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Convex uniform polytopes
2D
3D
Truncated triangle or uniform hexagon, with Coxeter diagram .
Truncated octahedron,
4D
5D
Truncated 16-cell,
Truncated 5-orthoplex,
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges).
This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions. A strict definition requires uniform polytopes to be finite, while a more expansive definition allows uniform honeycombs (2-dimensional tilings and higher dimensional honeycombs) of Euclidean and hyperbolic space to be considered polytopes as well.
geometry, a uniformpolytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniformpolytopes in two dimensions...
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any...
geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The...
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset...
122 polytope is a uniformpolytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named...
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset...
Mathematics 15:179–89. Norman Johnson UniformPolytopes, Manuscript (1991) N.W. Johnson: The Theory of UniformPolytopes and Honeycombs, Ph.D. (1966) Victor...
In geometry, 1k2 polytope is a uniformpolytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter...
In 7-dimensional geometry, 231 is a uniformpolytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin...
In geometry, 2k1 polytope is a uniformpolytope in n dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter...
antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniformpolytope) bounded by 320 cells: 20 pentagonal antiprisms, and...
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its...
In 7-dimensional geometry, 132 is a uniformpolytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin...
considered to be prismatic 4-polytopes. A duoprism constructed from two regular polygons of the same edge length is a uniform duoprism. A duoprism made of...
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