600-cell | |
---|---|

Type | Convex regular 4-polytope |

Schläfli symbol | {3,3,5} |

Coxeter diagram | |

Cells | 600 (3.3.3) |

Faces | 1200 {3} |

Edges | 720 |

Vertices | 120 |

Vertex figure | icosahedron |

Petrie polygon | 30-gon |

Coxeter group | H_{4}, [3,3,5], order 14400 |

Dual | 120-cell |

Properties | convex, isogonal, isotoxal, isohedral |

Uniform index | 35 |

In geometry, the **600-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the **C _{600}**,

- Geometry
- Coordinates
- Structure
- Constructions
- As a configuration
- Symmetries
- Visualization
- 2D projections
- 3D projections
- Diminished 600-cells
- Related complex polygons
- Related polytopes and honeycombs
- See also
- Notes
- Citations
- References
- External links

The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex.^{ [lower-alpha 1] } Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-dimensional analogue of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. Its dual polytope is the 120-cell.

The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).^{ [lower-alpha 2] } It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the 24-cell,^{ [4] } as the 24-cell can be deconstructed into three overlapping instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into two 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 3] } The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius. The 600-cell's radius and edge length are in the golden ratio.

Regular convex 4-polytopes | |||||||
---|---|---|---|---|---|---|---|

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 vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length 1/φ ≈ 0.618 (where φ = 1 + √5/2 ≈ 1.618 is the golden ratio), can be given^{ [6] } as follows:

8 vertices obtained from

- (0, 0, 0, ±1)

by permuting coordinates, and 16 vertices of the form:

- (±1/2, ±1/2, ±1/2, ±1/2)

The remaining 96 vertices are obtained by taking even permutations of

- (±φ/2, ±1/2, ±φ
^{−1}/2, 0)

Note that the first 8 are the vertices of a 16-cell, the second 16 are the vertices of a tesseract, and those 24 vertices together are the vertices of a 24-cell. The remaining 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.^{ [7] }

When interpreted as quaternions,^{ [lower-alpha 4] } these are the unit icosians.

In the 24-cell, there are squares, hexagons and triangles that lie on great circles (in central planes through four or six vertices).^{ [lower-alpha 5] } In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each square unique to one 24-cell, each hexagon or triangle shared by two 24-cells, and each vertex shared among five 24-cells.^{ [lower-alpha 7] }

In the 600-cell there are also great circle pentagons and decagons (in central planes through ten vertices).^{ [lower-alpha 9] } Five 24-cells meet at each 600-cell vertex,^{ [lower-alpha 10] } so all 25 24-cells are linked by each great pentagon. Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and also one fifth of all the vertices in each great pentagon in the 600-cell. The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),^{ [lower-alpha 11] } and also into 24 disjoint pentagons (the same 6 different ways for each 24-cell) by choosing appropriately one pentagon of the six that intersect each 24-cell vertex. Appropriately chosen the 24 disjoint pentagons will lie on 12 disjoint Clifford parallel^{ [lower-alpha 12] } decagonal great circles (with two disjoint pentagons inscribed in each decagon) and comprise one of the 6 decagonal fibrations of the 600-cell.</ref>

Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).

By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of completely orthogonal^{ [lower-alpha 14] } squares which do not share any vertices, or as 100 *dual pairs* of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600-cell is captured by the set of Hopf coordinates ^{ [lower-alpha 16] } Hopf coordinates are a natural alternative to Cartesian coordinates^{ [lower-alpha 17] } for framing regular convex 4-polytopes, because the group of 4-dimensional rotations , denoted SO(4), generates those polytopes.</ref> (𝜉_{i}, 𝜂, 𝜉_{j}) given as:

- ({<10}𝜋/5, {≤5}𝜋/10, {<10}𝜋/5)

where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5). The 𝜉_{i} and 𝜉_{j} coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.^{ [lower-alpha 18] }

The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = 𝜋/5, 60° = 𝜋/3, 72° = 2𝜋/5, 90° = 𝜋/2, 108° = 3𝜋/5, 120° = 2𝜋/3, 144° = 4𝜋/5, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron,^{ [lower-alpha 1] } at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V.^{ [9] } These can be seen in the H3 Coxeter plane projections with overlapping vertices colored.^{ [10] }^{ [11] }

These polyhedral sections are *solids* in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface. V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell. Thus V is the apex of a 4-pyramid based on the polyhedron.

The 120 vertices are distributed^{ [12] } at eight different chord lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons.^{ [13] } In ascending order of length, they are √0.𝚫, √1, √1.𝚫, √2, √2.𝚽, √3, √3.𝚽, and √4.^{ [lower-alpha 22] }

Notice that the four hypercubic chords of the 24-cell (√1, √2, √3, √4) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new chord lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio^{ [lower-alpha 19] } including the two golden sections of √5, as shown in the diagram.^{ [lower-alpha 20] }

The 600-cell *rounds out* the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,^{ [lower-alpha 24] } in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.^{ [lower-alpha 25] } The new surface thus formed is a tessellation of smaller, more numerous cells^{ [lower-alpha 26] } and faces: tetrahedra of edge length 1/φ ≈ 0.618 instead of octahedra of edge length 1. It encloses the √1 edges of the 24-cells, which become invisible interior chords in the 600-cell, like the √2 and √3 chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of 1/φ, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not radially equilateral. Like them it is radially triangular in a special way, but one in which golden triangles rather than equilateral triangles meet at the center.^{ [lower-alpha 21] }

The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes). The shape of those interstices must be an octahedral 4-pyramid of some kind, but in the 600-cell it is not regular.^{ [lower-alpha 28] }

The vertex chords of the 600-cell are arranged in geodesic great circle polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.^{ [16] }

The √0.𝚫 = 𝚽 edges form 72 flat regular central decagons, 6 of which cross at each vertex.^{ [lower-alpha 1] } Just as the icosidodecahedron can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 √0.𝚫 edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, √3.𝚽 apart. As in the decagon and the icosidodecahedron, the edges occur in golden triangles ^{ [lower-alpha 27] } which meet at the center of the polytope.^{ [lower-alpha 21] } The 72 great decagons can be divided into 6 sets of 12 non-intersecting Clifford parallel geodesics,^{ [lower-alpha 12] } such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.^{ [18] }

The √1 chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),^{ [lower-alpha 6] } 10 of which cross at each vertex^{ [lower-alpha 32] } (4 from each of five 24-cells, with each hexagon in two of the 24-cells). Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells. The √1 chords join vertices which are two √0.𝚫 edges apart. Each √1 chord is the long diameter of a face-bonded pair of tetrahedral cells (a triangular bipyramid), and passes through the center of the shared face. As there are 1200 faces, there are 1200 √1 chords, in 600 parallel pairs, √3 apart. The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 *dual pairs* in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.^{ [20] } The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.^{ [21] }

The √1.𝚫 chords form 144 central pentagons, 6 of which cross at each vertex.^{ [lower-alpha 9] } The √1.𝚫 chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon. The √1.𝚫 chords join vertices which are two √0.𝚫 edges apart on a geodesic great circle. The 720 √1.𝚫 chords occur in 360 parallel pairs, √2.𝚽 = φ apart.

The √2 chords form 450 central squares (25 disjoint sets of 18), 15 of which cross at each vertex (3 from each of five 24-cells). Each set of 18 squares consists of the 72 √2 chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The √2 chords join vertices which are three √0.𝚫 edges apart (and two √1 chords apart). Each √2 chord is the long diameter of an octahedral cell in just one 24-cell. There are 1800 √2 chords, in 900 parallel pairs, √2 apart. The 450 great squares (225 completely orthogonal^{ [lower-alpha 14] } pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares in each set reach all 120 vertices.^{ [22] }

The √2.𝚽 = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length √3.𝚽. The √2.𝚽 chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three √0.𝚫 edges apart on a geodesic great circle. There are 720 distinct √2.𝚽 chords, in 360 parallel pairs, √1.𝚫 apart.

The √3 chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five 24-cells, with each triangle in two of the 24-cells). Each set of 32 triangles consists of the 96 √3 chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The √3 chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The √3 chords join vertices which are four √0.𝚫 edges apart (and two √1 chords apart on a geodesic great circle). Each √3 chord is the long diameter of two cubic cells in the same 24-cell.^{ [lower-alpha 33] } There are 1200 √3 chords, in 600 parallel pairs, √1 apart.

The √3.𝚽 chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length √1.𝚫, so these are golden triangles.^{ [lower-alpha 27] } The √3.𝚽 chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four √0.𝚫 edges apart on a geodesic great circle. There are 720 distinct √3.𝚽 chords, in 360 parallel pairs, √0.𝚫 apart.

