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Augmentation involves attaching the Johnson solids onto one or more faces of polyhedra, while elongation or gyroelongation involve joining them onto the bases of a prism or antiprism, respectively. Some others are constructed by diminishment, the removal of one of the first six solids from one or more of a polyhedron's faces. [6]
15 are in the regular H 4 [3,3,5] group (120-cell/600-cell) family. 1 special snub form in the [3,4,3] group family. 1 special non-Wythoffian 4-polytope, the grand antiprism. TOTAL: 68 − 4 = 64; These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.
3.4 Catalan solids. ... 6.1 Honeycombs. ... A polytope is a geometric object with flat sides, which exists in any general number of dimensions.
[W] Wenninger, 1974, has 119 figures: 1–5 for the Platonic solids, 6–18 for the Archimedean solids, 19–66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67–119 for the nonconvex uniform polyhedra.
6: 10: 18: 1: 7: 4: ... There are 4 regular projective polyhedra related to 4 of 5 Platonic solids. ... There are two main geometric classes of apeirotope: ...
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 ...
In geometry, a 4-polytope (sometimes also called a polychoron, [1] polycell, or polyhedroid) is a four-dimensional polytope. [ 2 ] [ 3 ] It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices , edges , faces ( polygons ), and cells ( polyhedra ).
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.In particular, all its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.