Convex regular 4-polytope Totally Explained

Everything about Convex Regular 4-polytope totally explained

In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both regular and convex. These are the four-dimensional analogs of the Platonic solids (in three dimensions) and the regular polygons (in two dimensions). These polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. Schläfli discovered that there are precisely six such figures. Five of these may be thought of as higher dimensional analogs of the Platonic solids. There is one additional figure (the 24-cell) which has no three-dimensional equivalent.
Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces in a regular fashion.

Properties

The following tables lists some properties of the six convex regular polychora. The symmetry groups of these polychora are all Coxeter groups and given in the notation described in that article. The number following the name of the group is the order of the group.

Name	Family	Schläfli symbol	Vertices	Edges	Faces
pentachoron	simplex
Wireframe orthographic projections

Solid orthographic projections (cell-centered)
tetrahedral envelope	cubic envelope	octahedral envelope	cuboctahedral envelope	truncated rhombic triacontahedron envelope	pentakis dodecahedral envelope
Wireframe Schlegel diagrams (Perspective projection)
(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Cell-centered)	(Vertex-centered)
Wireframe stereographic projections (Hyperspherical)

Further Information

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