Editing 600-cell (section)

==== Geodesics ====
The vertex chords of the 600-cell are arranged in [[geodesic]] [[great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|loc=§4 The planes of the 600-cell|pp=437-439}}

[[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[stereographic projection]] of the 600-cell's 72 central decagons onto their great circles.
Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]]
The {{radic|0.𝚫}} = 𝚽 edges form 72 flat regular central [[decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}}
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 {{radic|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, {{radic|3.𝚽}} apart.
As in the decagon and the icosidodecahedron, the edges occur in [[Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope.
The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[Clifford parallel]] geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|150px|Two [[Clifford parallel]] great circles spanned by a twisted [[Annulus (mathematics)|annulus]].]][[Clifford parallel]]s 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]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} 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 fibration|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{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[Hopf fibration]]s, each filling the whole 600-cell. Each [[#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} 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.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}}

The {{radic|1}} chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{Efn|1=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.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}}
Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The {{radic|1}} chords join vertices which are two {{radic|0.𝚫}} edges apart.
Each {{radic|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 {{radic|1}} chords, in 600 parallel pairs, {{radic|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.{{Sfn|Waegell|Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; 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.}}
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.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}}

The {{radic|1.𝚫}} chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}}
The {{radic|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 {{radic|1.𝚫}} chords join vertices which are two {{radic|0.𝚫}} edges apart on a geodesic great circle.
The 720 {{radic|1.𝚫}} chords occur in 360 parallel pairs, {{radic|2.𝚽}} = φ apart.

The {{radic|2}} chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex).
The {{radic|2}} chords join vertices which are three {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart).
There are 600 {{radic|2}} chords, in 300 parallel pairs, {{radic|2}} apart.
The 450 great squares (225 [[completely orthogonal]] 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 (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}}

The {{radic|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 {{radic|3.𝚽}}.
The {{radic|2.𝚽}} chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 720 distinct {{radic|2.𝚽}} chords, in 360 parallel pairs, {{radic|1.𝚫}} apart.

The {{radic|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-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells).
Each set of 32 triangles consists of the 96 {{radic|3}} chords and 24 vertices of one of the 25 overlapping inscribed 24-cells.
The {{radic|3}} chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The {{radic|3}} chords join vertices which are four {{radic|0.𝚫}} edges apart (and two {{radic|1}} chords apart on a geodesic great circle).
Each {{radic|3}} chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 {{radic|1}} cubic cells.
The 1200 {{radic|3}} chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each {{radic|3}} chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}}
There are 1200 {{radic|3}} chords, in 600 parallel pairs, {{radic|1}} apart.

The {{radic|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 {{radic|1.𝚫}}, so these are [[Golden triangle (mathematics)|golden triangles]]. The {{radic|3.𝚽}} chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four {{radic|0.𝚫}} edges apart on a geodesic great circle.
There are 720 distinct {{radic|3.𝚽}} chords, in 360 parallel pairs, {{radic|0.𝚫}} apart.

The {{radic|4}} chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell.
The {{radic|4}} chords join opposite vertices which are five {{radic|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.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells.

The sum of the squared lengths{{Efn|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.}} of all these distinct chords of the 600-cell is 14,400 = 120<sup>2</sup>.{{Efn|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.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}}
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 (a 0-gon).{{Efn|Each great decagon central plane is [[completely orthogonal]] to a great 30-gon{{Efn|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°).|name=triacontagon}} 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 helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} 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;{{Efn|name=0-gon central planes}} its curved segments are not chords.
It does not touch any edges or vertices, but it does hit faces.
It is the central axis of a spiral skew 30-gram, the [[Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}}
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|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}}

All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|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 orthogonal 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.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; "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.)"}}
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° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens|Seidel|1973}}
Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|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''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}}
Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration.
Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles.
Vectors in 4-space may be combined by [[Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[William Rowan Hamilton|Hamilton]].{{Sfn|Mamone|Pileio|Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0).
Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector.
Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br>
<small><math display=block>\begin{pmatrix}
w_2\\
x_2\\
y_2\\
z_2
\end{pmatrix}
*
\begin{pmatrix}
w_1\\
x_1\\
y_1\\
z_1
\end{pmatrix}
=
\begin{pmatrix}
{w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\
{w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\
{w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\
{w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1}
\end{pmatrix}
</math></small>}}
The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space.
The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}}
Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}}
Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|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° ({{sfrac|𝝅|3}}) apart form a 24-cell.
Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}}
Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}}
Planes which are isoclinic have [[Clifford parallel]] great circles.{{Efn|name=Clifford parallels}}
A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}}
Each great square plane is [[completely orthogonal]] to another great square plane.
Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one {{radic|4}} long diameter): a great [[digon]] plane.{{Efn|In the 24-cell each great square plane is [[completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[digon]] plane.|name=digon planes}}
Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, 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.
Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell.
In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}}