The √4 chords occur as 60 long diameters (75 sets of 4 orthogonal axes), the 120 long radii of the 600-cell. The √4 chords join opposite vertices which are five √0.𝚫 edges apart on a geodesic great circle. There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.^{ [lower-alpha 11] }

The sum of the squared lengths^{ [lower-alpha 34] } of all these distinct chords of the 600-cell is 14,400 = 120^{2}.^{ [lower-alpha 35] } These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices.^{ [lower-alpha 39] } Moreover, 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 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) rotations rather than simple rotations.^{ [lower-alpha 40] }

All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes (𝜋/5 apart), hexagon planes (𝜋/3 apart, also in the 25 inscribed 24-cells), and square planes (𝜋/2 apart, also in the 75 inscribed 16-cells and the 24-cells). These central planes of the 600-cell can be divided into 4 central hyperplanes (3-spaces) each forming an icosidodecahedron. There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.^{ [lower-alpha 42] } groups of 3, everywhere 3 Clifford parallel decagons 36° (𝝅/5) apart form a 30-cell Boerdijk–Coxeter triple helix ring .^{ [lower-alpha 30] } Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° (2𝝅/5) apart, 3 108° (3𝝅/5) apart, and 3 144° (4𝝅/5) apart, for a total of 12 Clifford parallel decagons (120 vertices) that comprise a discrete Hopf fibration.</ref> Great hexagons may be 60° (𝝅/3) apart in one or *both* angles, and may be a multiple (from 1 to 4) of 36° (𝝅/5) apart in one or *both* angles.^{ [lower-alpha 44] } Great squares may be 90° (𝝅/2) apart in one or both angles, may be 60° (𝝅/3) apart in one or both angles, and may be a multiple (from 1 to 4) of 36° (𝝅/5) apart in one or both angles.^{ [lower-alpha 45] } Planes which are separated by two equal angles are called * isoclinic *.^{ [lower-alpha 43] } Planes which are isoclinic have Clifford parallel great circles.^{ [lower-alpha 12] } A great hexagon and a great decagon are neither isoclinic nor Clifford parallel; they are separated by a 𝝅/3 (60°) angle *and* a multiple (from 0 to 4) of 𝝅/5 (36°) angle.</ref> Each great square plane is completely orthogonal^{ [lower-alpha 14] } to another great square plane. Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one √4 long diameter): a great digon plane. Each great decagon plane is completely orthogonal to a plane which intersects *no* vertices: a great 30-gon plane.^{ [lower-alpha 37] }

Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).^{ [lower-alpha 12] } Each fiber bundle of Clifford parallel great circles^{ [lower-alpha 43] } is a discrete Hopf fibration which fills the 600-cell, visiting all 120 vertices just once.^{ [25] }^{ [lower-alpha 46] } The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.^{ [26] }

The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.^{ [lower-alpha 31] } Each fiber bundle^{ [lower-alpha 41] } delineates 20 helical rings of 30 tetrahedral cells each,^{ [lower-alpha 30] } with five rings nesting together around each decagon.^{ [27] } Each tetrahedral cell occupies only one cell ring in each of the 6 fibrations. The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.^{ [lower-alpha 29] }

The fibrations of the 24-cell include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 4 fibrations. The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration.

The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.^{ [lower-alpha 9] } It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. Each fiber bundle^{ [lower-alpha 44] } delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 10 fibrations. The 20 helical rings belong to 5 disjoint 24-cells of 4 helical rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells.

The fibrations of the 16-cell include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each. Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations. The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration.

The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells. It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. Each fiber bundle^{ [lower-alpha 45] } delineates 150 helical rings of 8 tetrahedral cells each.^{ [lower-alpha 47] } Each tetrahedral cell occupies only one cell ring in each of the 15 fibrations.

Because each 16-cell constitutes an orthonormal basis for the choice of a coordinate reference frame, the fibrations of different 16-cells have different natural reference frames. The 15 fibrations of great squares in the 600-cell correspond to the 15 natural reference frames of the 600-cell. One or more of these reference frames is natural to each fibration of the 600-cell. Each fibration of great hexagons has three (equally natural) of these reference frames (as the 24-cell has 3 16-cells); each fibration of great decagons has all 15 (as the 600-cell has 15 disjoint 16-cells).

The densely packed helical cell rings^{ [28] } of fibrations are cell-disjoint, but they share vertices, edges and faces. Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other. The same fibration can also be seen as a minimal *sparse* arrangement of fewer *completely disjoint* cell rings that do not touch at all.^{ [lower-alpha 49] }

The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.^{ [29] } The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell). The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.^{ [lower-alpha 50] } This subset of 4 of 20 cell rings is dimensionally analogous to the subset of 12 of 72 decagons, in that both are sets of completely disjoint Clifford parallel polytopes which visit all 120 vertices.^{ [lower-alpha 51] } The subset of 4 of 20 cell rings is one of 5 fibrations *within* the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure with *subfibrations*.

The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon. Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings.

The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.^{ [lower-alpha 47] }

The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or 16-cell with cells of different colors to distinguish the cell rings from the spaces between them.^{ [lower-alpha 52] } The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous to the snub tetrahedron form of the icosahedron (which is the *base*^{ [lower-alpha 46] } of these fibrations on the 2-sphere). Each of the 20 Boerdijk-Coxeter cell rings^{ [lower-alpha 30] } is *lifted* from a corresponding *face* of the icosahedron.^{ [lower-alpha 53] }

The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.^{ [30] } Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial. The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize.

Thorold Gosset discovered the semiregular 4-polytopes, including the snub 24-cell with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form. In the first, more complex step (described elsewhere) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the golden sections of its edges.^{ [7] } In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.^{ [31] }

The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,^{ [lower-alpha 24] } leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.^{ [lower-alpha 1] } The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell.

Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires *three* steps. The 24-cell precursor to the snub-24 cell is *not* of the same radius: it is larger, since the snub-24 cell is its truncation. Starting with the unit-radius 24-cell, the first step is to reciprocate it around its midsphere to construct its outer canonical dual: a larger 24-cell, since the 24-cell is self-dual. That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell.

Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional surface envelope,^{ [lower-alpha 26] } or how they lie on the underlying surface envelope of the 24-cell's octahedral cells. For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.

Most of us have difficulty visualizing the 600-cell *from the outside* in 4-space, or recognizing an outside view of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,^{ [32] } but we should be able to visualize the surface envelope of 600 cells *from the inside* because that volume is a 3-dimensional space that we could actually "walk around in" and explore.^{ [33] } In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, closed curved space, in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point.

The vertex figure of the 600-cell is the icosahedron.^{ [lower-alpha 1] } Twenty tetrahedral cells meet at each vertex, forming an icosahedral pyramid whose apex is the vertex, surrounded by its base icosahedron. The 600-cell has a dihedral angle of 𝜋/3 + arccos(−1/4) ≈ 164.4775°.^{ [35] }

An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five). Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells. Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.^{ [lower-alpha 56] }.

The apexes of the 24 icosahedral pyramids are the vertices of 24-cells inscribed in the 600-cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed snub 24-cell, which has exactly the same structure of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells.

The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces. Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.^{ [lower-alpha 57] } Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,^{ [lower-alpha 10] } and the 120 vertices comprise 25 (not 5) 24-cells.^{ [lower-alpha 25] }

The icosahedra are face-bonded into geodesic "straight lines" by their opposite faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids. Their apexes are the vertices of a great circle hexagon. This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each triangular bipyramid) is a hexagon edge (a 24-cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking rings of 6 octahedra in the 24-cell (a hexagonal fibration).^{ [lower-alpha 59] }

The tetrahedral cells are face-bonded into triple helices, bent in the fourth dimension into rings of 30 tetrahedral cells.^{ [lower-alpha 30] } The three helices are geodesic "straight lines" of 10 edges: great circle decagons which run Clifford parallel^{ [lower-alpha 12] } to each other. Each tetrahedron, having six edges, participates in six different decagons^{ [lower-alpha 29] } and thereby in all 6 of the decagonal fibrations of the 600-cell.

The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same. One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.^{ [lower-alpha 55] } Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.^{ [lower-alpha 9] }

There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure^{ [39] } and a direct construction of the 600-cell from its predecessor the 24-cell.

Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells. The central cell is the first section of the 600-cell beginning with a cell. By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell.

First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra (triangular dipyramids) whose long diameter is a 24-cell edge (a hexagon edge) of length √1. Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,^{ [lower-alpha 60] } so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length √1. They form a tetrahedron of edge length √1, which is the second section of the 600-cell beginning with a cell.^{ [lower-alpha 61] } There are 600 of these √1 tetrahedral sections in the 600-cell.^{ [lower-alpha 62] }

With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster. The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length √1, obviously the cell of a 24-cell. As partially filled so far (by 17 tetrahedral cells), this √1 octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.^{ [lower-alpha 63] } Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.^{ [lower-alpha 64] }

Thus the unit-radius 600-cell may be constructed directly from its predecessor,^{ [lower-alpha 28] } the unit-radius 24-cell, by placing on each of its octahedral facets a truncated^{ [lower-alpha 65] } irregular octahedral pyramid of 14 vertices^{ [lower-alpha 66] } constructed (in the above manner) from 25 regular tetrahedral cells of edge length 1/φ ≈ 0.618.

There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure^{ [40] } and the decagonal fibrations of the 600-cell. An entire 600-cell can be assembled from 2 rings of 5 icosahedral pyramids, bonded vertex-to-vertex into geodesic "straight lines", plus 40 10-cell rings which fill the voids remaining between the icosahedra.

The 120-cell can be decomposed into two disjoint tori.^{ [41] } Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.

Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron. You can view this as five vertex stacked icosahedral pyramids, with the five extra annular ring gaps also filled in.^{ [lower-alpha 69] } The surface is the same as that of ten stacked pentagonal antiprisms: a triangular-faced column with a pentagonal cross-section.^{ [lower-alpha 70] } Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces, 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face.^{ [lower-alpha 72] } This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges. The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons). These decagons spiral around the center core decagon, but mathematically they are all equivalent (they all lie in central planes).

Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the duocylinder and form a Clifford torus. They can be "unrolled" into a square 10x10 array.^{ [lower-alpha 67] } Incidentally this structure forms one tetrahedral layer in the tetrahedral-octahedral honeycomb. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori. In this case into each recess, instead of an octahedron as in the honeycomb, fits a triangular bipyramid composed of two tetrahedra.^{ [lower-alpha 73] }

This decomposition of the 600-cell has symmetry [[10,2^{+},10]], order 400, the same symmetry as the grand antiprism. The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a pentagonal antiprism).^{ [lower-alpha 74] }

The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete fibration of 12 decagons that reaches all 120 vertices, despite filling only half the 600-cell with cells.

