A4 polytope explained

In 4-dimensional geometry, there are 9 uniform polytopes with A₄ symmetry. There is one self-dual regular form, the 5-cell with 5 vertices.

Symmetry

A₄ symmetry, or [3,3,3] is order 120, with Conway quaternion notation +¹/₆₀[I×{{overline|I}}].2₁. Its abstract structure is the symmetric group S₅. Three forms with symmetric Coxeter diagrams have extended symmetry, [<nowiki/>[3,3,3]] of order 240, and Conway notation ±¹/₆₀[I×{{overline|I}}].2, and abstract structure S₅×C₂.

Visualizations

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A₄ Coxeter group, and other subgroups. Three Coxeter plane 2D projections are given, for the A₄, A₃, A₂ Coxeter groups, showing symmetry order 5,4,3, and doubled on even A_k orders to 10,4,6 for symmetric Coxeter diagrams.

The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.