Editing 600-cell (section)

=== Rotations ===
The [[#Geometry|regular convex 4-polytopes]] are an [[Group action|expression]] of their underlying [[Symmetry (geometry)|symmetry]] which is known as [[SO(4)]], the [[Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨<sub>4</sub>.{{Efn|name=distinct rotations}}}} about a fixed point in 4-dimensional Euclidean space.{{Efn|A [[Rotations in 4-dimensional Euclidean space|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]] 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 of directions).
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.{{Efn|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: each moving vertex 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).{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles.
The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}}
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 [[Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}}
An '''isoclinic rotation''' is a different special case, similar but not identical to two simple rotations through the ''same'' angle.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[8-cell#Radial equilateral symmetry|4-dimensional diagonal]].{{Efn|name=isoclinic geodesic}}
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 at least two edge-lengths away.{{Efn|name=isoclinic rotation to non-adjacent vertices}}
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,{{Efn|name=non-vertex geodesic}} each vertex is displaced to another vertex {{radic|1}} (60°) distant, moving {{radic|1/4}} {{=}} 1/2 unit radius in four orthogonal directions.|name=isoclinic 4-dimensional diagonal}} 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 [[helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}}
But they are not like 3-dimensional [[screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}}
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.{{Efn|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'',{{Sfn|Stillwell|2001|p=24}} which are the 4-space analogues{{Efn|name=math of dimensional analogy}} of 2-dimensional great circles in 3-space (great 1-spheres).|name=4-dimensional great circles}}
They are true circles,{{Efn|name=one true 5𝝅 circle}} and even form [[Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.
These '''[[#Rotations on polygram isoclines|isoclines]]''' are geodesic 1-dimensional lines embedded in a 4-dimensional space.
On the 3-sphere{{Efn|All isoclines are [[geodesics]], and isoclines on the [[3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in [[chiral]] pairs as [[Villarceau circle]]s on the [[Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}}
A single black or white isocline forms a [[Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.{{Efn|name=Clifford polygon}}
The double loop is a true circle in four dimensions.{{Efn|name=one true 5𝝅 circle}}
Even and odd isoclines are also linked, not in a Möbius loop but as a [[Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} the geodesic paths traversed by vertices in an [[Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]].
They are [[Helix|helices]] bent into a [[Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[Winding number|winding route]] around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}|name=identical rotations}}

The 600-cell is generated by [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]]{{Efn|name=isoclinic geodesic}} of the 24-cell by 36° = {{sfrac|𝜋|5}} (the arc of one 600-cell edge length).{{Efn|In a ''[[William Kingdon Clifford|Clifford]] displacement'', also known as an [[Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes{{Efn|name=isoclinic invariant planes}} 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 [[William Kingdon Clifford|Clifford]] displacement is [[8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn||name=isoclinic 4-dimensional diagonal}}
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.{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}}
''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).|name=Clifford displacement}}

==== 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.

==== Rotations on polygram isoclines ====
The regular convex 4-polytopes each have their characteristic kind of right (and left) [[isoclinic rotation]], corresponding to their characteristic kind of discrete [[Hopf fibration]] of great circles.{{Efn|The poles of the invariant axis of a rotating 2-sphere are dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.{{Efn|name=Hopf fibration base}} The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,{{Efn|name=Clifford parallels}} but also [[completely orthogonal]]. ''The invariant great circles of the 4D rotation are its poles.'' In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on ''one'' such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, ''an isoclinic 4D rotation of a 3-sphere has nothing but poles'', an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. ''In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.'' The circles are arranged with a surprising symmetry, so that ''each pole-circle links with every other pole-circle'', like a maximally dense fabric of 4D [[chain mail]] in which all the circles are linked through each other, but no two circles ever intersect.|name=dense fabric of pole-circles}} For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel [[#Decagons|great decagons]], so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. We say these isoclinic rotations are ''characteristic'' of the 600-cell because the 600-cell's edges lie in their invariant planes. These rotations only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell.