A single 30-tetrahedron Boerdijk–Coxeter helix ring within the 600-cell, seen in stereographic projection. ^{ [lower-alpha 30] } | A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell. ^{ [lower-alpha 39] } | The 30 vertices of the 30-cell ring lie on a skew star 30-gon with a winding number of 11. ^{ [lower-alpha 38] } |

The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells, each ten edges long, forming a discrete Hopf fibration which fills the entire 600-cell.^{ [42] } These chains of 30 tetrahedra each form a Boerdijk–Coxeter helix.^{ [lower-alpha 30] } The center axis of each helix is a great 30-gon geodesic that does not intersect any vertices.^{ [lower-alpha 39] } The 30 vertices of the 30-cell ring form a skew compound 30-gon (a skew 30-gram) with a geodesic orbit that winds around the 600-cell twice.^{ [lower-alpha 38] } the winding 30-gram orbit through vertices 2 apart is a chiral geodesic isocline , the path of an isoclinic rotation .^{ [lower-alpha 40] }</ref> The dual of the 30-cell ring (the 30-gon made by connecting its cell centers) is a skew 30-gon Petrie polygon.^{ [lower-alpha 78] } Five of these 30-cell helices nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described above.^{ [43] } Thus *every* great decagon is the center core decagon of a 150-cell torus.^{ [lower-alpha 80] }

The 20 cell-disjoint 30-cell rings constitute four identical cell-disjoint 150-cell tori: the two described in the grand antiprism decomposition above, and two more that fill the middle layer of 300 tetrahedra occupied by 30 10-cell rings in the grand antiprism decomposition.^{ [lower-alpha 81] } The four 150-cell rings spiral around each other and pass through each other in much the same manner as the 20 30-cell rings or the 12 great decagons;^{ [lower-alpha 79] } these three sets of Clifford parallel polytopes are the same discrete decagonal fibration of the 600-cell.^{ [lower-alpha 73] }

The regular convex 4-polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations^{ [44] } about a fixed point in 4-dimensional Euclidean space.^{ [lower-alpha 84] } An **isoclinic rotation** is a different special case, similar but not identical to two simple rotations through the *same* angle.^{ [lower-alpha 40] } straight line of a single **isoclinic geodesic**, reaching its destination directly, instead of the bent line of two successive **simple geodesics**. 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 86] } But they are not like 3-dimensional screw threads either, because they form a closed loop like any circle (after *two* revolutions).^{ [lower-alpha 76] } it does not zig-zag between two planes: it always bends *either* right or left, along a chiral helical path that visits *three* planes in rotation. The Möbius helix is a geodesic "straight line" or * isocline *. The isocline connects the vertices of a lower frequency (longer wavelength) skew polygon than the Petrie polygon: a skew regular {30/2}=2{15} compound triacontagram rather than a regular skew triacontagon . The Petrie triacontagon has √0.𝚫 edges; the isoclinic triacontagram_{2} has √1 edges which join vertices which are two √0.𝚫 edges apart. But the two helical strands of its continuous double helix loop are √0.𝚫 apart at every pair of nearest vertices, just like the Petrie polygon: they are Clifford parallel.^{ [lower-alpha 12] } Each √1 edge belongs to a different great hexagon , and successive √1 edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.</ref> 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 87] } They are truly circles, and even form fibrations like ordinary 2-dimensional great circles. These ** isoclines ** are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere^{ [lower-alpha 88] } they always occur in chiral pairs and form Villarceau circles on the Clifford torus ,^{ [lower-alpha 89] } 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 vertices of a 4-polytope's skew polygon .</ref></ref>

The 600-cell is generated by isoclinic rotations ^{ [lower-alpha 40] } of the 24-cell by 36° = 𝜋/5 (the arc of one 600-cell edge length).^{ [lower-alpha 91] }

There are 25 inscribed 24-cells in the 600-cell. Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.^{ [lower-alpha 25] }

The 8-vertex 16-cell has 4 long diameters inclined at 90° = 𝜋/2 to each other, often taken as the 4 orthogonal axes or basis of the coordinate system.

The 24-vertex 24-cell has 12 long diameters inclined at 60° = 𝜋/3 to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by 𝜋/3 with respect to each other.^{ [lower-alpha 92] }

The 120-vertex 600-cell has 60 long diameters: *not just* 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.^{ [47] } There *are* 5 disjoint 24-cells in the 600-cell, but not *just* 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.^{ [lower-alpha 11] }

Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually isoclinic polytopes. The rotational distance between inscribed 24-cells is always an equal-angled rotation of 𝜋/5 in each pair of completely orthogonal invariant planes of rotation.^{ [lower-alpha 90] }

Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are 𝜋/5 apart on two non-intersecting Clifford parallel^{ [lower-alpha 12] } decagonal great circles (as well as 𝜋/5 apart on the same decagonal great circle).^{ [lower-alpha 31] } An isoclinic rotation of decagonal planes by 𝜋/5 takes each 24-cell to a disjoint 24-cell (just as an isoclinic rotation of hexagonal planes by 𝜋/3 takes each 16-cell to a disjoint 16-cell).^{ [lower-alpha 93] } Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the *left* of each 24-cell, and another 4 disjoint 24-cells to its *right*.^{ [lower-alpha 95] } and left or right to a more distant 24-cell from which it is completely disjoint.^{ [lower-alpha 49] } The four directions reach 8 different 24-cells^{ [lower-alpha 6] } because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once. Four paths are right-hand threaded (like most screws and bolts), moving along the circles in the "same" directions, and four are 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).</ref> The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells.

All Clifford parallel polytopes are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).^{ [lower-alpha 96] } Each 24-cell is isoclinic *and* Clifford parallel to 8 others, and isoclinic but *not* Clifford parallel to 16 others.^{ [lower-alpha 6] } With each of the 16 it shares 6 vertices: a hexagonal central plane.^{ [lower-alpha 10] } Non-disjoint 24-cells are related by a simple rotation by 𝜋/5 in an invariant plane intersecting only two vertices of the 600-cell, a rotation in which the completely orthogonal fixed plane is their common hexagonal central plane. They are also related by an isoclinic rotation in which both planes rotate by 𝜋/5.^{ [lower-alpha 98] }

There are two kinds of 𝜋/5 isoclinic rotations which take each 24-cell to another 24-cell.^{ [lower-alpha 93] } takes *every* central polygon , geodesic cell ring or inscribed 4-polytope^{ [lower-alpha 25] } in the 600-cell to a Clifford parallel polytope 𝜋/5 away.</ref>*Disjoint* 24-cells are related by a 𝜋/5 isoclinic rotation of an entire fibration of 12 Clifford parallel *decagonal* invariant planes. (There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.)^{ [lower-alpha 95] }*Non-disjoint* 24-cells are related by a 𝜋/5 isoclinic rotation of an entire fibration of 20 Clifford parallel *hexagonal* invariant planes.^{ [lower-alpha 100] } (There are 10 such sets of fibers, so there are 20 such distinct rotations.)^{ [lower-alpha 97] }

On the other hand, each of the 10 sets of five *disjoint* 24-cells is Clifford parallel because its corresponding great *hexagons* are Clifford parallel. (24-cells do not have great decagons.) The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel geodesics, each set of which covers all 24 vertices of the 24-cell. The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, each set of which covers all 120 vertices and constitutes a discrete hexagonal fibration. Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell. Similarly, the corresponding great *squares* of disjoint 24-cells are Clifford parallel.

Just as the geodesic *polygons* (decagons or hexagons or squares) in central planes form fiber bundles of Clifford parallel *great circles*, the corresponding geodesic skew * polygrams * (pentagrams or hexagrams or octagrams) on the Clifford torus form fiber bundles of Clifford parallel *isoclines*: helical circles which wind through all four dimensions. Each polygon fiber bundle has its corresponding polygram fiber bundle: they are two aspects of the same Hopf fibration, not two distinct fibrations, because they are both the expression of the same distinct isoclinic rotation.^{ [lower-alpha 40] }

Isoclinic rotations rotate a rigid object's vertices along parallel paths in two completely orthogonal directions, the way a loom weaves a piece of fabric from two orthogonal sets of parallel fibers. A set of Clifford parallel great circles and a set of Clifford parallel isoclines are the warp and woof of the same distinct isoclinic rotation, which takes Clifford parallel polygons to each other, flipping them like coins and rotating them through the Clifford parallel set of central planes. Meanwhile, because the polygons are also rotating individually, vertices are displaced along twisting Clifford parallel isoclines, through vertices which lie in successive Clifford parallel polygons.

In the 600-cell, each set of similar isoclinic skew polygrams (pentagrams or hexagrams or octagrams) can be divided into bundles of non-intersecting Clifford parallel isoclines (of 24 pentagrams or 20 hexagrams or 18 octagrams). Pairs of polygrams of *left* and *right* chirality occur in the same fibration. The polygrams are chiral objects but the fibration itself and its great circle polygons are not.^{ [lower-alpha 48] } Each fiber bundle of Clifford parallel isoclines is a discrete chiral Hopf fibration which fills the 600-cell, visiting all 120 vertices just once. It is a *different bundle of fibers* than the bundle of Clifford parallel great circles, but the two fiber bundles are the *same fibration* because they enumerate those 120 vertices together, by their intersection in the same fabric of woven parallel fibers.

Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle. There are two ways they can do this: by both rotating in the "same" direction, or by rotating in "opposite" directions (according to the right hand rule by which we conventionally say which way is "up" on each of the 4 coordinate axes). The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions. Left and right are different rotations that go to different places. In addition, each distinct isoclinic rotation (left or right) can be performed in a "forward" or "backward" direction along the parallel fibers.

The isoclines in each chiral bundle spiral around each other: they are axial geodesics around which the helical rings of face-bonded cells twist.^{ [lower-alpha 101] } Those Clifford parallel cell rings nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets. A fiber bundle of Clifford parallel isoclines and the corresponding bundle of cell rings belong to the same isoclinic fibration. The fibers are the parallel vertex circles of a particular left or right isoclinic rotation, in which a particular set of Clifford parallel central planes are the invariant planes of rotation.

A simple rotation is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane.^{ [lower-alpha 102] } In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration.

An isoclinic rotation is diagonal and global, taking *all* the vertices to *non-adjacent* vertices (two edges away along great circles, in an adjacent cell) along diagonal isoclines, and *all* the central plane polygons to Clifford parallel polygons (of the same kind). The invariant planes of the isoclinic rotation are a fibration of great circle fibers. The cell rings are a fibration of isocline fibers running through them. These two fiber bundles are the *same* discrete Hopf fibration. All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by *two* equal angles and lying in different hyperplanes.^{ [lower-alpha 42] } The diagonal isocline^{ [lower-alpha 85] } is a shorter route between the non-adjacent vertices than the multiple simple routes available along two edges on great circles: it is the *shortest route* by isoclinic rotation, the geodesic.

