Icosahedron
* No. of faces 
20
* No. of edges 
30
* No. of vertices/corner
points  12
* At each vortex, 5 edges
meet.
* Each face is an equilateral
triangle.
* Dihedral Angle 
138°11' 
The following Cartesian
coordinates define
the vertices of an icosahedron with edgelength 2, centered at the
origin:

(0, ±1, ±φ)

(±1, ±φ, 0)

(±φ, 0, ±1)

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%). Herpes Simplex Virus has the shape
of an icosahedron. These are identical protein subunits which
are used again and again to build the overall unit by the
genome. Icosahedron is also used as a 20 sided dice. Dihedral
angle is the angle formed by the intersections of 2 planes.
Buckminister fuller based his design of geodesic domes based on
icosahedron.
conjugacy classes
I 
I_{h} 
 identity
 12 × rotation by
72°, order 5
 12 × rotation by
144°, order 5
 20 × rotation by
120°, order 3
 15 × rotation by
180°, order 2

 12 ×
rotoreflection by 108°, order 10
 12 ×
rotoreflection by 36°, order 10
 20 ×
rotoreflection by 60°, order 6
 15 × reflection,
order 2

Facetoface
angles on the icosahedron.
The angle between faces on an icosahedron can be
found in the following way. Take the 5 sided pyramid formed by 5
of the triangular faces and remove it the rest of the shape
won’t be necesary for finding the facetoface angle. Label the
five vertices of the pentagonal base of the pyramid A,B,C,D, and
E. Label the tip of the pyramid X. The two sides between which
the angle will be found are ABX and AXE. The midpoint of AX is
F. So the triangle containing the facetoface angle is BFE.
Since the lines BF and FE are altitudes of the faces ABX and
AXE, they have length √3/2.
The line BE can be found with trigonometry it is equal to
2·sin(54°) or 1.6180339875. 3 Now that we know the three sides
of BFE, we can use trigonometry again to calculate the angle at
point F. The angle is approximately 138.189685104.
Icosahedral Symmetry
A body with cubic symmetry
possesses a number of axes about which it maybe rotated to give
a number of identical appearances. The occurrence of icosahedral
features in quite unrelated viruses is not a matter of chance,
but a preference. An ICOSAHEDRON is
composed of 20 facets, each an equilateral triangle, and 12
vertices, and because of the axes of rotational
symmetry is said to have 5:3:2 symmetry. There are, in fact, six
5fold axes of symmetry passing through the vertices, ten 3fold
axes extending through each face and fifteen 2fold axes passing
through the edges of an icosahedron.The simplest icosahedral
capsids with 5:3:2 symmetry are built up by using 3 identical
subunits to form each triangular face, thereby requiring 60
identical subunits to form a capsid. A few virus
particles are constructed in this way, e.g. bacteriophage ØX174.
Each unit would be related identically (equivalent) and
asymmetrically with its neighbours, and none of the units would
coincide with an axis of symmetry.
As soon as the first high resolution micrographs
of negatively stained icosahedral viruses were obtained (Horne
et al., 1959  adenovirus; and Huxley and Zubay, 1960  turnip
yellow mosaic virus) it seemed that there was a structural
paradox. The number of morphological units observed on the
surface of known icosahedral viruses at that time was never 60
or multiples of 60, and was often more than 60 (The major reason
is that the construction of a portein shell with a
simpleicosahedral design (with "only" 60 units) severly
restricts the size of the genome that can be packaged.); and more
than 60 subunits cannot be arranged in an equivalent fashion
around an icosahedron. Furthermore,
the capsomers themselves appeared to be symmetrical and were
located on symmetry axes, e.g. herpesvirus.
There was direct evidence that capsomers of herpesvirus
were hexagonal and pentagonal in section. It was therefore clear
that the capsomers were not equivalent to the subunits of Crick
and Watson (1956). An obvious solution to the problem was
provided by supposing that the symmetrical capsomers are built
from a number of ASYMMETRICAL SUBUNITS. In this way it is
possible to build a variety of complicated bodies in which 5:3:2
symmetry is preserved and in which the number of subunits is a
multiple of 60 as predicted by Crick and Watson.
In 1962 Donald Caspar and Aaron Klug developed
two theoretical framework accounting for the structural
properties of larger (> 60 units) particles with icosahedral
symmetry.
1. Caspar
and Klug proposed that when a capsid contains more than 60
subunits, each subunit occupies a quasiequivalent
position; that is, the bonding properties of subunits in
different structural environments are similar (but not
identical, as in the case of the simplest, 60structure).
2. The
second important idea was that of triangulation,
the description of the triangular face of a large icosahedral
structure in terms of its subdivision into smaller triangels
termed facets. This process is described by the TRIANGULATION
NUMBER T, which gives
the number of structural units per face. The icosahedron itself
has 20 equilateral triangular facets and therefore 20 T
structure units given by the rule: T=P x f(SQR) where P can be
any number of the series 1,3,7,13,19,21,31 and f is any integer.
Morphological units can be clustered as 20T trimers, 30T dimers
or separated as 60T monomers. The number of morphological units
that would be produced by a clustering into hexamers and
pentamers can be calculated as follows: There are 10(T1)
hexamers plus 12. (and only 12) pentamers. Example


