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The regular icosahedron can also be constructed starting from a regular octahedron. All triangular faces of a regular octahedron are breaking, twisting at a certain angle, and filling up with other equilateral triangles. This process is known as snub, and the regular icosahedron is also known as snub octahedron. [5]
The convex regular icosahedron is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex. Its dual polyhedron is the regular dodecahedron {5, 3} having three regular pentagonal faces ...
A regular polygon is a planar figure with all edges equal and all corners equal. A regular polyhedron is a solid (convex) figure with all faces being congruent regular polygons, the same number arranged all alike around each vertex.
The next most regular convex polyhedra after the Platonic solids are the cuboctahedron, which is a rectification of the cube and the octahedron, and the icosidodecahedron, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron).
For example, the icosahedron is {3,5+} 1,0, and pentakis dodecahedron, {3,5+} 1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles. The primary face of the subdivision is called a principal polyhedral triangle (PPT) or the breakdown structure. Calculating a single PPT allows the entire figure to be created.
Icosahedral symmetry fundamental domains A soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry. Rotations and reflections form the symmetry group of a great icosahedron. In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron.
A regular icosahedron colored in snub octahedron symmetry. [bi] Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces. The apex of the icosahedral pyramid (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).
The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.