Regular dodecahedron | Revolvy
A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges. Relation to the nested. 4, 4, 6, 2. Cube, Cube, 6, 8, 12, 2. Octahedron, Octahedron, 8, 6, 12, 2. Dodecahedron, Dodecahedron, 12, 20, 30, 2. Icosahedron, Icosahedron, 20, 12, 30, 2. The ratio edge of the dodecahedron / edge of the cube is the inverse of the golden Relations between dodecahedron and icosahedron.
Starting with any regular polyhedron, the dual can be constructed in the following manner: For example, starting with a cubewe 1 create six points in the centers of the six faces, 2 connect each new point to its four neighbors, creating 12 edges, and 3 erase the cube to find the result is an octahedronconsisting of eight triangular faces. This is an operation "of order 2" meaning that taking the dual of the dual of x gives back the original x. For example, take the dual of the octahedron and see that it is a cube.
Note that when taking the dual, a face with n sides transforms into a vertex where n faces meet, and vice versa. The six 4-sided faces of the cube transform into the six corners of the octahedron, with 4 faces meeting at each.
The eight 3-sided faces of the octahedron transform into the eight corners of the cube with 3 faces meeting at each. Also observe that the total number of edges remains unchanged, as each original edge crosses exactly one new edge.
The cube and octahedron each have twelve edges. All these properties can be easily seen if one makes a model of the cube and octahedron together.
The relative sizes of the two dual polyhedra can be adjusted as shown here, so that their edges are the same distance from their common center, and so cross through each other.
The twenty 3-sided faces and twelve 5-way corners of the icosahedron correspond to the twenty 3-way corners and twelve 5-sided faces of the dodecahedron. The Megaminx twisty puzzle, alongside its larger and smaller order analogues, is in the shape of a regular dodecahedron. In the children's novel The Phantom Tollbooththe regular dodecahedron appears as a character in the land of Mathematics.
Each of his faces wears a different expression — e. Dodecahedron is the name of an avant-garde black metal band from Netherlands. This was proposed by Jean-Pierre Luminet and colleagues in  and an optimal orientation on the sky for the model was estimated in The Vision of Professor Squarepunt," the number 5 said: I make pentagons and pentagrams.
And but for me dodecahedra could not exist; and, as everyone knows, the universe is a dodecahedron. So, but for me, there could be no universe.
Rhombicosidodecahedron - Wikipedia
The bilunabirotundae fill the rhombic gaps. Each cube meets six bilunabirotundae in three orientations. Block model 6 bilunabirotundae around a cube Related polyhedra and tilings[ edit ] The regular dodecahedron is topologically related to a series of tilings by vertex figure n3.