Editing 600-cell (section)

==== Twenty-five 24-cells ====
There are 25 inscribed 24-cells in the 600-cell.{{sfn|Denney|Hooker|Johnson|Robinson|2020}}{{Efn|The 600-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into sets of Clifford parallel invariant rotation planes of 25 distinct ''isoclinic'' rotations, and are usually given as those sets.{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}}
Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}}

The 8-vertex 16-cell has 4 long diameters inclined at 90° = {{sfrac|𝜋|2}} to each other, often taken as the 4 orthogonal axes or [[16-cell#Coordinates|basis]] of the coordinate system.

The 24-vertex 24-cell has 12 long diameters inclined at 60° = {{sfrac|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 {{sfrac|𝜋|3}} with respect to each other.{{Efn|The three 16-cells in the 24-cell are rotated by 60° ({{sfrac|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° ({{sfrac|2𝜋|3}}) apart.
In a unit-radius 4-polytope, vertices 120° apart are joined by a {{radic|3}} chord.|name=120° apart}}

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.{{Sfn|Waegell|Aravind|2009|loc=§3. The 600-cell|pp=2-5}}
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.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells|[[Pieter Hendrik Schoute|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.{{Sfn|Denney|Hooker|Johnson|Robinson|2020|p=434}}}}

Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually [[24-cell#Clifford parallel polytopes|isoclinic polytopes]].
The rotational distance between inscribed 24-cells is always {{sfrac|𝜋|5}} in each invariant plane of rotation.{{Efn|There is a single invariant plane in each simple rotation, and a completely orthogonal fixed plane.
There are an infinite number of pairs of [[completely orthogonal]] invariant planes in each isoclinic rotation, all rotating through the same angle;{{Efn|name=dense fabric of pole-circles}} nonetheless, not all [[#Geodesics|central planes]] are [[24-cell#Isoclinic rotations|invariant planes of rotation]].
The invariant planes of an isoclinic rotation constitute a [[#Fibrations of great circle polygons|fibration]] of the entire 4-polytope.{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}}
In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great [[#Decagons|decagons]], ''or'' 20 Clifford parallel great [[#Hexagons|hexagons]] ''or'' 30 Clifford parallel great [[#Squares|squares]] are invariant planes of rotation.|name=isoclinic invariant planes}}

Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are {{sfrac|𝜋|5}} apart on two non-intersecting Clifford parallel{{Efn|name=Clifford parallels}} decagonal great circles (as well as {{sfrac|𝜋|5}} apart on the same decagonal great circle).{{Efn|Two Clifford parallel{{Efn|name=Clifford parallels}} 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 [[Hopf fibration|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]]{{Efn|name=Boerdijk–Coxeter helix}} 30 tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} long.
The three Clifford parallel decagons can be seen as the cyan edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]].
Each magenta edge is one edge of another decagon linking two parallel decagons.|name=Clifford parallel decagons}}
An isoclinic rotation of decagonal planes by {{sfrac|𝜋|5}} takes each 24-cell to a disjoint 24-cell (just as an [[24-cell#Clifford parallel polytopes|isoclinic rotation of hexagonal planes]] by {{sfrac|𝜋|3}} takes each 16-cell to a disjoint 16-cell).{{Efn|name=isoclinic geodesic displaces every central polytope}}
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''.{{Efn|A ''disjoint'' 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation{{Efn|name=identical rotations}} takes it past (not through) the adjacent 24-cell it rotates toward,{{Efn|Five 24-cells meet at each vertex of the 600-cell,{{Efn|name=five 24-cells at each vertex of 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 [[24-cell#Isoclinic rotations|isoclinic rotation]]{{Efn|name=isoclinic geodesic displaces every central polytope}}) directly toward an adjacent 24-cell.|name=four directions toward another 24-cell}} and left or right to a more distant 24-cell from which it is completely disjoint.{{Efn|name=completely disjoint}}
The four directions reach 8 different 24-cells{{Efn|name=disjoint from 8 and intersects 16}} because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once.
Four paths are right-hand [[Screw thread#Handedness|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).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}|name=rotations to 8 disjoint 24-cells}}
The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells.

