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This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide. ... dodecahedron and icosahedron with 4, 6, 8, 12, and ...
In this tiling, there is a parallelogram for each pair of slopes of sides in the -sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling. [5] For instance, the regular octagon can be tiled by two squares and four 45° rhombi. [6]
Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface.
In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope). [1] [2] These polyhedra include:
A 4-polytope is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra. The faces of a uniform 4-polytope must be regular. A 4-polytope is scaliform if it is vertex-transitive, and has all equal length edges.
By a theorem of Descartes, this is equal to 4 π divided by the number of vertices (i.e. the total defect at all vertices is 4 π). The three-dimensional analog of a plane angle is a solid angle. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by
It can be embedded in four-dimensional space as the 4-permutahedron, whose vertices are all permutations of the counting numbers (1,2,3,4). In three-dimensional space, its most symmetric form is generated from six line segments parallel to the face diagonals of a cube. [2] It tiles space to form the bitruncated cubic honeycomb.
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface.