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In geometry, a dodecahedron (from Ancient Greek δωδεκάεδρον (dōdekáedron); from δώδεκα (dṓdeka) 'twelve' and ἕδρα (hédra) 'base, seat, face') or duodecahedron [1] is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid.
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 ...
In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron, one of the Archimedean solids. It has 60 vertices and 120 edges, and is a quartic graph Archimedean graph. [5] Square centered Schlegel diagram
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb.
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.
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.
Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property.
The defect of any of the vertices of a regular dodecahedron (in which three regular pentagons meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles measures 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.