Editing 24-cell (section)

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}