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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.
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.
A polygon is a 2-dimensional polytope. Polygons can be characterised according to various criteria. Some examples are: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (including both its boundary and its interior), and self-intersecting with varying densities of different regions.
The n dimensional real projective space is the quotient of the n sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1. The n dimensional torus is the product space of n circles. Its Euler characteristic is 0, by the product property.
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
Given a point A 0 in a Euclidean space and a translation S, define the point A i to be the point obtained from i applications of the translation S to A 0, so A i = S i (A 0).The set of vertices A i with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter.
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric ...
The cuboctahedron has 1 transitivity class of 12 vertices, 1 class of 24 edges, and 2 classes of faces: 8 triangular and 6 square; each element in a matrix's diagonal. [17] The 24 edges can be seen in 4 central hexagons.