Just as the [[#Geodesics|geodesic]] ''polygons'' (decagons or hexagons or squares) in the 600-cell's central planes form [[#Fibrations of great circle polygons|fiber bundles of Clifford parallel ''great circles'']], the corresponding geodesic [[Skew polygon|skew]] ''[[Polygram (geometry)|polygrams]]'' (which trace the paths on the [[Clifford torus]] of vertices under isoclinic rotation){{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} form [[fiber bundle]]s of Clifford parallel ''isoclines'': helical circles which wind through all four dimensions.{{Efn|name=isoclinic geodesic}}
Since isoclinic rotations are [[chiral]], occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.{{Sfn|Kim|Rote|2016|p=12-16|loc=§8 The Construction of Hopf Fibrations; see §8.3}}
All the fiber bundles are aspects of the same discrete [[Hopf fibration]], because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations.

Cell rings are another expression of the Hopf fibration.
Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating.
In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.|name=fibrations are distinguished only by rotations}}
The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells.
The [[#Clifford parallel cell rings|Clifford parallel cell rings]] of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets.

Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two orthogonal moving great circles, the way a [[loom]] weaves a piece of fabric from two orthogonal sets of parallel fibers.
A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the [[warp and woof]] of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes.
Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.{{Efn|name=helical geodesic}}

In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.{{Sfn|Perez-Gracia|Thomas|2017|loc=§1. Introduction|ps=; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."{{Efn|name=double rotation}}}}
The isocline bundles occur in pairs of ''left'' and ''right'' chirality; the isoclines in each rotation act as [[chiral]] objects, as does each distinct isoclinic rotation itself.{{Efn|The fibration's [[#Clifford parallel cell rings|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 acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.{{Efn|Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.|name=isoclines have no inherent chirality}}
The characteristic cell rings of the tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, 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 of a curve|torsion]]).
Pairs of left-handed and right-handed isoclines run through cell rings of this kind.
Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed [[#Geometry|predecessors]]' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.|name=directly congruent versus twisted cell rings}}
Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively.
Each left or right fiber bundle of isoclines ''by itself'' constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once.
It is a ''different bundle of fibers'' than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the ''same discrete fibration'' because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-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.{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}
Left and right isoclines are different paths that go to different places.
In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers.

A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation.
Each moving vertex travels along an isocline contained within a (moving) cell ring.
While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant [[24-cell#Planes of rotation|planes of rotation]], they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).{{Efn|name={2p} isoclinic rotations}}
The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is ''both'' a left and right bundle, but they have different bundles of {p} polygons{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.{{Efn|Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations.
Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines.
The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.}}

A [[24-cell#Simple rotations|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. (The 600-cell has four orthogonal [[#Polyhedral sections|central hyperplanes]], each of which is an icosidodecahedron.)
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 [[24-cell#Isoclinic rotations|isoclinic rotation]] is diagonal and global, taking ''all'' the vertices to ''non-adjacent'' vertices (two or more edge-lengths away){{Efn|Isoclinic rotations take each vertex to a non-adjacent vertex at least two edge-lengths away.
In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is exactly two edge-lengths away along one of several great circle geodesic routes: the opposite vertex of a neighboring cell.
In the 8-cell it is three zig-zag edge-lengths away in the same cell: the opposite vertex of a cube. In the 120-cell it is four zig-zag edges away in the same cell: the opposite vertex of a dodecahedron.
|name=isoclinic rotation to non-adjacent vertices}} along diagonal isoclines, and ''all'' the central plane polygons to Clifford parallel polygons (of the same kind).
A left-right pair of isoclinic rotations constitutes a discrete 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.{{Efn|name=two angles between central planes}}
The diagonal isocline{{Efn|name=isoclinic 4-dimensional diagonal}} is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the ''shortest route'' on the 3-sphere, the [[geodesic]].

==== Decagons and pentadecagrams ====
The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 6 [[#Decagons|fibrations of its 72 great decagons]]: 6 fiber bundles of 12 great decagons,{{Efn|name=Clifford parallel decagons}} each delineating [[#Boerdijk–Coxeter helix rings|20 chiral cell rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.{{Efn|name=equi-isoclinic decagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 12 great decagon invariant planes on 5𝝅 isoclines.