Three orthogonal projections of the isocline chords of the 30-cell ring^{ [lower-alpha 103] } | ||
---|---|---|

Triacontagram {30/2}=2{15} a compound of two 15-gons containing each other's antipodal vertices. Chords link every 2nd vertex of the 30-gram. | Triacontagram {30/12}=6{5/2} a 6-compound pentagram. Interior angles are 36°. The pentagrams (of different colors) link every 12th vertex of the 30-gram. | Triacontagram {30/11} 30 connected chords linking every 11th vertex in a single isocline circle. Equivalent to six open-ended pentagrams connected end-to-end. |

In the 600-cell, all three triacontagrams' 30 chords have length √1 and are also great hexagon edges (edges of inscribed 24-cells). In this article, they should all properly be drawn as dashed lines, because they are invisible interior chords of the 600-cell, not 600-cell edges. |

The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons,^{ [lower-alpha 31] } each delineating 20 helical cell rings (10 left-spiraling and 10 right-spiraling) of 30 tetrahedral cells each,^{ [lower-alpha 30] } with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagon fibers of this fibration are crossed by another set of Clifford parallel geodesic fibers that are a different kind of circle:^{ [lower-alpha 40] } namely, isocline compound pentagram fibers which are skew 30-gram helices, each consisting of 6 open pentagrams joined end-to-end, and each winding twice around the 600-cell through all four dimensions rather than lying flat in a central plane like a great circle polygon. The fibration is a fabric woven of these two different kinds of parallel circular fibers, which intersect each other, but nowhere intersect the parallel fibers of their own kind.

One 30-gram isocline is axial to each 30-cell ring; both are chiral (either left or right).^{ [lower-alpha 82] } The fibration's 20 30-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one √0.𝚫 edge-length apart). The 30 chords joining the isocline's 30 vertices are √1 hexagon edges (24-cell edges), connecting 600-cell vertices which are *two* 600-cell √0.𝚫 edges apart on a decagon great circle. These isocline chords are both hexa*gon* edges and penta*gram* edges.^{ [lower-alpha 103] }

Each fibration contains 20 30-cell rings, each with an axial 30-chord isocline, so the fibration contains 120 open-ended skew pentagrams. In all six fibrations, the 600-cell contains 120 30-cell rings and 720 skew pentagrams.^{ [lower-alpha 80] }

The 30-cell rings and their axial isoclines are chiral objects; 60 spiral clockwise (right) and 60 spiral counterclockwise (left).^{ [lower-alpha 79] } Each pair of left and right isoclinic rotations in decagon invariant planes partitions the vertices (and the cells) of the 600-cell into two disjoint sets of 60 vertices (and 300 cells) reached only by the left and right rotations respectively. With respect to that distinct isoclinic rotation, the vertices (and cells) alternate as left and right vertices (and left and right cells^{ [lower-alpha 105] }) like the black and white squares of the chessboard. At each of the 120 vertices, there are 6 great decagons and 6 isoclines (one of each from each fibration) that cross there.^{ [lower-alpha 106] }

The 600-cell can be constructed radially from 720 golden triangles of edge lengths √0.𝚫√1√1 which meet at the center of the 4-polytope, each contributing two √1 radii and a √0.𝚫 edge.^{ [lower-alpha 21] } They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral √0.𝚫 bases (the faces of the 600-cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular √0.𝚫 tetrahedron bases (the cells of the 600-cell).

Characteristics of the 600-cell^{ [49] } | |||||
---|---|---|---|---|---|

edge^{ [50] } | arc | dihedral^{ [51] } | |||

𝒍 | 36° | 164°29′ | |||

𝟀 | 22°15′20″ | 60° | |||

𝝓 | 18° | 36° | |||

𝟁 | 17°44′40″ | 60° | |||

22°15′20″ | 90° | ||||

18° | 90° | ||||

17°44′40″ | 90° | ||||

37°44′40″ |

Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell.^{ [lower-alpha 107] } The **characteristic 5-cell of the regular 600-cell** is represented by the Coxeter-Dynkin diagram , which can be read as a list of the dihedral angles between its mirror facets. It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.

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 600-cell).^{ [lower-alpha 108] } If the regular 600-cell has unit radius and edge length , 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 regular tetrahedron), plus , , , (edges which are the characteristic radii of the 600-cell). The 4-edge path along orthogonal edges of the orthoscheme is , , , , first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.

The 600-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).^{ [lower-alpha 109] } Reflections and rotations are related: a reflection in an *even* number of *intersecting* mirrors is a rotation.^{ [52] } For example, any 720° isoclinic rotation of the 600-cell in a decagonal invariant plane takes *each* of the 120 vertices to and through 29 other vertices and back to itself, on a skew triacontagram_{2} geodesic isocline that winds twice around the 3-sphere.^{ [lower-alpha 38] } Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells)^{ [lower-alpha 50] } performing *half* such an orbit visits 15 * 8 = 120 distinct vertices and generates the 600-cell sequentially by a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.

This configuration matrix ^{ [53] } represents the 600-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 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

Here is the configuration expanded with *k*-face elements and *k*-figures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal.

H_{4} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | k-fig | Notes | |
---|---|---|---|---|---|---|---|---|---|

H_{3} | ( ) | f_{0} | 120 | 12 | 30 | 20 | {3,5} | H_{4}/H_{3} = 14400/120 = 120 | |

A_{1}H_{2} | { } | f_{1} | 2 | 720 | 5 | 5 | {5} | H_{4}/H_{2}A_{1} = 14400/10/2 = 720 | |

A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 1200 | 2 | { } | H_{4}/A_{2}A_{1} = 14400/6/2 = 1200 | |

A_{3} | {3,3} | f_{3} | 4 | 6 | 4 | 600 | ( ) | H_{4}/A_{3} = 14400/24 = 600 |

The icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell.^{ [54] } The icosians lie in the *golden field*, (*a* + *b*√5) + (*c* + *d*√5)**i** + (*e* + *f*√5)**j** + (*g* + *h*√5)**k**, where the eight variables are rational numbers.^{ [55] } The finite sums of the 120 unit icosians are called the icosian ring.

When interpreted as quaternions,^{ [lower-alpha 4] } the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by *2I* as it is the double cover of the ordinary icosahedral group *I*.^{ [57] } It occurs twice in the rotational symmetry group *RSG* of the 600-cell as an invariant subgroup, namely as the subgroup *2I _{L}* of quaternion left-multiplications and as the subgroup

The binary icosahedral group is isomorphic to SL(2,5).

The full symmetry group of the 600-cell is the Weyl group of H_{4}.^{ [58] } This is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S.L. van Oss.^{ [59] }

The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,^{ [lower-alpha 26] } and the fact that the tetrahedron has no opposing faces or vertices.^{ [lower-alpha 48] } One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,^{ [30] } which with some effort can be seen in most of the below perspective projections.

The H3 decagonal projection shows the plane of the van Oss polygon.

H_{4} | - | F_{4} |
---|---|---|

[30] (Red=1) | [20] (Red=1) | [12] (Red=1) |

H_{3} | A_{2} / B_{3} / D_{4} | A_{3} / B_{2} |

[10] (Red=1,orange=5,yellow=10) | [6] (Red=1,orange=3,yellow=6) | [4] (Red=1,orange=2,yellow=4) |

A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by Man Ray, and formed part of two of his later "Shakesperean Equation" paintings.^{ [60] }

Frame synchronized orthogonal isometric (left) and perspective (right) projections |
---|

The snub 24-cell may be obtained from the 600-cell by removing the vertices of an inscribed 24-cell and taking the convex hull of the remaining vertices. This process is a * diminishing * of the 600-cell.

The grand antiprism may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.

A bi-24-diminished 600-cell, with all tridiminished icosahedron cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells.

There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.^{ [61] }

Diminished 600-cells | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Name | Tri-24-diminished 600-cell | Bi-24-diminished 600-cell | Snub 24-cell (24-diminished 600-cell) | Grand antiprism (20-diminished 600-cell) | 600-cell | ||||||

Vertices | 48 | 72 | 96 | 100 | 120 | ||||||

Vertex figure (Symmetry) | dual of tridiminished icosahedron ([3], order 6) | tetragonal antiwedge ([2] ^{+}, order 2) | tridiminished icosahedron ([3], order 6) | bidiminished icosahedron ([2], order 4) | Icosahedron ([5,3], order 120) | ||||||

Symmetry | Order 144 (48×3 or 72×2) | [3^{+},4,3]Order 576 (96×6) | [[10,2^{+},10]]Order 400 (100×4) | [5,3,3] Order 14400 (120×120) | |||||||

Net | |||||||||||

Ortho H _{4} plane | |||||||||||

Ortho F _{4} plane |

The regular complex polytopes _{3}{5}_{3}, and _{5}{3}_{5}, , in have a real representation as *600-cell* in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has Complex reflection group _{3}[5]_{3}, order 360, and the second has symmetry _{5}[3]_{5}, order 600.^{ [62] }

Regular complex polytope in orthogonal projection of H_{4} Coxeter plane | ||
---|---|---|

{3,3,5} Order 14400 | _{3}{5}_{3}Order 360 | _{5}{3}_{5}Order 600 |

The 600-cell is one of 15 regular and uniform polytopes with the same H_{4} symmetry [3,3,5]:^{ [63] }

H_{4} family polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

120-cell | rectified 120-cell | truncated 120-cell | cantellated 120-cell | runcinated 120-cell | cantitruncated 120-cell | runcitruncated 120-cell | omnitruncated 120-cell | ||||

{5,3,3} | r{5,3,3} | t{5,3,3} | rr{5,3,3} | t_{0,3}{5,3,3} | tr{5,3,3} | t_{0,1,3}{5,3,3} | t_{0,1,2,3}{5,3,3} | ||||

600-cell | rectified 600-cell | truncated 600-cell | cantellated 600-cell | bitruncated 600-cell | cantitruncated 600-cell | runcitruncated 600-cell | omnitruncated 600-cell | ||||

{3,3,5} | r{3,3,5} | t{3,3,5} | rr{3,3,5} | 2t{3,3,5} | tr{3,3,5} | t_{0,1,3}{3,3,5} | t_{0,1,2,3}{3,3,5} |

It is similar to three regular 4-polytopes: the 5-cell {3,3,3}, 16-cell {3,3,4} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cells.

