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The blue vertices lie at (± 1 / ϕ , 0, ±ϕ) and form a rectangle on the xz-plane. (The red, green and blue coordinate triples are circular permutations of each other.) The distance between adjacent vertices is 2 / ϕ , and the distance from the origin to any vertex is √ 3. ϕ = 1 + √ 5 / 2 is the golden ratio.
A regular dodecahedron or pentagonal dodecahedron [notes 1] is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron ...
The concave equilateral dodecahedron, called an endo-dodecahedron. [clarification needed] A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. A regular dodecahedron is an intermediate case with equal edge lengths. A rhombic dodecahedron is a degenerate case with the 6 crossedges reduced to ...
Regular star polygons are not convex, and their Schläfli symbols {p / q} contain irreducible fractions p / q, where p is the number of vertices, and q is their turning number. Equivalently, {p / q} is created from the vertices of {p}, connected every q. For example, {5 ⁄ 2} is a pentagram; {5 ⁄ 1} is a pentagon.
A vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. This vertex figure has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for vertex-uniform polyhedra all the neighboring vertices are in the same plane and so this plane projection can be used to visually represent the vertex configuration.
Among the fonts in widespread use, [6] [7] full implementation is provided by Segoe UI Symbol and significant partial implementation of this range is provided by Arial Unicode MS and Lucida Sans Unicode, which include coverage for 83% (80 out of 96) and 82% (79 out of 96) of the symbols, respectively.
The relations can be made apparent by examining the vertex figures obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example, the cube has vertex figure 4.4.4, which is to say, three adjacent square faces.
Net. In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C 120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron [1] and hecatonicosahedroid.