User:Dc.samizdat/Isoclines

= Isoclines in the sequence of regular convex 4-polytopes =

"Isoclinic rotations relate the regular convex 4-polytopes to each other as a sequence of increasingly 3-spherical convex hulls of isoclinic compounds of their predecessors. A general principle of the relationship of a 4-polytope to its adjacent 4-polytopes in the sequence emerges from a comparison of their characteristic isoclines, the circular helical geodesic paths traced by their vertices under isoclinic rotation. A regular 4-polytope's isocline chords are found in the chords of its predecessor's facets, and its isocline vertex figures are the vertex figures of its successor's facets."

Isoclinic compounds in sequence
The main sequence of the six regular convex 4-polytopes is evident in their vertex counts, and within that sequence is a subsequence of four with triangular faces representing the major Coxeter symmetry groups: the 5-cell (tetrahedral A4 symmetry), 16-cell (octahedral B4 symmetry), 24-cell (F4 symmetry), and 600-cell (icosahedral H4 symmetry).