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The regular dodecahedron can be interpreted as a truncated trapezohedron. It is the set of polyhedrons that can be constructed by truncating the two axial vertices of a trapezohedron. Here, the regular dodecahedron is constructed by truncating the pentagonal trapezohedron. The regular dodecahedron can be interpreted as the Goldberg polyhedron ...
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 ...
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.
It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. [1] It is given a Schläfli symbol sr{ 5 ⁄ 2 ,5}, as a snub great dodecahedron . Cartesian coordinates
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°.
In 4-dimensional geometry, the dodecahedral bipyramid is the direct sum of a dodecahedron and a segment, {5,3} + { }. Each face of a central dodecahedron is attached with two pentagonal pyramids, creating 24 pentagonal pyramidal cells, 72 isosceles triangular faces, 70 edges, and 22 vertices.
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.
It has 18 faces (12 pentagrams and 6 decagrams), 60 edges, and 30 vertices. [1] Its vertex figure is a crossed quadrilateral . Aside from the regular small stellated dodecahedron { 5 / 2 ,5} and great stellated dodecahedron { 5 / 2 ,3}, it is the only nonconvex uniform polyhedron whose faces are all non-convex regular polygons ( star polygons ...