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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]
Net 10, net 15, net 30 and net 60 (often hyphenated "net-" and/or followed by "days", e.g., "net 10 days") are payment terms for trade credit, which specify that the net amount (the total outstanding on the invoice) is expected to be paid in full by the buyer within 10, 15, 30 or 60 days of the date when the goods are dispatched or the service is completed.
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 ...
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.
A net of a 4-polytope, a four-dimensional polytope, is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane.
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.
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.
In 1858, Bertrand derived the regular star polyhedra (Kepler–Poinsot polyhedra) by faceting the regular convex icosahedron and dodecahedron. In 1974, Bridge enumerated the more straightforward facetings of the regular polyhedra, including those of the dodecahedron. In 2006, Inchbald described the basic theory of faceting diagrams for polyhedra.
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