A tesseract is the 4D analogue of a cube: 16 vertices connected by 32 edges, each vertex at (±1,±1,±1,±1). Rotating it through the 4th dimension (the xw-plane) and projecting to 3D makes the "inner" and "outer" cubes appear to pass through one another.
x' = x·cosθ - w·sinθ
w' = x·sinθ + w·cosθ
proj: (x,y,z,w) -> (x,y,z) / (2 - w/d) [perspective 4D→3D]
- 4D rotation angle — manual rotation in the xw-plane (drag to see the hypercube "turn inside out").
- Rotation speed — auto-advances the 4D angle over time when running.
- Slice plane (w) — shows the 3D cross-section of the tesseract at a fixed w value (a hyperplane slice), highlighted in a second colour.
Just as a 3D cube casts a 2D shadow, a 4D tesseract casts a 3D "shadow" — the shape you see distorting as it rotates.