6-cubeIn geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol {4,34}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets.
Semiregular polytopeIn geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition. In three-dimensional space and below, the terms semiregular polytope and uniform polytope have identical meanings, because all uniform polygons must be regular.
Uniform 5-polytopeIn geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.
Polygone de PetrieEn géométrie, un polygone de Petrie est donné par la projection orthogonale d'un polyèdre (ou même d'un polytope au sens général) sur un plan, de sorte à former un polygone régulier, avec tout le reste de la projection à l’intérieur. Ces polygones et graphes projetés sont utiles pour visualiser la structure et les symétries de polytopes aux nombreuses dimensions. Chaque paire de côtés consécutifs appartient à une même face du polyèdre, mais pas trois.
Hyperoctaèdrethumb|Diagramme de Schlegel de l'hexadécachore, hyperoctaèdre en dimension 4. Un hyperoctaèdre est, en géométrie, un polytope régulier convexe, généralisation de l'octaèdre en dimension quelconque. Un hyperoctaèdre de dimension n est également parfois nommé polytope croisé, n-orthoplexe ou cocube. Un hyperoctaèdre est l'enveloppe convexe des points formés par toutes les permutations des coordonnées (±1, 0, 0, ..., 0). En dimension 1, l'hyperoctaèdre est simplement le segment de droite [-1, +1] ; en dimension 2, il s'agit d'un carré de sommets {(1, 0), (-1, 0), (0, 1), (0, -1)}.