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

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

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

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

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

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

This 4-polytope is a part of a sequence of 4-polytope and honeycombs with icosahedron vertex figures:

{p,3,5} polytopes | |||||||
---|---|---|---|---|---|---|---|

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

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

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

Image | |||||||

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

- 24-cell, the predecessor 4-polytope on which the 600-cell is based
- 120-cell, the dual 4-polytope to the 600-cell, and its successor
- Uniform 4-polytope family with [5,3,3] symmetry
- Regular 4-polytope
- Polytope

- 1 2 3 4 5 6 7 8 9 10 In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center. Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there. However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices. Thus the vertex icosahedron is actually a canonical icosahedral pyramid, composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.
^{ [lower-alpha 55] } - ↑ 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^{ [3] }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 600-cell is the 120-point 4-polytope: fifth 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 are rare, the only such construction (in any dimension) is from the 8-cell to the 24-cell. - 1 2 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.
^{ [56] }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. - 1 2 The 600-cell geometry is based on the 24-cell. The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths.
- 1 2 3 4 5 6 A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.
^{ [19] }A 600-cell contains 25・16/2 = 200 such hexagons. - ↑ In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords and central polygons must likewise be disjoint. In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.
^{ [lower-alpha 6] } - ↑ Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.<ref name='FOOTNOTEDenneyHookerJohnsonRobinson2020438'>Denney et al. 2020, p. 438.
- 1 2 3 4 5 The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each of the 25 24-cells in the 600-cell intersects each of the 144 great pentagons in the 600-cell at just one vertex.
^{ [lower-alpha 8] }Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons. - 1 2 3 4 Five 24-cells meet at each icosahedral pyramid apex
^{ [lower-alpha 1] }of the 600-cell. Each 24-cell shares not just one vertex but 6 vertices (an entire hexagonal central plane) with each of the other five 24-cells.^{ [lower-alpha 6] } - 1 2 3 Schoute was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells. The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells. The rows and columns of the array are the only ten such partitions of the 600-cell.
^{ [15] } - 1 2 3 4 5 6 7 8 9 10 11 12 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. 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.
^{ [17] }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. They spiral around each other in Hopf fiber bundles which, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons^{ [lower-alpha 29] }belonging to 6 discrete Hopf fibrations, each filling the whole 600-cell. Each fibration is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells^{ [lower-alpha 30] }each bounded by three Clifford parallel great decagons.^{ [lower-alpha 31] } - ↑ 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. - 1 2 3 4 5 6 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.^{ [lower-alpha 13] } - ↑ The angles 𝜉
_{i}and 𝜉_{j}are angles of rotation in the two completely orthogonal<ref name='completely orthogonal planes' group='lower-alpha'> - ↑ The Hopf coordinates
^{ [8] }are triples of three angles:- (𝜉
_{i}, 𝜂, 𝜉_{j})

^{ [lower-alpha 15] }invariant planes which characterize rotations in 4-dimensional Euclidean space. The angle 𝜂 is the inclination of both these planes from the north-south pole axis, where 𝜂 ranges from 0 to 𝜋/2. The (𝜉_{i}, 0, 𝜉_{j}) coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_{i}, 𝜋/2, 𝜉_{j}) coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_{i}, 0 < 𝜂 < 𝜋/2, 𝜉_{j}) describe the great circles (*not*"lines of latitude") which cross an equator but do not pass through the north or south pole. - (𝜉
- ↑ The conversion from Hopf coordinates (𝜉
_{i}, 𝜂, 𝜉_{j}) to unit-radius Cartesian coordinates (w, x, y, z) is:- w = cos 𝜉
_{i}sin 𝜂 - x = cos 𝜉
_{j}cos 𝜂 - y = sin 𝜉
_{j}cos 𝜂 - z = sin 𝜉
_{i}sin 𝜂

Cartesian (1, 0, 0, 0) is Hopf (0, 𝜋/2, 0). - w = cos 𝜉
- ↑ There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell. These are actually the Hopf coordinates of the vertices of the 120-cell, which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.
- 1 2 3 The fractional-root
*golden chords*exemplify that the golden ratio φ is a circle ratio related to 𝜋:^{ [14] }- 𝜋/5 = arccos (φ/2)

- φ = 1 – 2 cos (3𝜋/5)

- 1 2 The 600-cell edges are decagon edges of length √0.𝚫, which is 𝚽, the
*smaller*golden section of √5; the edges are in the inverse golden ratio 1/φ to the √1 hexagon chords (the 24-cell edges). The other fractional-root chords exhibit golden relationships as well. The chord of length √1.𝚫 is a pentagon edge. The next fractional-root chord is a decagon diagonal of length √2.𝚽 which is φ, the*larger*golden section of √5; it is in the golden ratio^{ [lower-alpha 19] }to the √1 chord (and the radius).^{ [lower-alpha 23] }The last fractional-root chord is the pentagon diagonal of length √3.𝚽. The diagonal of a regular pentagon is always in the golden ratio to its edge, and indeed φ√1.𝚫 is √3.𝚽. - 1 2 3 4 5 The long radius (center to vertex) of the 600-cell is in the golden ratio to its edge length; thus its radius is φ if its edge length is 1, and its edge length is 1/φ if its radius is 1. Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.)
**Radially golden**polytopes are those which can be constructed, with their radii, from golden triangles^{ [lower-alpha 27] }which meet at the center, each contributing two radii and an edge. - ↑ The fractional square roots are given as decimal fractions where 𝚽 ≈ 0.618 is the inverse golden ratio 1/φ and 𝚫 ≈ 0.382 = 1 - 𝚽 = 𝚽
^{2}. For example:

√0.𝚫 = √0.382 = 0.618 = 𝚽 - ↑ Notice in the diagram how the φ chord (the
*larger*golden section) sums with the adjacent 𝚽 edge (the*smaller*golden section) to √5, as if together they were a √5 chord bent to fit inside the √4 diameter. - 1 2 Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell. The other 96 vertices constitute a snub 24-cell. Removing any one 24-cell from the 600-cell produces a snub 24-cell.
- 1 2 3 4 The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.
^{ [15] } - 1 2 3 4 Each tetrahedral cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.
- 1 2 3 A golden triangle is an isosceles triangle in which the duplicated side
*a*is in the golden ratio to the distinct side*b*:- a/b = φ = 1 + √5/2 ≈ 1.618