The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16}}
12 great polygons comprise a fiber bundle covering all 120 vertices in a discrete [[Hopf fibration]].
There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.{{Efn|name=completely disjoint}}
The 600-cell has six such discrete [[#Decagons|decagonal fibrations]], and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).{{Efn|There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons.
The 30-cell rings are the only chiral objects, other than the ''bundles'' of isoclines or pentagons.{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}}
A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great [[#Decagons|decagons]].
A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.|name=decagonal fibration of chiral bundles}} Each great decagon belongs to just one fibration,{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration).{{^|Efn|name=Schläfli double six}}
The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether.
Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices.

In these ''decagonal'' isoclinic rotations, vertices travel along isoclines which follow the edges of ''hexagons'',{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.{{Efn||name=apparent incongruity}}
In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon.
In a 60°×60° isoclinic rotation (as in the [[24-cell#Isoclinic rotations|24-cell's characteristic hexagonal rotation]], and [[#Hexagons and hexagrams|below in the ''hexagonal'' rotations of the 600-cell]]) this polygram is a [[hexagram]]: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes ''two'' revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are {{radic|3}} chords of the hexagon instead of {{radic|1}} hexagon edges.{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle. The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|name=4𝝅 rotation}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.|name=one true 4𝝅 circle}} But in the 600-cell's 36°×36° characteristic ''decagonal'' rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon ''edges'' is a pentadecagram (15-gram).{{Efn|name=one true 5𝝅 circle}} It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.{{Efn|The analogous relationships among three kinds of {2p} isoclinic rotations, in [[#Fibrations of great circle polygons|Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes]] respectively, are at the heart of the complex nested relationship among the [[#Geometry|regular convex 4-polytopes]].{{Efn|name=polytopes ordered by size and complexity}}
In the {{radic|1}} [[#Hexagons and hexagrams|hexagon {6} rotations characteristic of the 24-cell]], the [[#Rotations on polygram isoclines|isocline chords (polygram edges)]] are simply {{radic|3}} chords of the great hexagon, so the [[24-cell#Simple rotations|simple {6} hexagon rotation]] and the [[24-cell#Isoclinic rotations|isoclinic {6/2} hexagram rotation]] both rotate circles of 6 vertices.
The hexagram isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.{{Efn|name=one true 4𝝅 circle}}
The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,{{Efn|name=digon planes}} so an [[#Hexagons and hexagrams|isoclinic {6} rotation of hexagrams]] is also a {2} rotation of ''axes''.{{Efn|name=direct simple rotations}}
In the {{radic|2}} [[#Squares and octagrams|square {4} rotations characteristic of the 16-cell]], the isocline polygram is an [[16-cell#Helical construction|octagram]], and the isocline's chords are its {{radic|2}} edges and its {{radic|4}} diameters, so the isocline is a circle of circumference 4𝝅. In an isoclinic rotation, the eight vertices of the {8/3} octagram change places, each making one complete revolution through 720° as the isocline [[Winding number|winds]] ''three'' times around the 3-sphere.
The invariant central plane completely orthogonal to each {4} great square is another {4} great square {{radic|4}} distant, so a ''right'' {4} rotation of squares is also a ''left'' {4} rotation of squares.
The 16-cell's [[dual polytope]] the [[8-cell|8-cell tesseract]] inherits the same simple {4} and isoclinic {8/3} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great ''rectangle'' or a {2} great digon (from its successor the 24-cell). 
In the 8-cell this is a rotation of {{radic|1}} × {{radic|3}} great rectangles, and also a rotation of {{radic|4}} axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of {6} great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that [[24-cell#Geometry|the 8-cell and 24-cell have the same edge length]].
In the {{radic|0.𝚫}} [[#Decagons|decagon {10} rotations characteristic of the 600-cell]], the isocline ''chords'' are {{radic|1}} hexagon ''edges'', the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.{{Efn|name=one true 5𝝅 circle}}
The [[#Decagons and pentadecagrams|isoclinic {15/2} pentadecagram rotation]] rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices.
The invariant central plane completely orthogonal to each {10} great decagon is a {0} great 0-gon,{{Efn|name=0-gon central planes}} so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices.
The 600-cell's dual polytope the [[120-cell#Chords|120-cell inherits]] the same simple {10} and isoclinic {15/2} rotations, but [[120-cell#Chords|its characteristic isoclinic rotation]] takes place in completely orthogonal invariant planes which contain {2} great [[digon]]s (from its successor the 5-cell).{{Efn|120 regular 5-cells are inscribed in the 120-cell.
The [[5-cell#Geodesics and rotations|5-cell has digon central planes]], no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis.
The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes{{Efn|name=polytopes ordered by size and complexity}} nested like [[Russian dolls]] can also be seen as an infinite sequence.|name=infinite inscribed sequence}} 
This is a rotation of [[120-cell#Chords|irregular great hexagons]] {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three [[5-cell#Boerdijk–Coxeter helix|5-cell edges]].|name={2p} isoclinic rotations}}

