THE ICOSAHEDRON
Of the five platonic regular polyhedra, the twenty sided Icosahedron is one of the most
interesting having found application in the construction of Geodesic Domes and as aide in
understanding phenomena such as carbon 60 buckyballs. This polyhedron has a surface
composed of F=20 faces in the form of equilateral triangles, has a total of V=12 vertexes,
and E=30 edges. As all convex polyhedra, it satisfies the Euler Formula-
V + F − E = 2 and
here reads 12 + 20 − 30 = 2
To construct an Icosahedron graphically or mechanically one needs to locate the
coordinates of its vertexes. A convenient method for doing this in to start with three plates
of aspect ratio H/L=Φ=golden ratio placed in an orthogonal configuration as shown-
Note we have indicated in red the visible corners of the plates. What is very
interesting is that the total of the 12 corners of the plate represent the location of the
vertexes of an Icosahedron in 3D space. To convince yourself of this fact, look at the
triangle formed by points A, B, and C in the figure. By Pythagoras we have that the
distances-
Φ
1
Φ −1 2 1
d AB = d AC = ( ) 2 + ( ) 2 + (
) = (1 − Φ + Φ 2 ) = 1
2
2
2
2
Thus connecting points A, B, and C produces an equilateral triangle of side length one
and area sqrt(3)/4. Connecting any other neighboring points will also produce the
same area equilateral triangle. Hence, since there will be twenty of such distinct
triangles, it is clear that the surface area of an Icosahedron will be-
Area Icasohedron = 20 3 / 4 = 8.6602540... …
If one now places the largest sphere possible inside of the Icosahedron it will be one of
radius equal to δcenter=sqrt(3Φ2-1)/(2sqrt(3) and equals the distance from one of the
triangle centers to the origin O. Thus the area of the largest inscribed spheres will be-
(3Φ 2 − 1)
Area Inscribed Sphere = 4π
) = π (7 + 3 5 ) / 6 = 7.177598...
12
The two areas are observed to be fairly close to each other. This observation is the
clue which led to the development of Geodesic structures. It was realized that
triangles form rigid structures and that their concatenation approximating a
hemisphere can lead to a very stable structure requiring no column support as needed
for standard structures. This fact was first utilized by Walther Bauerfeld in the early
1920s in designing the world’s first geodesic planetarium at Zeiss in Jena using the
Icosahedron as a starting point.(as a small sidelight-I was born in Jena while my
father was working at the Zeiss Optical Works there before our family moved on to
Peenemuende in 1937 and eventually to the US as part of operation paperclip at the
end of WWII). Buckminster Fuller extended the work of Bauerfeld and his efforts led
to most of the Geodesic Structures extant in the world today. The very stable sixty
faced, nearly spherical, carbon structure based on the hexagonal carbon bond has
come to be known as a Bucky Ball after Fuller. Its discovery led to the Nobel Prize in
Chemistry for Smalley, Curl and Kroto in 1996.
One can readily plot the 20 equilateral triangles using the vertex point given in the
above figure. Many existing math programs already have these points stored. For
example, our MAPLE program yields-
In mechanically building an Icosahedron from separate equilateral triangles it
becomes necessary to know the angle between adjacent triangles. Referring back to
the three orthogonal plane figure shown above it is clear that the angle between
triangle ABC and BCD is as shown-
So the inside angle between the first and second triangle becomes-
2θ = 2 arctan(
5 +1
) = 138.18968.. deg
5 −1
By gluing a wedge of this angle between each neighboring triangle a stable
Icosahedron can be constructed. I show you here one I built in my garage workshop
using twenty equilateral triangles cut from 1/8”thick hardboard and then finished
with a birchwood cladding –
Next let us calculate the total volume of the Icosahedron. We do this by noting the
distance from points A, B, and C to the origin O in the above figure all have the
identical value-
1
δ AO = δ BO = δ CO = 1 + Φ 2
2
and that the distance from the center of the triangle ABC to the origin O is given as-
1
3Φ 2 − 1
2
δ center = (δ AO ) − (
) =
2 cos 30
2 3
2
Thus, recalling that the volume of a pyramid equals the product of the base and onethird of its height and that we are here dealing with a total of 20 such pyramids, one
finds the total volume of the Icosahedron to be-
Volume = 20
3 1
5
5
5
3Φ 2 − 1 =
7 + 3 5 = (3 + 5 )
 δ center =
4 3
6
12
6 2
Comparing this result with the volume of the largest inscribed sphere of radius δcenter
we find the inequality-
3
4  3Φ 2 − 1 
5
 <
π
(3 + 5 )
3  2 3 
12


which, when evaluated, yields 1.80818 < 2.18169. These numbers are again fairly
close to each other as expected , showing that the Icosahedron forms a pretty good
approximation to a sphere. In the construction of Geodesic Domes one usually starts
with an Icosahedron structure and compares it to the smallest radius external sphere
enclosing the polygon. One then builds up a tetrahedron structure for each of the
triangles so that their upper vertex just touches the imaginary outer spherical surface.
Other stable structural elements such as pentagons can also be used for Geodesic
Domes.
Finally, it is interesting to note that some viruses have the shape of Icosahedrons.Here
is a picture of the human rhino virus-
This shape contains within it the RNA strands which are injected into living cells to be
replicated until the cell dies. The ability of the body to manufacture the correct
antibodies on noticing a viral attack must somehow involve a recognition of the shape
of such invading viruses and explains also why vaccines using dead viruses of the
attacker are so effective in generating antibodies and thus aborting future attacks(
vaccines exist for small pox, polio, chicken pox, flu etc. but have not yet been
produced for the Human Rhino Virus, West-Nile virus , AIDS etc.).
November 2009