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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 ...
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.
Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the gyroelongated triangular bipyramid if it is made with equilateral triangles, but because it has coplanar faces is not strictly convex.
5 regular convex Platonic solids: regular tetrahedron, cube, regular octahedron, regular dodecahedron, and regular icosahedron. The regular octahedron is dual to the cube, and the regular icosahedron is dual to the regular dodecahedron. The regular tetrahedron is self-dual, meaning its dual is the regular tetrahedron itself. [1] 4 regular star ...
Coxeter, Regular Polytopes (1963), Macmillan Company Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (Table I: Regular Polytopes, (i) The nine regular polyhedra {p,q} in ordinary space) Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design.
The dissected regular icosahedron is a variant topologically equivalent to the sphenocorona with the two sets of 3 coplanar faces as trapezoids. This is the vertex figure of a 4D polytope, grand antiprism. It has 10 vertices, 22 edges, and 12 equilateral triangular faces and 2 trapezoid faces. [2]
In mathematics, an icosahedral number is a figurate number that represents an icosahedron. The nth icosahedral number is given by the formula (+). The first such numbers are: 1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, … (sequence A006564 in the OEIS).