Editing 600-cell (section)

=== Coordinates ===

==== Unit radius Cartesian coordinates ====
The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length {{sfrac|1|φ}} ≈ 0.618 (where φ = {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows:

8 vertices obtained from

:(0, 0, 0, ±1)

by permuting coordinates, and 16 vertices of the form:

:(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}})

The remaining 96 vertices are obtained by taking [[even permutation]]s of

:(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ<sup>−1</sup>|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.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}}

When interpreted as [[#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[icosian]]s.

In the 24-cell, there are [[24-cell#Squares|squares]], [[24-cell#Hexagons|hexagons]] and [[24-cell#Triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}}
In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|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, central polygons and cells 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.{{Efn|name=disjoint from 8 and intersects 16}}}}
In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}
Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]].

The 60 axes and 75 16-cells of the 600-cell constitute a [[Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60<sub>5</sub>75<sub>4</sub> to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.}}
Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs.

==== Hopf spherical coordinates ====
In the 600-cell there are also great circle [[pentagon]]s and [[decagon]]s (in central planes through ten vertices).{{Sfn|Denney|Hooker|Johnson|Robinson|2020}}

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).{{Efn|The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint{{Efn|name=completely disjoint}} 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell.
Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a [[#Diminished 600-cells|snub 24-cell]]) by the 600-cell's 144 pentagons.
Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.{{Efn|Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=438}}
Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.|name=distribution of pentagon vertices in 24-cells}}
Five 24-cells meet at each 600-cell vertex,{{Efn|name=five 24-cells at each vertex of 600-cell}} so all 25 24-cells are linked by each great pentagon.
The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 [[#Decagons|decagonal fibrations]]) by choosing a pentagon from the same fibration at each 24-cell vertex.|name=24-cells bound by pentagonal fibers}}

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]] 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 [[Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Sfn|Zamboj|2021|pp=10-11|loc=§Hopf coordinates}} (𝜉<sub>''i''</sub>, 𝜂, 𝜉<sub>''j''</sub>){{Efn|name=Hopf coordinates|The [[Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles:

: (𝜉<sub>''i''</sub>, 𝜂, 𝜉<sub>''j''</sub>)

that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles.
A Hopf coordinate describes a point as a rotation from a polar point (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉<sub>''i''</sub> and 𝜉<sub>''j''</sub> are angles of rotation in the two [[completely orthogonal]] invariant planes which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]].
The angle 𝜂 is the inclination of both these planes from the polar axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉<sub>''i''</sub>, 0, 𝜉<sub>''j''</sub>) coordinates describe the great circles which intersect at the north and south pole ("lines of longitude").
The (𝜉<sub>''i''</sub>, {{sfrac|𝜋|2}}, 𝜉<sub>''j''</sub>) 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<sub>''i''</sub>, 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉<sub>''j''</sub>) describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}}
Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉<sub>''i''</sub>, 𝜂, 𝜉<sub>''j''</sub>) to unit-radius Cartesian coordinates (w, x, y, z) is:<br>
: w {{=}} cos 𝜉<sub>''i''</sub> sin 𝜂
: x {{=}} cos 𝜉<sub>''j''</sub> cos 𝜂
: y {{=}} sin 𝜉<sub>''j''</sub> cos 𝜂
: z {{=}} sin 𝜉<sub>''i''</sub> sin 𝜂
The Hopf origin pole (0, 0, 0) is Cartesian (0, 1, 0, 0). The conventional "north pole" of Cartesian standard orientation is (0, 0, 1, 0), which is Hopf ({{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}). Cartesian (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|4-dimensional rotations]], denoted SO(4), generates those polytopes.}} given as:

: ({<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|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<sub>''i''</sub> and 𝜉<sub>''j''</sub> 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.{{Efn|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#Cartesian coordinates|120-cell]], which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.}}