Regular 4-polytope

He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2).

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 (face-to-face) in a regular fashion, forming the surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of the earth is a closed, curved 2-dimensional space).

Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius.

Each greater polytope in the sequence is rounder than its predecessor, enclosing more content within the same radius.

Hyper-tetrahedron 5-point Hyper-octahedron 8-point Hyper-cube 16-point 24-point Hyper-icosahedron 120-point Hyper-dodecahedron 600-point The following table lists some properties of the six convex regular 4-polytopes.

John Conway advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).

[3] Norman Johnson advocated the names n-cell, or pentachoron, hexadecachoron, tesseract or octachoron, icositetrachoron, hexacosichoron, and hecatonicosachoron (or dodecacontachoron), coining the term polychoron being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the Greek roots poly ("many") and choros ("room" or "space").

The Schläfli–Hess 4-polytopes are the complete set of 10 regular self-intersecting star polychora (four-dimensional polytopes).

Conway offered these operational definitions: John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral 600-cell), pI=polyicosahedron {3,5,⁠5/2⁠} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120-cell), with prefix modifiers: g, a, and s for great, (ag)grand, and stellated.

The tesseract is one of 6 convex regular 4-polytopes
This shows the relationships among the four-dimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a cuboctahedron . [ 9 ]
A subset of relations among 8 forms from the 120-cell, polydodecahedron (pD). The three operations {a,g,s} are commutable, defining a cubic framework. There are 7 densities seen in vertical positioning, with 2 dual forms having the same density.