All [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).{{Efn|All isoclinic ''polygons'' are Clifford parallels (completely disjoint).{{Efn||name=completely disjoint}}
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 {{sfrac|𝜋|3}} with respect to each other, yet are not disjoint: they share a [[16-cell#Octahedral dipyramid|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).|name=isoclinic and not disjoint}}
Each 24-cell is isoclinic ''and'' Clifford parallel to 8 others, and isoclinic but ''not'' Clifford parallel to 16 others.{{Efn|name=disjoint from 8 and intersects 16}}
With each of the 16 it shares 6 vertices: a hexagonal central plane.{{Efn|name=five 24-cells at each vertex of 600-cell}}
Non-disjoint 24-cells are related by a [[Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] by {{sfrac|𝜋|5}} in an invariant plane intersecting only two vertices of the 600-cell,{{Efn|name=digon planes}} a rotation in which the completely orthogonal [[24-cell#Simple rotations|fixed plane]] is their common hexagonal central plane.
They are also related by an [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in which both planes rotate by {{sfrac|𝜋|5}}.{{Efn|In the 600-cell, there is a [[24-cell#Simple rotations|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]],{{Efn|name=digon planes}} and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),{{Efn|name=non-vertex geodesic}} 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively.
Two ''non-disjoint'' 24-cells are related by a [[24-cell#Simple rotations|simple rotation]] through {{sfrac|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.{{Efn|name=rotations to 16 non-disjoint 24-cells}}|name=direct simple rotations}}

There are two kinds of {{sfrac|𝜋|5}} isoclinic rotations which take each 24-cell to another 24-cell.{{Efn|Any isoclinic rotation by {{sfrac|𝜋|5}} in decagonal invariant planes{{Efn|Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,{{Efn|name=isoclinic invariant planes}} and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.{{Efn|name=non-vertex geodesic}}
As the invariant planes rotate in two completely orthogonal directions at once,{{Efn|name=helical geodesic}} all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines{{Efn|name=isoclinic geodesic}} 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.}} takes ''every'' [[#Geodesics|central polygon]], [[#Clifford parallel cell rings|geodesic cell ring]] or inscribed 4-polytope{{Efn|name=4-polytopes inscribed in the 600-cell}} in the 600-cell to a [[24-cell#Clifford parallel polytopes|Clifford parallel polytope]] {{sfrac|𝜋|5}} away.|name=isoclinic geodesic displaces every central polytope}}
''Disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Decagons|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.){{Efn|name=rotations to 8 disjoint 24-cells}}
''Non-disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Hexagons|fibration of 20 Clifford parallel ''hexagonal'' invariant planes]].{{Efn|Notice the apparent incongruity of rotating ''hexagons'' by {{sfrac|𝜋|5}}, since only their opposite vertices are an integral multiple of {{sfrac|𝜋|5}} apart.
However, [[#Icosahedra|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 [[#Hexagons|10 discrete fibrations]] and [[#Clifford parallel cell rings|cell rings]], not Clifford parallel to the [[#Decagons|decagonal fibrations]] but also by fives{{Efn|name=24-cells bound by pentagonal fibers}} in that five 24-cells meet at each vertex, each pair sharing a hexagon.{{Efn|name=five 24-cells at each vertex of 600-cell}}
Each hexagon rotates ''non-invariantly'' by {{sfrac|𝜋|5}} in a [[#Hexagons and hexagrams|hexagonal isoclinic rotation]] between ''non-disjoint'' 24-cells.{{Efn|name=rotations to 16 non-disjoint 24-cells}} Conversely, in all [[#Decagons and pentadecagrams|{{sfrac|𝜋|5}} isoclinic rotations in ''decagonal'' invariant planes]], all the vertices travel along isoclines{{Efn|name=isoclinic geodesic}} which follow the edges of ''hexagons''.|name=apparent incongruity}}
(There are 10 such sets of fibers, so there are 20 such distinct rotations.){{Efn|At each vertex, a 600-cell has four adjacent (non-disjoint){{Efn||name=completely disjoint}} 24-cells that can each be reached by a simple rotation in that direction.{{Efn|name=four directions toward another 24-cell}}
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.{{Efn|name=disjoint from 8 and intersects 16}}
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 {{sfrac|𝜋|5}}.
The double rotation reaches an adjacent 24-cell ''directly'' as if indirectly by two successive simple rotations:{{Efn|name=double rotation}} first to one of the ''other'' adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).|name=rotations to 16 non-disjoint 24-cells}}

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 [[24-cell#Geodesics|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|geodesics]], each set of which covers all 120 vertices and constitutes a discrete [[#Hexagons|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.