyle 24}$	${\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}$	${\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}$	${\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}$
Volume	${\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}$	${\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}$	${\displaystyle 8}$	${\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}$	${\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}$	${\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}$
4-Content	${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}$	${\displaystyle {\tfrac {2}{3}}\approx 0.667}$	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}$	${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}$

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,^{[lower-alpha 17]} four of which intersect^{[lower-alpha 11]} at each vertex.^{[lower-alpha 19]} 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 20]}

The 12 axes and 16 hexagons of the 24-cell constitute a Reye configuration, which in the language of configurations is written as 12₄16₃ to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.^[11]

Triangles

The 24 vertices form 32 equilateral great triangles, of edge length √3 in the unit-radius 24-cell,^{[lower-alpha 23]} inscribed in the 16 great hexagons.^{[lower-alpha 24]} Each great triangle is a ring linking three completely disjoint^{[lower-alpha 25]} great squares.^{[lower-alpha 28]}

Hypercubic chords

Vertex geometry of the radially equilateral^{[lower-alpha 2]} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.

The 24 vertices of the 24-cell are distributed^[12] at four different chord lengths from each other: √1, √2, √3 and √4.

Each vertex is joined to 8 others^{[lower-alpha 29]} by an edge of length 1, spanning 60° = π/3 of arc. Next nearest are 6 vertices^{[lower-alpha 30]} 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.^{[lower-alpha 31]} 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 32]} as a 25th canonical apex vertex,^{[lower-alpha 33]} 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 34]} 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.

Geodesics

Stereographic projection of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.

The vertex chords of the 24-cell are arranged in geodesic great circle polygons.^{[lower-alpha 36]} 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 37]}

The √1 edges occur in 16 hexagonal great circles (in planes inclined at 60 degrees to each other), 4 of which cross^{[lower-alpha 19]} at each vertex.^{[lower-alpha 18]} 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 20]}

Orthogonal projections of the 24-cell

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

The √2 chords occur in 18 square great circles (3 sets of 6 orthogonal planes^{[lower-alpha 10]}), 3 of which cross at each vertex.^{[lower-alpha 40]} 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 41]} 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 35]} 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 12]}

The √3 chords occur in 32 triangular great circles in 16 planes, 4 of which cross at each vertex.^{[lower-alpha 31]} The 96 distinct √3 chords^{[lower-alpha 23]} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.^{[lower-alpha 24]} 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 22]}

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 33]}

The sum of the squared lengths^{[lower-alpha 45]} of all these distinct chords of the 24-cell is 576 = 24².^{[lower-alpha 46]} 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 47]}

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 48]}

The central planes of the 24-cell can be divided into 4 orthogonal 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 50]} 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 51]} 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 11]} with the other great circles to which it is not Clifford parallel at one √4 diameter of the 24-cell.^{[lower-alpha 52]} Great circles which are completely orthogonal^{[lower-alpha 7]} or otherwise Clifford parallel^{[lower-alpha 35]} do not intersect at all: they pass through disjoint sets of vertices.^{[lower-alpha 53]}

Reciprocal constructions from 8-cell and 16-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).^[21] The tesseract gives Gosset's construction^[22] 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 55]} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,^[23] 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.^[24]

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.^[25] 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 27]}

Diminishings

We can facet the 24-cell by cutting^{[lower-alpha 56]} 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 57]} of interior polytopes.^{[lower-alpha 58]}

Tetrahedral constructions

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 63]}

The 24-cell can be constructed directly from its characteristic simplex , the irregular 5-cell which is the fundamental region of its symmetry group F₄, by reflection of that 4-orthoscheme in its own cells (which are 3-orthoschemes).^{[lower-alpha 64]}

Relationships among interior polytopes

The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.^{[lower-alpha 61]} The tesseracts and the 16-cells are rotated 60° isoclinically^{[lower-alpha 15]} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are √3 (120°) apart.^{[lower-alpha 22]}

The tesseracts are inscribed in the 24-cell^{[lower-alpha 65]} 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 66]} 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 67]} 16-cell edges have length √2.^{[lower-alpha 28]}

Kepler's drawing of tetrahedra in the cube.^[28]

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.^{[29][lower-alpha 27]} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.^[28] In fact it is the exact dimensional analogy (the demihypercubes), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.^{[30][lower-alpha 63]}

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^[7] 4-dimensional interstices^{[lower-alpha 68]} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are 4-pyramids, alluded to above.^{[lower-alpha 69]}

Boundary cells

Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where 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 71]} 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 70]}

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,^[32] the apexes of which also lie at a vertex of a cube.^{[lower-alpha 72]} 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 71]}

Simple rotations

A 3D projection of a 24-cell performing a simple rotation.^{[lower-alpha 58]}

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 and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed axis of rotation perpendicular to the invariant plane), but the circles do not lie within a central plane.

When a 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. 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 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 53]}

Double rotations

A 3D projection of a 24-cell performing a double rotation.