In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.{{Efn|In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it.
Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first [[#Polyhedral sections|surrounding shell]], and one in the second surrounding shell.}}
The two non-adjacent vertices are linked by a {{radic|1}} chord of the isocline which is a great hexagon edge (a 24-cell edge).
The {{radic|1}} chords of the 30-cell ring (without the {{radic|0.𝚫}} 600-cell edges) form a skew [[triacontagram]]<sub>{30/4}=2{15/2}</sub> which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of [[pentadecagram]]<sub>2</sub> isoclines.
Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers.
Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous{{Efn|name=math of dimensional analogy}} to the way the [[Bishop (chess)|bishops]]' diagonal moves restrict them to the white or the black squares of the [[chessboard]].{{Efn|Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}}
The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors.
Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[chessboard]], '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. (Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.)
Things which have [[chirality]] come in '''''right or left''''' enantiomorphous forms: '''isoclinic rotations''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''pairs of Clifford parallel great polygon planes''',{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} '''[[fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings found in the [[16-cell#Helical construction|16-cell]] and [[#Boerdijk–Coxeter helix rings|600-cell]].
Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]].
Some things have '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.{{Efn|name=isoclines have no inherent chirality}}
Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}} The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.|name=black and white}} The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.{{Efn|name=isoclines have no inherent chirality}}
The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations.

At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.{{Efn|Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration 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).}}
Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation.
Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines.
There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell.
Each distinct isocline belongs to 5 fiber bundles.

{| class="wikitable" width="450"
!colspan=4|Three sets of 30-cell ring chords from the same [[orthogonal projection]] viewpoint
|-
![[Pentadecagon#Pentadecagram|Pentadecagram {15/2}]]
![[Triacontagon#Triacontagram|Triacontagram {30/4}=2{15/2}]]
![[Triacontagon#Triacontagram|Triacontagram {30/6}=6{5}]]
|-
|colspan=2 align=center|All edges are [[pentadecagram]] isocline chords of length {{radic|1}}, which are also [[24-cell#Hexagons|great hexagon]] edges of 24-cells inscribed in the 600-cell.
|colspan=1 align=center|Only [[#Golden chords|great pentagon edges]] of length {{radic|1.𝚫}} ≈ 1.176. 
|-
|[[File:Regular_star_polygon_15-2.svg|200px]]
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_6(5,1).svg|200px]]
|-
|valign=top|A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.{{Efn|name=one true 5𝝅 circle}} The {{radic|1}} chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the [[#Boerdijk–Coxeter helix rings|30-cell ring illustration]], where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart.
|valign=top|The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).{{Efn|name=black and white}} The {{radic|1}} chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra.
|valign=top|Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are ''not'' flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great ''hexagon'' planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly congruent, each ''acting'' as a left or right isocline in different fibrations. 
|- 
|colspan=3|No 600-cell edges appear in these illustrations, only [[#Hopf spherical coordinates|invisible interior chords of the 600-cell]]. In this article, they should all properly be drawn as dashed lines. 
|}

Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings, but the two isoclines in each 3-cell ring are directly congruent.{{Efn|The chord-path of an isocline may be called the 4-polytope's ''Clifford polygon'', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}}
The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit.
The double loop is entirely contained within a single [[#Boerdijk–Coxeter helix rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}}
Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell.
Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[Möbius strip]], exactly one edge length apart.
Thus each cell has two helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points.
Globally these two helices are a single connected circle of ''both'' chiralities,{{Efn|An isoclinic rotation by 36° is two simple rotations by 36° at the same time.{{Efn|The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in ''twelve'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.}} It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an ''isocline'', winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).{{Efn|name=double threaded}} The helical double loop 5𝝅 isocline is just a special kind of ''single'' full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is ''one'' true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent. Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isoclinic geodesic}}|name=one true 5𝝅 circle}} with no net [[Torsion of a curve|torsion]].
An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).|name=Clifford polygon}} Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.{{Efn|name=isoclines have no inherent chirality}}
The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one {{radic|0.𝚫}} edge-length apart).
The 30 chords joining the isocline's 30 vertices are {{radic|1}} hexagon edges (24-cell edges), connecting 600-cell vertices which are ''two'' 600-cell {{radic|0.𝚫}} edges apart on a decagon great circle.
{{Efn|Because the 600-cell's [[#Decagons and pentadecagrams|helical pentadecagram<sub>2</sub> geodesic]] 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 follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions.
The Möbius helix is a geodesic "straight line" or ''[[#Decagons and pentadecagrams|isocline]]''.
The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon.
The Petrie triacontagon has {{radic|0.𝚫}} edges; the isoclinic pentadecagram<sub>2</sub> has {{radic|1}} edges which join vertices which are two {{radic|0.𝚫}} edges apart.
Each {{radic|1}} edge belongs to a different [[#Hexagons|great hexagon]], and successive {{radic|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.|name=double threaded}}
These isocline chords are both hexa''gon'' edges and penta''gram'' edges.

The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other.
Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the [[chessboard]].{{Efn|name=isoclinic chessboard}}
In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring.

==== Hexagons and hexagrams ====
[[File:Regular_star_figure_2(10,3).svg|thumb|[[Icosagon#Related polygons|Icosagram {20/6}{{=}}2{10/3}]] contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 10 [[#Hexagons|fibrations of its 200 great hexagons]]: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 20 great hexagon invariant planes on 4𝝅 isoclines.

Each fiber bundle delineates 20 disjoint directly congruent [[24-cell#6-cell rings|cell rings of 6 octahedral cells]] each, with three cell rings nesting together around each hexagon.
The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black {{radic|3}} [[24-cell#Triangles|great triangle]] fibers and a bundle of 20 white great triangle fibers, with a black and a white triangle inscribed in each hexagon and 6 black and 6 white triangles in each 6-octahedron ring.
The black or white triangles are joined by three intersecting black or white isoclines, each of which is a special kind of helical great circle{{Efn|name=one true 4𝝅 circle}} through the corresponding vertices in 10 Clifford parallel black (or white) great triangles. The 10 {{radic|1.𝚫}} chords of each isocline form a skew [[Decagon#decagram|decagram {10/3}]], 10 great pentagon edges joined end-to-end in a helical loop, [[Winding number|winding]] 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane. Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon [[Icosagon#Related polygons|icosagram {20/6}{{=}}2{10/3}]].

Notice the relation between the [[24-cell#Helical hexagrams and their isoclines|24-cell's characteristic rotation in great hexagon invariant planes]] (on hexagram isoclines), and the 600-cell's own version of the rotation of great hexagon planes (on decagram isoclines). They are exactly the same isoclinic rotation: they have the same isocline. They have different numbers of the same isocline, and the 600-cell's {{radic|1.𝚫}} isocline chord is shorter than the 24-cell's {{radic|3}} isocline chord because the isocline intersects more vertices in the 600-cell (10) than it does in the 24-cell (6), but both Clifford polygrams have a 4𝝅 circumference.{{Efn|The 24-cell rotates hexagons on [[24-cell#Helical hexagrams and their 4𝝅 isoclines|hexagrams]], while the 600-cell rotates hexagons on decagrams, but these are discrete instances of the same kind of isoclinic rotation in hexagon invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅.{{Efn|All 3-sphere isoclines{{Efn|name=isoclinic geodesic}} of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference 2𝝅; simple rotations take place on 2𝝅 isoclines.  Double rotations may have isoclines of other than <math>2\pi r</math> circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference.}}|name=4𝝅 rotation}} They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell.{{Efn|The 600-cell's helical {20/6}{{=}}2{10/3} [[20-gon|icosagram]] is a compound of the 24-cell's helical {6/2} hexagram, which is inscribed within it just as the 24-cell is inscribed in the 600-cell.}}

==== Squares and octagrams ====
[[File:Regular_star_polygon_24-5.svg|thumb|The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular [[24-gon#Related polygons|{24/5} 24-gram]], with <big>φ</big> {{=}} {{radic|2.𝚽}} edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell ({{radic|1}} edges not shown).]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 15 [[#Squares|fibrations of its 450 great squares]]: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 30 great square invariant planes (15 completely orthogonal pairs) on 4𝝅 isoclines.

Each fiber bundle delineates 30 chiral [[16-cell#Helical construction|cell rings of 8 tetrahedral cells]] each,{{Efn|name=two different tetrahelixes}} with a left and right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. Axial to each 8-tetrahedron ring is a special kind of helical great circle, an isocline.{{Efn|name=isoclinic geodesic}} In a left (or right) isoclinic rotation of the 600-cell in great square invariant planes, all the vertices circulate on one of 15 Clifford parallel isoclines.

The 30 Clifford parallel squares in each bundle are joined by four Clifford parallel 24-gram isoclines (one through each vertex), each of which intersects one vertex in 24 of the 30 squares, and all 24 vertices of just one of the 600-cell's 25 24-cells. Each isocline is a 24-gram circuit intersecting all 25 24-cells, 24 of them just once and one of them 24 times. The 24 vertices in each 24-gram isocline comprise a unique 24-cell; there are 25 such distinct isoclines in the 600-cell. Each isocline is a skew {24/5} 24-gram, 24 <big>φ</big> {{=}} {{radic|2.𝚽}} chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane. Adjacent vertices of the 24-cell are one {{radic|1}} chord apart, and 5 <big>φ</big> chords apart on its isocline. A left (or right) isoclinic rotation through 720° takes each 24-cell to and through every other 24-cell.

Notice the relations between the [[16-cell#Helical construction|16-cell's rotation of just 2 invariant great square planes]], the [[24-cell#Helical octagrams and their isoclines|24-cell's rotation in 6 Clifford parallel great squares]], and this rotation of the 600-cell in 30 Clifford parallel great squares. These three rotations are the same rotation, taking place on exactly the same kind of isocline circles, which happen to intersect more vertices in the 600-cell (24) than they do in the 16-cell (8).{{Efn|The 16-cell rotates squares on [[16-cell#Helical construction|{8/3} octagrams]], the 24-cell rotates squares on [[24-cell#Helical octagrams and their isoclines|{24/9}=3{8/3} octagrams]], and the 600 rotates squares on {24/5} 24-grams, but these are discrete instances of the same kind of isoclinic rotation in great square invariant planes. In particular, their congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅. They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell or the 16-cell. The 600-cell's helical {24/5} 24-gram is a compound of the 24-cell's helical {24/9} octagram, which is inscribed within the 600-cell just as the 16-cell's helical {8/3} octagram is inscribed within the 24-cell.}} In the 16-cell's rotation the distance between vertices on an isocline curve is the {{radic|4}} diameter. In the 600-cell vertices are closer together, and its {{radic|2.𝚽}} {{=}} <big>φ</big> chord is the distance between adjacent vertices on the same isocline, but all these isoclines have a 4𝝅 circumference.