A polychoron (/ˌpälēˈkôrən/ or /ˌpälēˈkôrän/), plural polychora, is a 4-dimensional polytope composed of points, line segments, faces, cells, and a single teron, all of which are polytopes. Polychora can be considered subsets of a Euclidean flune.
Polychora make up the boundary of 5D polytera and define their surteron bulk.
Uniform polychora[]
A uniform polychoron is a polychoron that are vertex-transitive and bound by uniform facets. Each face is a regular polygon and each cell is a uniform polyhedron. As of June 2025, there are 2191 known uniform polychora (excluding the infinite duoprism and antiduoprism families), most of which were discovered by the two amateur mathematicians Jonathan Bowers and George Olshevsky. Uniform polychora whose cells are all congruent and angles are all equal are known as regular polychora. All regular polychora have a three-entry Schläfli symbol.
There are sixteen regular polychora, six convex and ten star polychora (known as the Schläfli-Hess polytopes).
The six regular convex polychora are the pentachoron (pen), the tesseract (tes), the hexadecachoron (hex), the icositetrachoron (ico), the hecatonicosachoron (hi), and the hexacosichoron (ex).
The ten regular star polychora are the faceted hexacosichoron (fix), the great hecatonicosachoron (gohi), the grand hecatonicosachoron (gahi), the small stellated hecatonicosachoron (sishi), the great grand hecatonicosachoron (gaghi), the great stellated hecatonicosachoron (gishi), the grand stellated hecatonicosachoron (gashi), the great faceted hexacosichoron (gofix), the grand hexacosichoron (gax), and the great grand stellated hecatonicosachoron (gogishi).
See Also[]
| Polytope |
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| Null polytope · Point · Polytelon · Polygon · Polyhedron · Polychoron · Polyteron · Polypeton · Polyecton · Polyzetton · Polyyotton |