4D Immersions

4D Immersions: Tesseract

A tesseract (hypercube) is the 4D analogue of a cube. It has 16 vertices, 32 edges, 24 square faces, and 8 cubic cells. What you see is its stereographic projection into 3D as it rotates in the XW, YW, and ZW planes.

Auto-rotate (4D)

Projection

Stereographic: ℝ⁴ → ℝ³

(x,y,z,w) ↦ d/(d−w) · (x,y,z)

d = 4 (projection distance)

4D Rotation Planes

XW plane — rotates x against the 4th dimension

YW plane — rotates y against the 4th dimension

ZW plane — rotates z against the 4th dimension

In 4D there are 6 independent rotation planes (vs 3 axes in 3D)

Key Properties

  • • 16 vertices, 32 edges, 24 faces, 8 cells
  • • Symmetry group: B₄ (order 384)
  • • Vertex figure: regular tetrahedron
  • • Schläfli symbol: {4,3,3}