In 3D there are exactly five perfect solids — the Platonic solids. In 4D there are exactly six, and this is all of them, proven complete in the 1850s. Their "faces" are whole 3D solids (cells): the 120-cell is built from 120 dodecahedra glued together in 4D.
Top: shadows (projections) through an orbiting 4D camera. Bottom: live slices — each solid shape is where that polychoron crosses our 3D world right now.
Turn on color cells: every colored patch on a slice is the cut through one individual cell, so you can watch single dodecahedra enter and leave our world as the shape rotates. Click any slice to light up the source cell in the shadow above it.