Search results
Results from the WOW.Com Content Network
In the above coordinates, the first 12 vertices form a regular icosahedron, the next 20 vertices (those with R) form a regular dodecahedron, and the last 30 vertices (those with S) form an icosidodecahedron. Normalizing all vertices to the unit sphere gives a spherical disdyakis triacontahedron, shown in the adjacent figure.
It is a polyhedra in a sequence defined by the face configuration V4.6.2n.This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any n ≥ 7.
Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √ φ 6 +2 = √ 8φ+7 for edge length 2. For unit edge length, R must be halved, giving R = √ 8φ+7 / 2 = √ 11+4 √ 5 / 2 ≈ 2.233.
Let φ be the golden ratio.The 12 points given by (0, ±1, ±φ) and cyclic permutations of these coordinates are the vertices of a regular icosahedron.Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points (±1, ±1, ±1) together with the 12 points (0, ±φ, ± 1 / φ ) and cyclic permutations of these coordinates.
It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. ... 6: 12: 7: 1: No 4{3}+3{4}
The vertices can be colored with 3 colors, as can the edges, and the diameter is five. [37] The dodecahedral graph is Hamiltonian, meaning a path visits all of its vertices exactly once. The name of this property is named after William Rowan Hamilton, who invented a mathematical game known as the icosian game.
There are 34 topologically distinct convex heptahedra, excluding mirror images. [2] ( Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by (a,b), where b and a depend on each other such that the tetrahedron defined by the four vertices of a face has volume zero, i.e. is a planar face. (1,1) is the rhombic solution.