# Dictionary Definition

icosahedral adj : of or relating to an
icosahedron

# User Contributed Dictionary

## English

### Adjective

- Of, relating to, or having the shape of an icosahedron.

# Extensive Definition

An icosahedron (Greek:
eikosaedron, from eikosi twenty + hedron seat; /ˌaɪ.kəʊ.sə.ˈhi.dɹən/; plural: -drons, -dra
/-dɹə/) is any polyhedron having 20 faces,
but usually a regular icosahedron is implied, which has equilateral
triangles
as faces.

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 and 12 vertices. Its dual
polyhedron is the dodecahedron.

## Dimensions

If the edge length of a regular icosahedron is a, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is- r_u = \frac \sqrt = \frac \sqrt \approx 0.9510565163 \cdot a

and the radius of an inscribed sphere (tangent to each of the
icosahedron's faces) is

- r_i = \frac = \frac \sqrt \left(3+ \sqrt \right) a \approx 0.7557613141\cdot a

while the midradius, which touches the
middle of each edge, is

- r_m = \frac = \frac \left(1+\sqrt\right) a \approx 0.80901699\cdot a

where \varphi (also called \tau) is the golden
ratio.

## Area and volume

The surface area A and the volume V of a regular icosahedron of edge length a are:- A = 5\sqrta^2 \approx 8.66025404a^2
- V = \frac (3+\sqrt5)a^3 \approx 2.18169499a^3.

## Cartesian coordinates

The following Cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin:- (0, ±1, ±φ)
- (±1, ±φ, 0)
- (±φ, 0, ±1)

The 12 edges of a regular 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, as do the two icosahedra that can be defined in this
way from any given octahedron.

## 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 Th-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 polyhedra 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°. Its dihedral angle is approximately
138.19°. 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 peak 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).

A rectified
icosahedron forms an icosidodecahedron.

### Icosahedron vs dodecahedron

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 1904, Ernst
Haeckel described a number of species of Radiolaria,
including Circogonia icosahedra, whose skeleton is shaped like a
regular icosahedron. A copy of Haeckel's illustration for this
radiolarian appears in the article on regular
polyhedra.

In some roleplaying
games, the twenty-sided die (for short, d20)
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 (in
which form it usually serves as a ten-sided die, or d10),
but most modern versions are labeled from "1" to "20". See d20
System.

An icosahedron is the three-dimensional game
board for Icosagame,
formerly known as the Ico Crystal Game.

An icosahedron is used in the board game Scattergories
to choose a letter of the alphabet. Six little-used letters, such
as X, Q, and Z, are omitted.

Inside a Magic
8-Ball, various answers to yes-no questions are printed on a
regular icosahedron.

The icosahedron displayed in a functional form is
seen in the Sol de la
Flor light shade. The rosette formed by the overlapping pieces
show a resemblance to the Frangipani
flower.

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

- Truncated icosahedron
- Regular polyhedron
- Geodesic grids use an iteratively bisected icosahedron to generate grids on a sphere

## References

## External links

- The Uniform Polyhedra
- K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
- Interactive Icosahedron model - works right in your web browser
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Tulane.edu A discussion of viral structure and the icosahedron
- Paper Models of Polyhedra Many links
- Origami Polyhedra - Models made with Modular Origami

icosahedral in Azerbaijani: İkosaedr

icosahedral in Catalan: Icosàedre

icosahedral in Czech: Dvacetistěn

icosahedral in Danish: Ikosaeder

icosahedral in German: Ikosaeder

icosahedral in Estonian: Ikosaeeder

icosahedral in Spanish: Icosaedro

icosahedral in Esperanto: Dudekedro

icosahedral in French: Icosaèdre

icosahedral in Korean: 정이십면체

icosahedral in Italian: Icosaedro

icosahedral in Hebrew: איקוסהדרון

icosahedral in Latvian: Ikosaedrs

icosahedral in Hungarian: Ikozaéder

icosahedral in Dutch: Icosaëder

icosahedral in Japanese: 正二十面体

icosahedral in Norwegian: Ikosaeder

icosahedral in Polish: Dwudziestościan
foremny

icosahedral in Portuguese: Icosaedro

icosahedral in Russian: Икосаэдр

icosahedral in Simple English: Icosahedron

icosahedral in Serbian: Икосаедар

icosahedral in Finnish: Ikosaedri

icosahedral in Swedish: Ikosaeder

icosahedral in Tamil: இருபதுமுக முக்கோணகம்

icosahedral in Thai: ทรงยี่สิบหน้า

icosahedral in Ukrainian: Ікосаедр

icosahedral in Chinese: 正二十面體