The vertex angle is:- 𝛉 = arccos(φ/2) = 𝜋/5 = 36°

- 1 2 Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.
^{ [5] }Therefore the successor may be constructed by placing 4-pyramids of some kind on the cells of its predecessor. Between the 16-cell and the tesseract, we have 16 right tetrahedral pyramids, with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical cubic pyramids. But if we place 24 canonical octahedral pyramids on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell. Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base. - 1 2 3 4 5 6 7 The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex. Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular edges of the tetrahedron. Each tetrahedron links three pairs of decagons which do
*not*intersect at a vertex of the tetrahedron. However, none of the six decagons are Clifford parallel;^{ [lower-alpha 12] }each belongs to a different Hopf fiber bundle of 12. Only one of the tetrahedron's six edges may be part of a helix in any one Boerdijk–Coxeter triple helix ring.^{ [lower-alpha 30] }Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons^{ [lower-alpha 31] }that all reference each other. - 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Since tetrahedra
^{ [lower-alpha 29] }do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a Boerdijk-Coxeter helix. This is a Clifford parallel^{ [lower-alpha 12] }triple helix as shown in the illustration. In the 600-cell we find them bent in the fourth dimension into geodesic rings. Each ring has 30 cells and touches 30 vertices. The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.^{ [lower-alpha 31] }5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge). A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete Hopf fibration. There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions". - 1 2 3 4 5 6 7 Two Clifford parallel
^{ [lower-alpha 12] }great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon. The two parallel decagons and the ten linking edges form a double helix ring. Three decagons can also be parallel (decagons come in parallel fiber bundles of 12) and three of them may form a triple helix ring. If the ring is cut and laid out flat in 3-space, it is a Boerdijk–Coxeter helix^{ [lower-alpha 30] }30 tetrahedra^{ [lower-alpha 29] }long. The three Clifford parallel decagons can be seen as the cyan edges in the triple helix illustration. Each magenta edge is one edge of another decagon linking two parallel decagons. - ↑ The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.
^{ [lower-alpha 1] } - ↑ The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 √1 cubic cells. The 1200 √3 chords are the 4 long diameters of these 600 cubes; the 3 tesseracts overlap and each chord is the long diameter of a cube in two different tesseracts.
- ↑ The sum of 0.𝚫・720 + 1・1200 + 1.𝚫・720 + 2・1800 + 2.𝚽・720 + 3・1200 + 3.𝚽・720 + 4・60 is 14,400.
- ↑ 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.
^{ [23] } - ↑ A
*triacontagon*or 30-gon is a thirty-sided polygon. The triacontagon is the largest regular polygon whose interior angle is the sum of the interior angles of smaller polygons: 168° is the sum of the interior angles of the equilateral triangle (60°) and the regular pentagon (108°). - 1 2 The 600-cell has 72 30-gon central planes: 6 sets of 12 Clifford parallel 30-gon great circles, each completely orthogonal to a decagon central plane. Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere. Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere.
- 1 2 3 4 5 The 30-cell Boerdijk–Coxeter helix
^{ [lower-alpha 30] }ring can be seen as a skew {30/11} star polygon (a star 30-gon called a triacontagram with a winding number of 11), because its 30 vertices lie on a skew triacontagram_{11}. As such the 30-cell ring can be characterized as a continuous*tight*corkscrew helix bent into a loop of 30 edges (the magenta edges in the triple helix illustration) comprising the Petrie polygon^{ [lower-alpha 75] }of the 600-cell which winds 11 times around itself in the course of a single revolution around the 600-cell accompanied by a single 360 degree twist of the 30-cell ring.^{ [43] }Equivalently, the twisted 30-cell ring can also be seen as a skew {30/2}=2{15} compound 30-gon (or triacontagram_{2}) helix which winds twice around the 600-cell, connecting vertices which are 2 skew 30-gon edges apart (on the 30-gram Möbius double helix of the 15 orange and 15 yellow edges in the triple helix illustration), before completing one orbit through all 30 vertices and one 360 degree twist of the 30-cell ring.^{ [lower-alpha 76] }Although the 30 vertices do not lie in one great 30-gon central plane,^{ [lower-alpha 77] }We can visualize how these ordinary 2-dimensional great circles (they are*not*helices) spiral around each other in the curved 3-dimensional surface volume of the 600-cell. From inside that closed curved 3-space at a point on the center axis of the Boerdijk–Coxeter triple-helix we would see the 3 great circle geodesics nearby, passing by us seemingly in parallel; but in the distance we would see them beginning to spiral around each other (either leftwards or rightwards) in this strangely curved 3-space before the horizon eclipsed our view. - 1 2 3 4 Each great decagon central plane is completely orthogonal
^{ [lower-alpha 14] }to a great 30-gon^{ [lower-alpha 36] }central plane which does not intersect any vertices of the 600-cell. The 72 30-gons are each the center axis of a 30-cell Boerdijk–Coxeter triple helix ring,^{ [lower-alpha 30] }with each segment of the 30-gon passing through a tetrahedron similarly. The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;^{ [lower-alpha 37] }its curved segments are not chords. It does not touch any edges or vertices, but it does hit faces. Nearby is a skew 30-gon which does pass through all 30 vertices of the Boerdijk–Coxeter helix, linking vertices 2 apart in a geodesic orbit that winds around the 600-cell twice.^{ [lower-alpha 38] } - 1 2 3 4 5 6 7 A point under isoclinic rotation traverses the diagonal
^{ [lower-alpha 85] }each vertex is displaced to another vertex √1 (60°) distant, moving √1/4 = 1/2 unit radius in four orthogonal directions. - 1 2 The decagonal planes in the 600-cell occur in equi-isoclinic and all the same angular distance apart.<ref name='FOOTNOTELemmensSeidel1973'>Lemmens & Seidel 1973.
- 1 2 Two angles are required to fix the relative positions of two planes in 4-space.
^{ [24] }Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great decagons are a multiple (from 0 to 4) of 36° (𝝅/5) apart in each angle, and*may*be the same angle apart in*both*angles.^{ [lower-alpha 41] }Such central planes are mutually*isoclinic*: each pair of planes is separated by two*equal*angles, and an isoclinic rotation by that angle will bring them together. Where three or four such planes are all separated by the*same*angle, they are called*equi-isoclinic*. - 1 2 3 4
- 1 2 The hexagonal planes in the 600-cell occur in equi-isoclinic
^{ [lower-alpha 43] }groups of 4, everywhere 4 Clifford parallel hexagons 60° (𝝅/3) apart form a 24-cell. Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° (𝝅/5) apart, 4 72° (2𝝅/5) apart, 4 108° (3𝝅/5) apart, and 4 144° (4𝝅/5) apart, for a total of 20 Clifford parallel hexagons (120 vertices) that comprise a discrete Hopf fibration. - 1 2 The square planes in the 600-cell occur in an equi-isoclinic
^{ [lower-alpha 43] }group of 2, everywhere 2 Clifford parallel squares 90° (𝝅/2) apart form a 16-cell. Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° (𝝅/3) apart form a 24-cell. Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° (𝝅/5) apart, 3 72° (2𝝅/5) apart, 3 108° (3𝝅/5) apart, and 3 144° (4𝝅/5) apart, for a total of 30 Clifford parallel squares (120 vertices) that comprise a discrete Hopf fibration. - 1 2 Each Hopf fibration of the 3-sphere into Clifford parallel great circle fibers has a map (called its
*base*) which is an ordinary 2-sphere. On this map each great circle fiber appears as a single point. The base of a great decagon fibration of the 600-cell is an icosahedron in which each vertex represents one of the 12 great decagons. To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere, useful for reasoning about the fibration. But in fact the 600-cell does have icosahedra in it: 120 icosahedral vertex figures,^{ [lower-alpha 1] }any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell. Each 3-dimensional vertex icosahedron is*lifted*to the 4-dimensional 600-cell by a 720 degree isoclinic rotation,^{ [lower-alpha 40] }which takes each of its 4 disjoint triangular faces (each disjoint set of 3 of its 12 vertices) in a circuit around one of 4 disjoint 30-vertex rings of tetrahedral cells (each composed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell. Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel decagons of the same fibration, we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration is an expression of an isoclinic symmetry. - 1 2 These are the √2 tetrahedral cells of the 75 16-cells,
*not*the √0.𝚫 tetrahedral cells of the 600-cell. - 1 2 3 4 The fibration's Clifford parallel cell rings may or may not be chiral objects, depending upon whether the 4-polytope's cells have opposing faces or not. The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise. Isoclines of the same chirality run through these chiral cell rings. The characteristic cell rings of the tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no torsion). Both left-handed and right-handed isoclines run through it. Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed predecessors' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.
- 1 2 3 4 5 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 The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings
^{ [lower-alpha 30] }is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell. The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells. In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring. - ↑ Unlike their bounding decagons, the 20 cell rings themselves are
*not*all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.^{ [lower-alpha 49] }The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings. Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration). In fact each of the 5 different subsets of 4 cell rings is bounded by the*same*12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings). - ↑ Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration,
*not*the different fibrations of the 4-polytope. Each fibration is the entire 4-polytope. - ↑ The 4 red faces of the snub tetrahedron correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its
*subfibration*). The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron. - 1 2 Because the octahedron can be snub truncated yielding an icosahedron,
^{ [34] }another name for the icosahedron is snub octahedron. This term refers specifically to a lower symmetry arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another). - 1 2 3 The 120-point 600-cell has 120 overlapping icosahedral pyramids.
^{ [lower-alpha 1] } - ↑ An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)
- ↑ The pentagonal pyramids around each vertex of the "snub octahedron" icosahedron all look the same, with two yellow and three blue faces. Each pentagon has five distinct rotational orientations. Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.
- ↑ Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.
- ↑ There is a vertex icosahedron
^{ [lower-alpha 1] }inside each 24-cell octahedral central section (not inside a √1 octahedral cell, but in the larger √2 octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron. The two different-sized icosahedra are the second and fourth sections of the 600-cell (beginning with a vertex). The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).^{ [36] }The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like Russian dolls and are related by a helical contraction.^{ [37] }The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.^{ [lower-alpha 58] }The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a Jessen's icosahedron; they continue to spiral toward each other until they merge into the 8 vertices of the octahedron; and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).^{ [lower-alpha 54] }The geometry of this sequence of transformations in S^{3}is similar to the kinematics of the cuboctahedron and the tensegrity icosahedron in R^{3}. The twisting, expansive-contractive transformations between these polyhedra were named Jitterbug transformations by Buckminster Fuller.^{ [38] } - ↑ These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs. They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.
- ↑ The √1 tetrahedron has a volume of 9 √0.𝚫 tetrahedral cells. In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it. The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the √1 tetrahedron. The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.
- ↑ We also find √1 tetrahedra as the cells of the unit-radius 5-cell, and radially around the center of the 24-cell (one behind each of the 96 faces). Those radial √1 tetrahedra also occur in the 600-cell (in the 25 inscribed 24-cells), but note that those are not the same tetrahedra as the 600 √1 tetrahedral sections.
- ↑ Each √1 edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells). In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces. Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.
- ↑ A √1 octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell. The same √1 octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point √1 octahedral section, a 4-point √1 tetrahedral section, and a 4-point √0.𝚫 tetrahedral section. In the curved three-dimensional space of the 600-cell's surface, the √1 octahedron surrounds the √1 tetrahedron which surrounds the √0.𝚫 tetrahedron, as three concentric hulls. This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical octahedral pyramid with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid
*lies on*the surface of the 24-cell. - ↑ The apex of a canonical √1 octahedral pyramid has been truncated into a regular tetrahedral cell with shorter √0.𝚫 edges, replacing the apex with four vertices. The truncation has also created another four vertices (arranged as a √1 tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with √0.𝚫 edges. The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all. The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two √1 edges (and just one of those routes ran through the single apex). The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three √0.𝚫 edges (and pass through two 'apexes').
- ↑ The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell rectified 5-cell and its dual (it has characteristics of both).
- 1 2 The 100 tetrahedral voids on the exact boundary of the duocylinder are filled by another 10 rings of 10 tetrahedra. These 10-cell rings can be seen (lying vertex-to-vertex) along the diagonals of the square 10x10 array. They have a different topology
^{ [lower-alpha 71] }than either the annular 10-cell rings around one vertex (lying face-to-face) or the Clifford parallel 10-cell rings banding the 150-cell torus (lying edge-to-edge^{ [lower-alpha 68] }). The cells on each diagonal of the 10x10 array form a vertex-bonded 10-cell ring along a geodesic "straight line" around the 600-cell (a great decagon). Note that the great decagon runs along one edge of each tetrahedron,^{ [lower-alpha 29] }not through the interior of the cell or any of its faces. Each great decagon ring of 10 vertex-bonded tetrahedra is one of the helices, and 10 of the cells, of a 30 tetrahedron Boerdijk–Coxeter triple helix ring.^{ [lower-alpha 30] } - 1 2 In the 10x10 Clifford duocylinder, the horizontal and vertical lines are decagonal great circles (central planes seen edge-on). The horizontal lines belong to one decagonal fibration, and the vertical lines belong to the completely orthogonal fibration. There are only 10 horizontal and 10 vertical lines (the opposite edges of the duocylinder are identified) but there are 12 decagons in each complete fibration; where are the other 2 decagons of each fibration in this projection?
^{ [lower-alpha 67] } - 1 2 The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet. This vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.
^{ [lower-alpha 55] }Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of the icosahedral pyramid. - 1 2 The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10x10 parallelogram of triangles.
- 1 2 3 Each tetrahedral cell participates in multiple 10-cell rings of three different topologies. Each tetrahedron is face-bonded to two others in 3 different annular 10-cell rings around each of its 4 vertices; it is edge-bonded to two others in 2 different 10-cell rings around a 150-cell torus; and it is vertex-bonded to two others in 6 different geodesic "straight line" 10-cell rings around the 600-cell (great decagons).
- ↑ The 100 cells stacked as peaks on the exposed faces of the 150-cell torus can be seen as 10 Clifford parallel 10-cell rings banding the torus. These rings have a different topology than the annular 10-cell rings around one vertex: their cells are edge-bonded, rather than face bonded.
^{ [lower-alpha 71] }Each ring is a double helix of two Clifford parallel great decagons; it is two of the helices, and 10 of the cells, of a 30 tetrahedron Boerdijk–Coxeter triple helix ring.^{ [lower-alpha 30] } - 1 2 The cells in the 10x10 Clifford torus boundary array are vertex-bonded to each other along the diagonals of the array; the array consists of 10 parallel vertex-bonded 10-cell rings. The 50 "egg crate" recesses and peaks on each side of the array mate with 10 parallel edge-bonded 10-cell rings (on the surface of a 150-cell torus) along the 10
*rows*of the array (on one side of the array), and along the 10*columns*of the array (on the other side of the array).^{ [lower-alpha 71] }Thus the edge-bonded 10-cell rings (or more precisely, their Clifford parallel great decagon edges) on one side of the 10x10 array lie perpendicular to those on the other side of the boundary. Nevertheless the two 150-cell tori, their 20 combined 30-cell rings, and their 12 combined great decagons are all Clifford parallel parts of the same fibration of 12 decagons. - ↑ The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.
^{ [lower-alpha 69] } - 1 2 3 The Petrie polygons of 3-dimensional polyhedra are zig-zag polygons (so-called because adjacent skew vertices lie in two parallel planes), but in 4-polytopes regular skew polygons are found which lie in more than two (Clifford) parallel planes, including chiral skew polygons which lie in an
*odd*number of planes. For example, in the 600-cell the Petrie polygon is a regular skew {30/11} 30-gon that connects the 30 vertices of each chiral Boerdijk–Coxeter 30-cell ring, which lie in three different central planes. Equivalently, these same 30 vertices are connected by a regular skew {30/2}=2{15} compound 30-gon (or triacontagram_{2}) that winds around the 600-cell twice, connecting vertices 2 apart on a chiral 30-gon geodesic that is a helix not a simple great circle. This latter isoclinic geodesic or*isocline*is the path of a chiral isoclinic rotation in decagonal invariant planes.<ref name='isoclinic geodesic' group='lower-alpha'> - 1 2 Because the 600-cell's helical triacontagram
_{2}geodesic^{ [lower-alpha 38] }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 30-vertex isoclinic path forms a Möbius double helix (like a 3-dimensional double helix but with*opposite*ends of its two 15-vertex helices joined to form a single continuous 30-vertex loop). The helix passes through non-adjacent vertices of a skew 30-gon of the 4-polytope, as the Petrie 30-gon passes through adjacent vertices. Unlike the Petrie polygon^{ [lower-alpha 75] } - ↑ The 30 vertices of the Boerdijk–Coxeter triple-helix ring lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all (they are π/5 apart). Their decagonal great circles are Clifford parallel: one 600-cell edge length apart at every point.<ref name='Clifford parallels' group='lower-alpha'>
- ↑ The regular skew 30-gon is the Petrie polygon
^{ [lower-alpha 75] }of the 600-cell and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell Boerdijk–Coxeter helix rings: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell). - 1 2 3 The 20 30-cell rings are chiral objects; 10 spiral clockwise (right) and 10 spiral counterclockwise (left). The 150-cell torus formed by five cell-disjoint 30-cell rings (of the same chirality) surrounding a great decagon is likewise a chiral object, although the great decagon itself is not.
^{ [lower-alpha 82] }Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; left-handed and right-handed 30-cell rings are*not*cell-disjoint and belong to different fibrations. Finally the 300-cell torus formed by the union of two 150-cell tori is*not*a chiral object; the two 300-cell tori are directly congruent, because each is formed by the union of a left and a right 150-cell torus, which are able to counter-spiral around each other and pass through each other in 4 dimensions without intersecting in any of their cells (an act which would not be possible in 3 dimensions). Thus the 300-cell tori, the 150-cell tori, the 30-cell rings, and the great decagons all occur as sets of Clifford parallel interlinked circles,^{ [lower-alpha 12] }though the way they nest together, avoid intersecting each other, and pass through each other to form a Hopf link is not*exactly*the same geometric relationship for these four different kinds of Clifford parallel polytopes, since the linked pairs are variously of the same chirality, opposite chirality, or no chirality. - 1 2 The 600-cell contains 120 overlapping 30-cell rings (6 fibrations of 20 disjoint 30-cell rings), with five left rings and five right rings around each of 72 decagons, and each 30-cell ring adjacent to 3 decagons. These form 24 overlapping 150-cell tori (6 fibrations of 4 disjoint tori), with one left torus and one right torus around each of 72 decagons, and each 150-cell torus adjacent to 6 decagons. Each chiral pair
^{ [lower-alpha 79] }of left and right 150-cell tori combine to form one of 12 300-cell tori: 6 fibrations of 2 non-chiral 300-cell tori which meet at the Clifford torus boundary. - ↑ The middle layer of 300 cells can be seen as an adjacent pair of face-bonded 150-cell tori. Their shared surface of 100 triangular faces constitutes a flat 2-dimensional 10x10 Clifford torus boundary separating them.
^{ [lower-alpha 70] }The four 150-cell tori filling the 600-cell are separated into pairs by 6 such 2-dimensional 100-triangle boundaries (as the 6 edges of a tetrahedron separate its pairs of faces); each 150-cell torus shares a boundary with each of the other three (as each face of a tetrahedron does). The four 150-cell tori meet three around each great decagon (as three faces meet around each vertex of a tetrahedron), and are separated by a single curved 3-dimensional 100-cell 10x10 Clifford torus boundary (as the tetrahedron's four faces are separated by the tetrahedron itself).^{ [lower-alpha 29] } - 1 2 The three great decagons bounding each cell ring are not themselves chiral objects. Although in a 30-cell ring cut and laid out in flat 3-dimensional space they form helices which spiral in the same left or right direction as the cell ring itself, in 4 dimensional space they are ordinary flat circles not chiral helices, and in the curved 3-dimensional space of the 3-sphere they are straight lines (geodesics).
- ↑ 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. - 1 2 3 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
^{ [lower-alpha 14] }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 83] } - 1 2 3 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 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,<ref name='non-vertex geodesic' group='lower-alpha'>
- ↑
- ↑ Isoclinic geodesics 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*,^{ [45] }which are the 4-space analogues of great 1-spheres (2-dimensional great circles in 3-space). - ↑ All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in all four dimensions), but not all isoclines on 3-manifolds in 4-space are circles.
- ↑ Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a Hopf link called the {1,1} torus knot
^{ [46] }in which*each*of the two linked circles traverses all four dimensions. - 1 2 There are an infinite number of pairs of completely orthogonal
^{ [lower-alpha 14] }invariant planes in each isoclinic rotation, all rotating through the same angle; nonetheless, not all central planes are invariant planes of rotation. The invariant planes of an isoclinic rotation constitute a fibration of the entire 4-polytope.^{ [48] }In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great decagons,*or*20 Clifford parallel great hexagons*or*30 Clifford parallel great squares are invariant planes of rotation. - ↑ In a
*Clifford displacement*, also known as an isoclinic rotation, all the Clifford parallel^{ [lower-alpha 12] }invariant planes^{ [lower-alpha 90] }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 85] }Every plane that is Clifford parallel to one of the completely orthogonal planes 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 rotates sideways.*All*central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind). - ↑ The three 16-cells in the 24-cell are rotated by 60° (𝜋/3) isoclinically with respect to each other. Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° (2𝜋/3) apart. In a unit-radius 4-polytope, vertices 120° apart are joined by a √3 chord.
- 1 2 Any isoclinic rotation by 𝜋/5 in decagonal invariant planes
^{ [lower-alpha 99] }through 4-space. Note however that in a*discrete*decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain*no*points. - 1 2 Five 24-cells meet at each vertex of the 600-cell, so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an isoclinic rotation<ref name='isoclinic geodesic displaces every central polytope' group='lower-alpha'>
- 1 2 A
*disjoint*24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation^{ [lower-alpha 84] }takes it past (not through) the adjacent 24-cell it rotates toward,^{ [lower-alpha 94] }) directly toward an adjacent 24-cell. - ↑ All isoclinic
*polygons*are Clifford parallels (completely disjoint).^{ [lower-alpha 49] }Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and*not*disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared). For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by 𝜋/3 with respect to each other, yet are not disjoint: they share a 16-cell (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint). - 1 2 3 At each vertex, a 600-cell has four adjacent (non-disjoint)
^{ [lower-alpha 49] }24-cells that can each be reached by a simple rotation in that direction.^{ [lower-alpha 94] }Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates*around*the common hexagonal plane. The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.^{ [lower-alpha 6] }In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is*not*fixed: it rotates (non-invariantly) through 𝜋/5. The double rotation reaches an adjacent 24-cell*directly*as if indirectly by two successive simple rotations:^{ [lower-alpha 84] }first to one of the*other*adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them). - ↑ In the 600-cell, there is a simple rotation which will take any vertex
*directly*to any other vertex, also moving most or all of the other vertices but leaving 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 decagon, a great hexagon, a great square or a great digon, and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),^{ [lower-alpha 39] }2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively. Two*non-disjoint*24-cells are related by a simple rotation through 𝜋/5 of the digon central plane completely orthogonal to their common hexagonal central plane. In this simple rotation, the hexagon does not move. The two*non-disjoint*24-cells are also related by an isoclinic rotation in which the shared hexagonal plane*does*move.^{ [lower-alpha 97] } - ↑ Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes, and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes. As the invariant planes rotate in two completely orthogonal directions at once, all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines<ref name='isoclinic geodesic' group='lower-alpha'>
- ↑ Notice the apparent incongruity of rotating
*hexagons*by 𝜋/5, since only their opposite vertices are an integral multiple of 𝜋/5 apart. However, recall that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart. The hexagons have their own 10 discrete fibrations and cell rings, not Clifford parallel to the decagonal fibrations but also by fives^{ [lower-alpha 9] }in that five 24-cells meet at each vertex, each pair sharing a hexagon;^{ [lower-alpha 10] }each hexagon rotates*non-invariantly*by 𝜋/5 during an isoclinic rotation between*non-disjoint*24-cells.^{ [lower-alpha 97] } - ↑ Although not all cell rings are chiral,
^{ [lower-alpha 48] }all are helical in that they have chiral isoclines running through them. They also have non-chiral great circle polygons bounding them (running along their edges). In non-chiral cell rings (such as the 24-cell's 6-cell rings of octahedra, but*not*the 600-cell's chiral 30-cell rings of tetrahedra), the great circle fibers occur in pairs which intersect, running around the ring on its edges as counter-rotating helices. The isocline fibers also occur in counter-rotating pairs within the non-chiral ring, but*they*do not intersect. Left and right isoclines are Clifford parallel (part of the same fibration) and spiral around each other forming a special kind of double helix which cannot occur in three dimensions (where counter-rotating helices of the same radius must intersect). Because non-chiral cell rings contain pairs of helices of opposite chirality, they have no net torsion. Chiral rings have just one axial isocline running through them, and they have the same torsion as that isocline. - ↑ The 600-cell has four orthogonal central hyperplanes, each of which is an icosidodecahedron.
^{ [lower-alpha 21] } - 1 2 In the same sense that the 3-sphere is the product of two completely orthogonal cylinders, each cell ring rotates along two completely orthogonal circular cylindrical spirals at once.
^{ [lower-alpha 104] }In one circular cylindrical spiral it winds twice around the 600-cell (in a 720° rotation) visiting 15 of its 30 vertices in the first 360° revolution and their antipodal 15 vertices in the second revolution. (The isoclinic rotation as a whole visits 60 of the 120 vertices on the first revolution, and their antipodal 60 vertices on the second.) Another circular cylindrical spiral winds 6 times around the 600-cell in the same 720° rotation, circling through 6 concatenated open-ended pentagrams linked end-to-end.^{ [21] }From yet another perspective viewpoint, the 30-chord isocline is seen as a single connected circular cylindrical spiral through every 11th vertex, linking all 30 vertices in the same 720° rotation.^{ [27] } - ↑ Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders in order to move the short distance between Clifford parallel subspaces.
- ↑ Tetrahedral cells in contact only at a vertex
^{ [lower-alpha 26] }belong to rotations of opposite chirality, though of course the cells themselves are not chiral objects. - ↑ Each vertex is visited only by left rotations or right rotations. Six isoclines (all left-handed or all right-handed) from the six different fibrations cross at each vertex. Each axis of the 600-cell touches left isoclines at one end and right isoclines at the other end. Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).
- ↑ 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. - ↑ The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.
- ↑ The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) polychoron consists of 3-dimensional cells.

