Editing 600-cell (section)

==== 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.