Icosahedron
An icosahedron
noun 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
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.
Encyclopedia
An
icosahedron noun 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
adjectiveIn
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 , the radius of a circumscribed
sphere is
-
and the radius of an inscribed sphere is
-
where τ is the
golden ratio.
Area and volume
The surface area
A and the volume
V of a regular icosahedron of edge length
a are:
Cartesian coordinates
The following
Cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin:
where φ = /2 is the
golden ratio . Note that these vertices form five 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
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 T
h-symmetry, i.e. have different planes of symmetry from 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 . However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora , just as hexagons can be used as faces in semi-regular polyhedra . 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 and a snub dodecahedron as a snub icosidodecahedron , one may call the icosahedron the snub octahedron .
Icosahedron vs dodecahedron
Despite appearances, when an icosahedron is inscribed in a
sphere, it occupies less of the sphere's volume
than a
dodecahedron inscribed in the same sphere .
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 some roleplaying games, the twenty-sided die is used 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 "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.
The
symmetry group of the icosahedron is isomorphic to the alternating group on five letters. This
nonabelian simple group is the only nontrivial normal subgroup of the symmetric group on five letters. Since the Galois group of the general
quintic equation is isomorphic to the symmetric group on five letters, and the fact that the icosahedral group is simple and nonabelian means that quintic equations need not have a solution in radicals. The proof of the Abel-Ruffini theorem uses this simple fact, and Felix Klein wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.
See also
References
External links
- The Encyclopedia of Polyhedra
- A discussion of viral structure and the icosahedron
- Many links
