Patron (géométrie)En géométrie, le patron d'un polyèdre est une figure géométrique plane en un seul morceau qui permet de reconstituer le polyèdre après plusieurs pliages (au niveau de certaines arêtes, les autres apparaissant par jonction des bords du patron). Le terme de patron est à prendre ici dans son deuxième sens : celui de modèle pour construire un objet. Développer un polyèdre consiste à rabattre les différentes faces du polyèdre dans un même plan par découpage selon les arêtes.
Rotation en quatre dimensionsEn mathématiques, les rotations en quatre dimensions (souvent appelées simplement rotations 4D) sont des transformations de l'espace euclidien , généralisant la notion de rotation ordinaire dans l'espace usuel ; on les définit comme des isométries directes ayant un point fixe (qu'on peut prendre comme origine, identifiant les rotations aux rotations vectorielles) ; le groupe de ces rotations est noté SO(4) : il est en effet isomorphe au groupe spécial orthogonal d'ordre 4.
Hyperoctahedral groupIn mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the hypercube. As a Coxeter group it is of type B_n = C_n, and as a Weyl group it is associated to the symplectic groups and with the orthogonal groups in odd dimensions. As a wreath product it is where S_n is the symmetric group of degree n.
RhomboèdreEn géométrie, un rhomboèdre est un polyèdre ressemblant au cube, excepté que ses faces ne sont pas carrées mais en forme de losanges. C'est un des cas particuliers d'un parallélépipède où toutes les arêtes sont de la même longueur. En général, le rhomboèdre peut avoir trois types de faces rhombiques par faces opposées congrues. Si tous les angles internes non-obtus des faces sont égaux, il peut être appelé un trapézoèdre trigonal.
Truncated tesseractIn geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the truncated 16-cell. The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. Truncated tesseract (Norman W. Johnson) Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers) The truncated tesseract may be constructed by truncating the vertices of the tesseract at of the edge length.
Rectified tesseractIn geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a runcic tesseract. It has two uniform constructions, as a rectified 8-cell r{4,3,3} and a cantellated demitesseract, rr{3,31,1}, the second alternating with two types of tetrahedral cells. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC8.
Tesseractic honeycombIn four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets. Its vertex figure is a 16-cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex. It is an analog of the square tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3-space.
HyperrectangleIn geometry, an orthotope (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all of the edges are equal length, it is a hypercube. A hyperrectangle is a special case of a parallelotope. A three-dimensional orthotope is also called a right rectangular prism, rectangular cuboid, or rectangular parallelepiped. A four-dimensional orthotope is likely a hypercuboid.
Space diagonalIn geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the same face (but not on the same edge) as each other. For example, a pyramid has no space diagonals, while a cube (shown at right) or more generally a parallelepiped has four space diagonals. An axial diagonal is a space diagonal that passes through the center of a polyhedron.
Runcinated tesseractsIn four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract. There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations. The runcinated tesseract or (small) disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
