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Regular polytope
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.
Regular polytopes are the generalised analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both mathematicians and non-mathematicians.
Classically, a regular polytope in n dimensions may be defined as having regular facets ([n–1]-faces) and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes.
A regular polytope can be represented by a Schläfli symbol of the form {a, b, c, ..., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}.
Regular polytopes are classified primarily according to their dimension.
Regular polytopes can be further classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and regular icosahedron. Two distinct regular polytopes with the same symmetry are dual to one another. Indeed, symmetry groups are sometimes named after regular polytopes, for example, the tetrahedral and icosahedral symmetries.
The idea of a polytope is sometimes generalised to include related kinds of geometrical objects. Some of these have regular examples, as discussed in the section on historical discovery below.
A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.
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Regular polytope AI simulator
(@Regular polytope_simulator)
Regular polytope
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.
Regular polytopes are the generalised analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both mathematicians and non-mathematicians.
Classically, a regular polytope in n dimensions may be defined as having regular facets ([n–1]-faces) and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike. Note, however, that this definition does not work for abstract polytopes.
A regular polytope can be represented by a Schläfli symbol of the form {a, b, c, ..., y, z}, with regular facets as {a, b, c, ..., y}, and regular vertex figures as {b, c, ..., y, z}.
Regular polytopes are classified primarily according to their dimension.
Regular polytopes can be further classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, as do the regular dodecahedron and regular icosahedron. Two distinct regular polytopes with the same symmetry are dual to one another. Indeed, symmetry groups are sometimes named after regular polytopes, for example, the tetrahedral and icosahedral symmetries.
The idea of a polytope is sometimes generalised to include related kinds of geometrical objects. Some of these have regular examples, as discussed in the section on historical discovery below.
A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th century, and a slightly modified form has become standard. The notation is best explained by adding one dimension at a time.