PermutohedronIn mathematics, the permutohedron of order n is an (n − 1)-dimensional polytope embedded in an n-dimensional space. Its vertex coordinates (labels) are the permutations of the first n natural numbers. The edges identify the shortest possible paths (sets of transpositions) that connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one transposition), and the numbers on these places are neighbors (differ in value by 1).
BitruncationIn geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular polytopes can be represented by an extended Schläfli symbol notation t_1,2{p,q,...} or 2t{p,q,...}. For regular polyhedra (i.e. regular 3-polytopes), a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.
TétrakihexaèdreUn tétrakihexaèdre est un solide de Catalan (le dual d'un solide d'Archimède). Son dual est l'octaèdre tronqué. Il peut être vu comme un cube dont chaque face (de côté a) est couverte par une pyramide carrée (de hauteur a/4). Cette interprétation est exprimée dans le nom, d'origine grecque : = « hexaèdre » (six faces) = cube, = « quatre fois » = faces partagées en 4). Le rapport entre les longueurs des deux types d'arêtes est de 3/4.
Elongated dodecahedronIn geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism. Along with the rhombic dodecahedron, it is a space-filling polyhedron, one of the five types of parallelohedron identified by Evgraf Fedorov that tile space face-to-face by translations.
Coupole hexagonaleIn geometry, the triangular cupola is one of the Johnson solids (J_3). It can be seen as half a cuboctahedron. The following formulae for the volume (), the surface area () and the height () can be used if all faces are regular, with edge length a: The dual of the triangular cupola has 6 triangular and 3 kite faces: The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. This isn't a Johnson solid because of its coplanar faces.
Chamfer (geometry)In geometry, chamfering or edge-truncation is a topological operator that modifies one polyhedron into another. It is similar to expansion, moving faces apart and outward, but also maintains the original vertices. For polyhedra, this operation adds a new hexagonal face in place of each original edge. In Conway polyhedron notation it is represented by the letter c. A polyhedron with e edges will have a chamfered form containing 2e new vertices, 3e new edges, and e new hexagonal faces.
Pyramide à base carréeEn géométrie, une pyramide à base carrée est une pyramide avec une base carrée et quatre faces latérales triangulaires. Si les quatre faces triangulaires sont équilatérales, alors la pyramide est un solide de Johnson (J1), et peut être pensée comme la moitié d'un octaèdre. D'autres pyramides carrées ne sont pas semblables à ce solide de Johnson ; la pyramide de Khéops, par exemple, possède quatre faces triangulaires isocèles non équilatérales.
Runcinated 5-cellIn four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell. There are 3 unique degrees of runcinations of the 5-cell, including with permutations, truncations, and cantellations. The runcinated 5-cell or small prismatodecachoron is constructed by expanding the cells of a 5-cell radially and filling in the gaps with triangular prisms (which are the face prisms and edge figures) and tetrahedra (cells of the dual 5-cell).