* Volume  L^{3}5(3+√5)/12
= (√5/6)φ^{2}L^{3}
where φ is the Golden number
whose value is 1.618034
* Area of each equlateral triangle 
√3 L^{2}/ 4
* Surface Area  5√3 L^{3}
^{* Each face has fcc111
structure}
* Icosahedron is one of the 5 Platonic
solids consisting of 20 equilateral triangles.
* symmetries  i_{h}(*532) . A
regular icosahedron has 60 rotational or orientation preserving
symmetries and a symmetry order of 120 including transformations
that combine a reflection and a rotation.
Why? What is the basis for icosahedral symmetry being so strongly
preferred by viruses? Following up on the implicit conclusions drawn
by Crick and Watson and by Casper and Klug almost 50 years ago, we
have recently argued (2003
PRL and 2004
PNAS)
that there is indeed a simple physical basis for this special
symmetry shown by viruses of so many different kinds, involving so
many different capsid proteins. In particular we demonstrate that
icosahedral symmetry allows for the lowestenergy configuration of
particles interacting isotropically on the surface of a sphere. More
explicitly, we find that the energyperparticle is a minimum for
configurations that involve 12 fivefold defects at the vertices of
an icosahedron, and that these configurations are especially favored
for "magic" numbers of particles corresponding to the
"triangulation" ("T") numbers of Casper and Klug.
Numerical Characterstics of Regular Polyhedra

n 
m 
f 
e 
v 
Tetrahedron 
3 
3 
4 
6 
4 
Octahedron 
3 
4 
8 
12 
6 
Icosahedron 
3 
5 
20 
30 
12 
Hexahedron 
4 
3 
6 
12 
8 
Dodecahedron 
5 
3 
12 
30 
20 
m: no. of polygons meeting at one vertex
n: no. of vertices of each polygon
f: no. of faces of each polyhedron
e: no. of edges of a polyhedron
v: no. of vertices of a polyhedron
e= nf/2 ; v=nf/m;
f = 2+ev ..... Euler's polyhedron theorem
* The number of dodecahedron faces is equal
to the number of icosahedron vertices and no. of icosahedron faces
is equal to the no. of dodecahedron vertices. The no. of planar
angles on the surfaces of both are also same, i.e. 60 = 3x20=5x12
* Ri = radius of insphere that touches
centroids of its faces.
Rm = radius of mid sphere that
touches centroids of its edges.
Rc = radius of circumsphere that
passes through the vertices.
(side length is 1) 
Ri 
Rm 
Rc 
Icosahedron 
(φ/2)√(3φ) 
φ/2 
φ^{2 }/ 2√3 
Dodecahedron 
(φ/2)√3 
φ^{2 }/ 2 
φ^{2 }/ 2√[3φ] 
* Rc / Ri are
equal for both icosahedron and dedecahedron.
* In Plato's
cosmology, icosahedron symbolized water as the most fluid polyhedron
whereas dedecahedron symbolized "The Real World" 