24-cell
Isoclinic rotations

When the angles of rotation in the two invariant planes are exactly the same, a remarkably symmetric transformation occurs: all the great circle planes Clifford parallel to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates isoclinically in many directions at once. Each vertex moves an equal distance in four orthogonal directions at the same time. In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates all 16 hexagons by 60 degrees, and takes every great circle polygon (square, hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a Clifford displacement, after its discoverer.

The 24-cell in the double rotation animation appears to turn itself inside out. It appears to, because it actually does, reversing the chirality of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in two completely orthogonal planes one of which is a great hexagon, each vertex rotates first to a vertex two edge lengths away (√ 3 and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between. Each √ 3 chord of the helical geodesic crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both. The √ 3 chords meet at a 60° angle, but since they lie in different planes they form a helix not a triangle. Three √ 3 chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of √ 3 chords closes into a loop only after six √ 3 chords: a 720° rotation twice around the 24-cell on a skew hexagram with √ 3 edges. Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has not arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees and been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's orientation in the 4-space in which it is embedded is now different. Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the same direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic will arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each hexagram geodesic to complete a circuit through every second vertex of its six vertices by winding around the 24-cell twice, returning the 24-cell to its original chiral orientation.

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a Möbius ring, so that the two strands of the double helix form a continuous single strand in a closed loop. In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius ring is a closed spiral not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.
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