- ↑ N.W. Johnson:
*Geometries and Transformations*, (2018) ISBN 978-1-107-10340-5 Chapter 11:*Finite Symmetry Groups*, 11.5*Spherical Coxeter groups*, p.249 - ↑ Matila Ghyka,
*The Geometry of Art and Life*(1977), p.68 - ↑ Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {
*p,q,r*} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius. - ↑ Coxeter 1973, p. 153, §8.51; "In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s."
- 1 2 Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
- ↑ Coxeter 1973, pp. 156–157, §8.7 Cartesian coordinates.
- 1 2 Coxeter 1973, pp. 151–153, §8.4 The snub {3,4,3}.
- ↑ Zamboj 2021, pp. 10–11, §Hopf coordinates.
- ↑ Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏
^{−1}) beginning with a vertex. - ↑ Oss 1899; van Oss does not mention the arc distances between vertices of the 600-cell.
- ↑ Buekenhout & Parker 1998.
- ↑ Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏
^{−1}) beginning with a vertex; see column*a*. - ↑ Steinbach 1997, p. 23, Figure 3; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.
- ↑ Baez, John (7 March 2017). "Pi and the Golden Ratio".
*Azimuth*. Retrieved 10 October 2022. - 1 2 Denney et al. 2020, p. 434.
- ↑ Denney et al. 2020, pp. 437–439, §4 The planes of the 600-cell.
- ↑ Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism.
- ↑ Sadoc 2001, p. 576, §2.4: the ten-fold screw axis.
- ↑ Denney et al. 2020, p. 438.
- ↑ Waegell & Aravind 2009, p. 5, §3.4. The 24-cell: points, lines, and Reye's configuration; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon
*planes*are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell. - 1 2 Sadoc 2001, pp. 576–577, §2.4: the six-fold screw axis.
- ↑ Sadoc 2001, p. 577, §2.4: the four-fold screw axis.
- ↑ Copher 2019, p. 6, §3.2 Theorem 3.4.
- ↑ Kim & Rote 2016, p. 7, §6 Angles between two Planes in 4-Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally,
*k*angles are defined between*k*-dimensional subspaces.)" - ↑ Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5} polytope.
- ↑ Tyrrell & Semple 1971, pp. 6–7, §4. Isoclinic planes in Euclidean space E
_{4}. - 1 2 Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries.
- ↑ Coxeter 1970, studied cell rings in the general case of their geometry and group theory, identifying each cell ring as a polytope in its own right which fills a three-dimensional manifold (such as the 3-sphere) with its corresponding honeycomb. He found that some (but not all) cell rings and their honeycombs are
*twisted*, occuring in left- and right-handed chiral forms.^{ [lower-alpha 48] } - ↑ Sadoc 2001, p. 578, §2.6 The {3, 3, 5} polytope: a set of four helices.
- 1 2 Coxeter 1973, p. 303, Table VI (iii): 𝐈𝐈 = {3,3,5}.
- ↑ Coxeter 1973, p. 153, §8.5 Gosset's construction for {3,3,5}.
- ↑ Borovik 2006; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences.... [Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains. Coxeter made full use of it, and expected the reader to use it.... Visualization is one of the most powerful interiorization techniques. It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module. Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."
- ↑ Miyazaki 1990; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).
- ↑ Coxeter 1973, pp. 50–52, §3.7.
- ↑ Coxeter 1973, p. 293; 164°29'
- ↑ Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections.
- ↑ Coxeter 1973, pp. 50–52, §3.7: Coordinates for the vertices of the regular and quasi-regular solids.
- ↑ Verheyen, H. F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion".
*Computers and Mathematics with Applications*.**17**(1–3): 203–250. doi: 10.1016/0898-1221(89)90160-0 . MR 0994201. - ↑ Coxeter 1973, p. 299, Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell.
- ↑ Sadoc 2001, p. 576-577, §2.4 Discretising the fibration for the {3, 3, 5} polytope.
- ↑ Banchoff T.F. (2013) Torus Decompostions of Regular Polytopes in 4-space. In: Senechal M. (eds) Shaping Space. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92714-5_20
- ↑ Zamboj 2021, pp. 6–12, §2 Mathematical background.
- 1 2 Sadoc 2001, p. 577-578, §2.5 The 30/11 symmetry.
- ↑ Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨
_{4}. - ↑ Stillwell 2001, p. 24.
- ↑ Dorst 2019, p. 44, §1. Villarceau Circles; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a Villarceau circle. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a Hopf fibration.... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."
- ↑ Waegell & Aravind 2009, pp. 2–5, §3. The 600-cell.
- ↑ Kim & Rote 2016, pp. 13–14, §8.2 Equivalence of an Invariant Family and a Hopf Bundle.
- ↑ Coxeter 1973, pp. 292–293, Table I(ii); "600-cell".
- ↑ Coxeter 1973, p. 139, §7.9 The characteristic simplex.
- ↑ Coxeter 1973, p. 290, Table I(ii); "dihedral angles".
- ↑ Coxeter 1973, pp. 33–38, §3.1 Congruent transformations.
- ↑ Coxeter 1973, p. 12, §1.8. Configurations.
- ↑ van Ittersum 2020, pp. 80–95, §4.3.
- ↑ Steinbach 1997, p. 24.
- ↑ Stillwell 2001, p. 18-21.
- ↑ Stillwell 2001, pp. 22–23, The Poincaré Homology Sphere.
- ↑ Denney et al. 2020, §2 The Labeling of H
_{4}. - ↑ Oss 1899, pp. 1–18.
- ↑ Grossman, Wendy A.; Sebline, Edouard, eds. (2015),
*Man Ray Human Equations: A journey from mathematics to Shakespeare*, Hatje Cantz. See in particular*mathematical object mo-6.2*, p. 58;*Antony and Cleopatra*, SE-6, p. 59;*mathematical object mo-9*, p. 64;*Merchant of Venice*, SE-9, p. 65, and "The Hexacosichoron", Philip Ordning, p. 96. - ↑ Sikiric, Mathieu; Myrvold, Wendy (2007). "The special cuts of 600-cell".
*Beiträge zur Algebra und Geometrie*.**49**(1). arXiv: 0708.3443 . - ↑ Coxeter 1991, pp. 48–49.
- ↑ Denney et al. 2020.

