Semiregular polytope

A semiregular polytope is a isogonal polytope that only contains regular facets. The semiregular polytopes are a subset of the uniform polytopes, and the convex semiregular polytopes are a subset of the Blind polytopes.

The concept was created as a generalization of uniformity in 3D to higher dimensions, but a broader definition of uniforms for 4D and above gained more traction in the enthusiast community.

The convex semiregular polytopes were fully enumerated in all dimensions by the Blinds, but already in 4D the full set of nonconvex semiregular polytopes is not known.

Convex examples

• All convex regular polytopes (sometimes excluded)
• All convex uniform polyhedra (Archimedean solids, prisms, and antiprisms)
• The k₂₁ family in 4 to 8 dimensions
  • The rectified pentachoron (0₂₁) "tetroctahedric"
  • The demipenteract (1₂₁) "5-ic semi-regular"
  • The 27-72-peton (2₂₁) "6-ic semi-regular"
  • The 126-576-exon (3₂₁) "7-ic semi-regular"
  • The 2160-17280-zetton (4₂₁) "8-ic semi-regular"
• The rectified hexacosichoron "octicosahedric"
• The snub disicositetrachoron "tetricosahedric"

Nonconvex examples

• All nonconvex regular polytopes (sometimes excluded)
• All nonconvex uniform polyhedra
• Conjugates of convex semiregulars
  • The rectified grand hexacosichoron
  • The retrosnub disicositetrachoron
• Facetings of regulars with uniform vertex figures
  • The tesseractihemioctachoron and other demicrosses
  • The small ditrigonary hexacosihecatonicosachoron
  • The ditrigonary dishecatonicosachoron
  • The great ditrigonary hexacosihecatonicosachoron
• Snub facetings of the small stellated hecatonicosachoron
  • The retroantiprismatosnub disicositetrachoron
  • The snub hexecontatetrasnub-snub disoctachoron
  • The snub hecatonicosoctasnub disoctachoron
  • The snub triacontadihexecontatetrasnub disoctachoron
  • The snub hexecontatetrasnub dishexadecachoron
• Ionic decachoric partial faceting of the hecatonicosachoron
  • The tetrasnub antipodic disdecachoron
• Blends of semiregulars
  • The small disnub dishexacosichoron
  • The great disnub dishexacosichoron

Euclidean honeycombs

• All regular honeycombs (sometimes excluded)
• All uniform tilings of 2-space
• Convex cases
  • The tetrahedral-octahedral honeycomb "simple tetroctahedric check"
  • The gyrated tetrahedral-octahedral honeycomb "complex tetroctahedric check"
  • The gosset octacomb "9-ic check"
• Slab cases
  • Any prism product of a hypercube and a hypercubic honeycomb
  • The triangular tiling antiprism
• Nonconvex cases
  • The cubihemisquare honeycomb and its analogs
  • The tetrahedral-hemitriangular honeycomb
  • The octahedral-hemitriangular honeycomb