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In more than three dimensions, polyhedra generalize to polytopes, with higher-dimensional convex regular polytopes being the equivalents of the three-dimensional Platonic solids. In the mid-19th century the Swiss mathematician Ludwig Schläfli discovered the four-dimensional analogues of the Platonic solids, called convex regular 4-polytopes.
Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of ...
The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ({n/m}#{} where n is odd) or an antiprism ({n/m}#{} where n is even). All polygons in 3 space have an even number of vertices and edges.
The convex regular 4-polytopes are the four-dimensional analogues of the Platonic solids. The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube. The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius.
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.
In three dimensions, there are 5 regular polyhedra known as the Platonic solids. In four dimensions, there are 6 convex regular 4-polytopes, the analogs of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform 4-polytopes, analogous to the 13 semi-regular Archimedean solids in three dimensions ...
But in higher dimensions there are no other regular polytopes. [2] In three dimensions the convex Platonic solids include the fivefold-symmetric dodecahedron and icosahedron, and there are also four star Kepler-Poinsot polyhedra with fivefold symmetry, bringing the total to nine regular polyhedra.
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.