The six regular 4D polytopes, built and checked, to turn through w and slice.
Molecules and crystals are built on regular shapes, so first the shapes. Our space has five regular solids. Four dimensions have six regular polytopes, found by Ludwig Schläfli around 1852 and again by Washington Stringham in 1880. Five are the cousins of ours. The sixth, the 24-cell, has no 3D cousin: 24 octahedra, each vertex shared by six, and it's its own dual. From five dimensions up there are only three, so four is the richest dimension there is. Four is also the only dimension past two that can be tiled by more than one regular shape: by tesseracts, by 16-cells, or by 24-cells (Foam⁴'s 24-cell foam).
Turn spins one in our three directions. Through w turns it into the fourth: its shadow turns inside out, the way a cube's does when it rolls. Slice shows what a 3D creature sees, the 3D cross-section, and the slider moves the slice along w. Colour is how far along w a part sits: kata, mid, ana. Every vertex, edge, face and cell here is computed. The counts come out as Schläfli's, and V − E + F − C = 0 for all six.
ATOM⁴: the position · how it works · Lumen⁴ · IRIS⁴ · KATABASIS · Four · the arcade