A **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, an **icosidodecahedron** is a polyhedron with twenty (*icosi*) triangular faces and twelve (*dodeca*) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, the **24-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called **C _{24}**, or the

In geometry, the **5-cell** is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a **C _{5}**,

In geometry, the **16-cell** is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In geometry, the **120-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a **C _{120}**,

In four-dimensional geometry, a **runcinated 5-cell** is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

In geometry, a **truncated tesseract** is a uniform 4-polytope formed as the truncation of the regular tesseract.

In geometry, a **truncated 5-cell** is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In geometry, the **grand antiprism** or **pentagonal double antiprismoid** is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope, discovered in 1965 by Conway and Guy. Topologically, under its highest symmetry, the pentagonal antiprisms have *D _{5d}* symmetry and there are two types of tetrahedra, one with

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In four-dimensional geometry, a **runcinated 120-cell** is a convex uniform 4-polytope, being a runcination of the regular 120-cell.

In geometry, a **skew polygon** is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The *interior* surface of such a polygon is not uniquely defined.

In geometry, a **complex polytope** is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos^{−1}(1/5), or approximately 78.46°.

In five-dimensional geometry, a **rectified 5-simplex** is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

The **Boerdijk–Coxeter helix**, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.

In geometry, the **simplectic honeycomb** is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of *n* + 1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-*simplex honeycomb* is an expanded n-simplex.

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