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Peak, an (n-3)-dimensional element; For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak. Vertex figure: not itself an element of a polytope, but a diagram showing how the elements meet.
Each Platonic solid can therefore be assigned a pair {p, q} of integers, where p is the number of edges (or, equivalently, vertices) of each face, and q is the number of faces (or, equivalently, edges) that meet at each vertex. This pair {p, q}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli ...
Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate.
v3.3.3.3.3 The above shapes may also be realized as slices orthogonal to the long diagonal of a tesseract . If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights r , 3 / 8 , 1 / 2 , 5 / 8 , and s , where r is any number in the range 0 < r ≤ 1 / 4 , and s ...
A realization of the Borromean rings by three mutually perpendicular golden rectangles can be found within a regular icosahedron by connecting three opposite pairs of its edges. [2] Every three unknotted polygons in Euclidean space may be combined, after a suitable scaling transformation, to form the Borromean rings. If all three polygons are ...
[2] [3] This lattice is known as the face-centered cubic lattice in crystallography and is also referred to as the cubic close packed lattice as its vertices are the centers of a close-packing with equal spheres that achieves the highest possible average density. The tetrahedral-octahedral honeycomb is the 3-dimensional case of a simplectic ...
Polyhedra are the three dimensional correlates of polygons. They are built from flat faces and straight edges and have sharp corner turns at the vertices. Although a cube is the only regular polyhedron that admits of tessellation, many non-regular 3-dimensional shapes can tessellate, such as the truncated octahedron.
A 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of [isometries of 3-space that preserve the tiling] is transitive on vertices). There are 28 convex examples in Euclidean 3-space, [1] also called the Archimedean honeycombs.