Editing 24-cell (section)

==== Helical hexagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[hexagram]]<sub>2</sub>, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) 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 [[#Cell 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 both 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, 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).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}}

Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite "edges" of a [[Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram<sub>2</sub> path. Followed along the column of six octahedra (and "around the end" where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h<sub>1</sub> is {12}, h<sub>2</sub> is {12/5}''}} In contrast, the skew hexagram<sub>2</sub> isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram<sub>2</sub> has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[Reye configuration]] 12<sub>4</sub>16<sub>3</sub>, just the way the 24-cell's 12 axes and [[#Hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}