This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Regular icosahedron" – news · newspapers · books · scholar · JSTOR(October 2023) (Learn how and when to remove this message)
Polyhedron with 20 regular triangular faces
Regular icosahedron
Type
Gyroelongated bipyramid Deltahedron
Faces
20
Edges
30
Vertices
12
Vertex configuration
Symmetry group
Icosahedral symmetry
Dihedral angle (degrees)
138.190 (approximately)
Dual polyhedron
Regular dodecahedron
Net
In geometry, the regular icosahedron[1] (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of the Platonic solid and of the deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.
Many polyhedrons are constructed from the regular icosahedron. For example, most of the Kepler–Poinsot polyhedron is constructed by faceting. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron has many relations with other Platonic solids, one of them is the regular dodecahedron as its dual polyhedron and has the historical background on the comparison mensuration. It also has many relations with other polytopes.
The appearance of regular icosahedron can be found in nature, such as the virus with icosahedral-shaped shells and radiolarians. Other applications of the regular icosahedron are the usage of its net in cartography, twenty-sided dice that may have been found in ancient times and role-playing games.
^ See Jones 2003 for the pronunciation: (/ˌaɪkɒsəˈhiːdrən,-kə-,-koʊ-/ or /aɪˌkɒsəˈhiːdrən/)
and 20 Related for: Regular icosahedron information
In geometry, the regularicosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two...
"regularicosahedron" generally refers to the convex variety, while the nonconvex form is called a great icosahedron. The convex regularicosahedron is...
regular dodecahedron has edge length 1, its dual icosahedron has edge length ϕ. If the five Platonic solids are built with same volume, the regular dodecahedron...
water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron...
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin...
icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation...
truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons....
to the great icosahedron {3, 5/2}. All of these regular star dodecahedra have regular pentagonal or pentagrammic faces. The convex regular dodecahedron...
icosahedron can be formed by gluing triangular pyramids to each face of a regularicosahedron. Depending on the height of these pyramids relative to their base...
tetrahedra is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regularicosahedron. It was first described...
may be realized by linked ellipses, or (using the vertices of a regularicosahedron) by linked golden rectangles. It is impossible to realize them using...
Map, a version of the Dymaxion map that used a modified but mostly regularicosahedron as the base for the projection. The version most commonly referred...
vertex arrangement as the convex regularicosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms a degenerate...
The square bipyramid is the regular octahedron, while the gyroelongated pentagonal bipyramid is the regularicosahedron, so they are not Johnson solids...
of the regularicosahedron, the Jessen's icosahedron, and the regular octahedron, in accordance with the pyritohedral symmetry of the icosahedron. When...
hull of two opposite edges of a regularicosahedron forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into...
with a regular pentagonal base and 5 isosceles triangle sides of any height. The pentagonal pyramid can be seen as the "lid" of a regularicosahedron; the...
is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that...
symmetries as a regularicosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the...