Icosahedron
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An icosahedron [ˌaıkəsə'hiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant, which has faces which are equilateral triangles. [Etymology: 16th Century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedral adjective
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In geometry, the regular icosahedron is one of the five Platonic solids. It is a convex regular polyhedron composed of twenty triangular faces, with five meeting at each of the twelve vertices. It has 30 edges. Its dual polyhedron is the dodecahedron.
Dimensions
If the edge length of a regular icosahedron is <math>a</math>, the radius of a circumscribed sphere (one which touches the icosahedron at all vertices) is
- <math>r_u = \frac{a}{2} \sqrt{\tau \sqrt{5}} = \frac{a}{4} \sqrt{10 +2\sqrt{5}} \approx 0.9510565163 \cdot a </math>
and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is
- <math>r_i = \frac{\tau^2 a}{2 \sqrt{3}} = \frac{a}{12} \sqrt{3} \left(3+ \sqrt{5} \right) \approx 0.7557613141\cdot a </math>
where τ (also called φ) is the golden ratio.
Area and volume
The surface area A and the volume V of a regular icosahedron of edge length a are:
- <math>A=5\sqrt3a^2</math>
- <math>V=\begin{matrix}{5\over12}\end{matrix}(3+\sqrt5)a^3</math>
Cartesian coordinates
The following Cartesian coordinates define the vertices of an icosahedron centered at the origin:
- (0, ±1, ±φ)
- (±1, ±φ, 0)
- (±φ, 0, ±1)
where φ = (1+√5)/2 is the golden ratio (also written τ). Note that these vertices form sets of three mutually orthogonal golden rectangles.
The 12 edges of an octahedron can be partitioned in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. The five octahedra defining any given icosahedron form a regular polyhedral compound.
Geometric relations
Image:Snub tetrahedron.png There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with Th-symmetry, i.e. have different planes of symmetry than the tetrahedron. The icosahedron has a large number of stellations, including one of the Kepler-Poinsot solids and some of the regular compounds, which could be discussed here.
The icosahedron is unique among the Platonic solids in possessing a dihedral angle not less than 120°. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in n dimensions, at least three facets must meet at a ridge and leave a positive defect for folding in n-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the snub 24-cell), just as hexagons can be used as faces in semi-regular polyhedra (for example the truncated icosahedron). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120-cell, one of the ten non-convex regular polychora.
An icosahedron can also be called a gyroelongated pentagonal bipyramid. It can be decomposed into a gyroelongated pentagonal pyramid and a pentagonal pyramid or into a pentagonal antiprism and two equal pentagonal pyramids.
The icosahedron can also be called a snub tetrahedron, as snubification of a regular tetrahedron gives a regular icosahedron. Alternatively, using the nomenclature for snub polyhedra that refers to a snub cube as a snub cuboctahedron (cuboctahedron = rectified cube) and a snub dodecahedron as a snub icosidodecahedron (icosidodecahedron = rectified dodecahedron), one may call the icosahedron the snub octahedron (octahedron = rectified tetrahedron).
Icosahedron vs dodecahedron
Despite appearances, when an icosahedron is inscribed in a sphere, it occupies less of the sphere's volume (60.54%) than a dodecahedron inscribed in the same sphere (66.49%).
Natural forms and uses
Many viruses, e.g. herpes virus, have the shape of an icosahedron. Viral structures are built of repeated identical protein subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome.
In several roleplaying games, such as D&D, the twenty-sided die (for short, d20) plays a vital role in determining success or failure of an action. This die is in the form of a regular icosahedron. It may be numbered from "0" to "9" twice, but most modern versions are labeled from from "1" to "20".
The die inside of a Magic 8-Ball that has printed on it 20 answers to yes-no questions is a regular icosahedron.
If each edge of an icosahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 0.5 ohms, and that between adjacent vertices 11/30 ohms.
See also
External links
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- [1] A discussion of viral structure and the icosahedron
- Paper Models of Polyhedra Many linksca:Icosàedre
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