User:Dc.samizdat/sandbox/scratched

Great squares of the 16-cell
The 8 vertices of the 16-cell lie on the 4 coordinate axes and form 6 great squares in the 6 orthogonal central planes. By convention rotations are always specified in two completely orthogonal invariant planes xy (whose vertices are numbered by 𝜉xy) and wz (whose vertices are numbered by 𝜉wz). The rotation in the xy plane does not move points in the wz plane, and vice versa. In the 16-cell these two simple rotations rotate disjoint sets of 4 vertices each (because completely orthogonal planes intersect only at the origin and share no vertices). The 𝜂 coordinate of the 4 vertices in the wz plane is 1 ($𝜋⁄8$) and the 𝜂 coordinate of the 4 vertices in the xy plane is 0. The x and y coordinates of the vertices in the wz plane are 0 regardless of the rotational position of the xy plane (the 𝜉xy coordinate), and the w and z coordinates of the vertices in the xy plane are 0 regardless of the rotational position of the wz plane (the 𝜉wz coordinate); thus there are 4 Hopf coordinate synonyms for each vertex.

Great circle hexagons of the 24-cell
{| class="wikitable" style="white-space:nowrap;" !colspan=1|Great circle hexagons of the 24-cell Cartesian[ 0, ±$𝜋⁄8$, ±$𝜋⁄8$, 0 ]( ±$𝜋⁄2$, ±$𝜋⁄2$, ±$𝜋⁄2$, ±$𝜋⁄2$ ) Hopf({ <6 }$𝜋⁄2$, {≤3}$𝜋⁄2$, { <6 }$𝜋⁄2$)6


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