Petrie polygonIn geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides (but no n) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides (but no three) belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.
Quasiregular polyhedronIn geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive. Their dual figures are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular.
Abstract polytopeIn mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a realization of an abstract polytope in some real N-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.
Harold Scott MacDonald CoxeterHarold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts.
Coxeter notationIn geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram.
Isogonal figureIn geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second.
Complex polytopeIn geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Precise definitions exist only for the regular complex polytopes, which are configurations.
Prism (geometry)In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.
