quality:correction, structural:expansion
4-cell rings

Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4 √ 2. The axis can then be bent into a square of edge length √ 2. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the √ 2 axes of the four octahedra occupy the same plane, forming one of the 18 √ 2 great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane, and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these links all have a common center). An isoclinic rotation in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.