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Regular Polytopes · 4D

24-Cell

{3,4,3} · the octaplex

24-Cell, drawn as a glowing line figure

Twenty-four octahedra, fitted perfectly. It has no equivalent in any other dimension: a shape that exists only in 4D.

Vertices
24
Edges
96
Faces
96
Cells
24

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Watch it turn through the fourth dimension, hear the chord of its edges, and slice it with our space to see it pass through.

Facts

  • The only regular polytope in any dimension with no 3D or 2D counterpart.
  • It is self-dual, and its vertices are the tesseract’s plus the 16-cell’s.
  • It tiles 4D space by itself, as squares tile the plane.
  • Its 24 vertices form a group: the unit Hurwitz quaternions.

Across the dimensions

  • Its equator Vector EquilibriumCut the 24-cell through its middle and the vector equilibrium appears: it is the 24-cell’s equator.
  • Its equator Sphere PackingIn 4D, twenty-four spheres can touch one. Their centres are the vertices of the 24-cell.
  • Its equator Flower of Life in 3DOne dimension up, the same packing is centred on the 24-cell.
  • Its equator Fruit of Life in 3DIn four dimensions the twelve rays become twenty-four, pointing to the corners of the 24-cell.
  • Dual Itself: it is self-dual.

Learn it step by step

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