Search results
Results from the WOW.Com Content Network
An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa. [17] An icosahedron can be inscribed in an octahedron by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two golden sections. Because the golden sections are ...
Construction from the vertices of a truncated octahedron, showing internal rectangles. The Cartesian coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.
The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio [11]
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.
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 regular icosahedron can be constructed by intersecting three golden rectangles perpendicularly, arranged in two-by-two orthogonal, and connecting each of the golden rectangle's vertices with a segment line. There are 12 regular icosahedron vertices, considered as the center of 12 regular dodecahedron faces. [13]
The surface area and the volume of the truncated icosahedron of edge length are: [2] = (+ +) = +. The sphericity of a polyhedron describes how closely a polyhedron resembles a sphere. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1.
Truncated icosahedron, one of the Archimedean solids illustrated in De quinque corporibus regularibus. The five Platonic solids (the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron) were known to della Francesca through two classical sources: Timaeus, in which Plato theorizes that four of them correspond to the classical elements making up the world (with the fifth, the ...