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 81]} 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.^{[lower-alpha 83]} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.^{[lower-alpha 79]} 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).^[46]

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.^[47]

Isoclinic rotations

When the angles of rotation in the two invariant planes are exactly the same, a remarkably symmetric transformation occurs:^[48] all the great circle planes Clifford parallel^{[lower-alpha 35]} 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.^[49] Each vertex moves an equal distance in four orthogonal directions at the same time.^{[lower-alpha 15]} 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 78]}

The 24-cell in the double rotation animation appears to turn itself inside out.^{[lower-alpha 86]} 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).^[50]

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 53]} 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 21]} Each √3 chord of the helical geodesic^{[lower-alpha 92]} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.^{[lower-alpha 97]} 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^{[lower-alpha 80]} on a skew hexagram with √3 edges.^{[lower-alpha 96]} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation hits only half the vertices in the 24-cell.^{[lower-alpha 91]} 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.^[52] Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic will arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each hexagram₂ 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 104]}

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 99]} 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.^{[lower-alpha 47]}

4-cell rings

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 112]} 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). An isoclinic rotation in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

6-cell rings

A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.^{[lower-alpha 113]}

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 115]} 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 113]} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.^{[lower-alpha 117]} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A simple rotation in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An isoclinic rotation by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a non-adjacent octahedron in the ring.^{[lower-alpha 119]}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original orientation.

Four great hexagons comprise a discrete fiber bundle covering all 24 vertices in a Hopf fibration. Four 6-cell rings comprise the same discrete fibration. The 24-cell has four such discrete hexagonal fibrations, and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles). Each great hexagon belongs to just one fibration,^[59] but each 6-cell ring belongs to three fibrations. The 24-cell contains 16 great hexagons, divided among four fibrations, each of which is a set of four 6-cell rings, but the 24-cell has only four distinct 6-cell rings. Each 6-cell ring contains 3 of the great hexagons in each of three fibrations: only 3 of the 4 Clifford parallel hexagons of each of the three fibrations, and only 18 of the 24 vertices.^{[lower-alpha 108]}

Helical hexagrams and their 4𝝅 isoclines

Another kind of geodesic fiber, the helical hexagram isoclines, can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every second vertex of a skew hexagram₂, which in the unit-radius, unit-edge-length 24-cell has six √3 edges. The hexagram does not lie in a single central plane, but is composed of six linked √3 chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,^{[lower-alpha 47]} 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 85]} 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 99]}

Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.^{[lower-alpha 116]} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.^{[lower-alpha 21]}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of √3 chords of an isocline from octahedron to octahedron. In the 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 97]} At each vertex the isoclinic path of √3 chords bends 60 degrees in two central planes^{[lower-alpha 120]} 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 123]} 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 hexagram₂ 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 three adjacent parallel hexagonal central planes (like a Petrie polygon), but it is not: any isoclinic path we can pick out always zig-zags between two sets of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.^{[lower-alpha 84]} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons^{[lower-alpha 125]} and four hexagram isoclines (all black or all white) that cross at the vertex.^{[lower-alpha 126]} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.^{[lower-alpha 127]} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).^{[lower-alpha 128]} Each isocline is a skew Clifford polygon of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.^{[lower-alpha 85]}

Helical octagrams and their 8𝝅 isoclines

The 24-cell contains 18 helical octagram isoclines, 9 left-handed and 9 right-handed. 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 can be decomposed (three different ways) into a left-right pair of 8-cell rings of √2-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. The chords of these 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, these 18 helical octagram isoclines 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, and 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 each distinct square isoclinic rotation. 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 70]} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.^{[lower-alpha 60]} In the 24-cell, the 16-cells' isocline chords describe an octagram_12{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 12 √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° 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 six 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 six octagram isoclines (three left-right pairs) that cross at the vertex.^{[lower-alpha 88]}

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on 8-chord helical isoclines characteristic of the 16-cell.^{[lower-alpha 129]}

This is the characteristic rotation of the 16-cell, not the 24-cell's characteristic rotation, and it does not take whole 16-cells of the 24-cell to each other the way the 24-cell's rotation in great hexagon planes does.^{[lower-alpha 130]}

Five ways of looking at a skew 24-gram
Edge path	Petrie polygons	In a 600-cell	Discrete fibration	Isocline chords
16-cells3{3/8}	Dodecagons2{12}	24-gram{24/5}	Squares6{4}	{24/12}={12/2}
The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with √2 edges.	2 skew polygons of 12 √1 edges each. The 24-cell can be decomposed into 2 disjoint zig-zag dodecagons (4 different ways).[63]	In compounds of 5 24-cells, isoclines with golden chords of length φ = √2.𝚽 connect all 24-cells in 24-chord circuits.[64]	Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with √2 edges to each other.	Adjacent vertices on an isocline are antipodal vertices of the 24-cell. The isocline chord is the √4 axis diameter.[lower-alpha 129]