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Icosahedron

A Platonic Solid ($P_5$) with 12 Vertices, 30 Edges, and 20 equivalent Equilateral Triangle faces $20\{3\}$. It is described by the Schläfli Symbol $\{3,5\}$. It is also Uniform Polyhedron $U_{22}$ and has Wythoff Symbol $5\,\vert\,2\,3$. The icosahedron has the Icosahedral Group $I_h$ of symmetries.


A plane Perpendicular to a $C_5$ axis of an icosahedron cuts the solid in a regular Decagonal Cross-Section (Holden 1991, pp. 24-25).


A construction for an icosahedron with side length $a=\sqrt{50-10\sqrt{5}}/5$ places the end vertices at $(0,0,\pm 1$) and the central vertices around two staggered Circles of Radii ${\textstyle{2\over 5}}\sqrt{5}$ and heights $\pm {\textstyle{1\over 5}}\sqrt{5}$, giving coordinates

\begin{displaymath}
\pm \left({{\textstyle{2\over 5}}\sqrt{5}\cos({\textstyle{2\...
...style{2\over 5}} i\pi), {\textstyle{1\over 5}}\sqrt{5}}\right)
\end{displaymath} (1)

for $i=0$, 1, ..., 4, where all the plus signs or minus signs are taken together. Explicitly, these coordinates are
$\displaystyle {\bf x}_0^\pm$ $\textstyle =$ $\displaystyle \pm({\textstyle{2\over 5}}\sqrt{5}, 0, {\textstyle{1\over 5}}\sqrt{5})$ (2)
$\displaystyle {\bf x}_1^\pm$ $\textstyle =$ $\displaystyle \pm({\textstyle{1\over 10}}(5-\sqrt{5}), {\textstyle{1\over 10}}\sqrt{50+10\sqrt{5}}, {\textstyle{1\over 5}}\sqrt{5})$ (3)
$\displaystyle {\bf x}_2^\pm$ $\textstyle =$ $\displaystyle \pm(-{\textstyle{1\over 10}}(\sqrt{5}+5), {\textstyle{1\over 10}}\sqrt{50-10\sqrt{5}}, {\textstyle{1\over 5}}\sqrt{5})$ (4)
$\displaystyle {\bf x}_3^\pm$ $\textstyle =$ $\displaystyle \pm(-{\textstyle{1\over 10}}(\sqrt{5}+5), -{\textstyle{1\over 10}}\sqrt{50-10\sqrt{5}}, {\textstyle{1\over 5}}\sqrt{5})$ (5)
$\displaystyle {\bf x}_4^\pm$ $\textstyle =$ $\displaystyle \pm({\textstyle{1\over 10}}(5-\sqrt{5}), -{\textstyle{1\over 10}}\sqrt{50+10\sqrt{5}}, {\textstyle{1\over 5}}\sqrt{5}).$ (6)

By a suitable rotation, the Vertices of an icosahedron of side length 2 can also be placed at $(0,\pm \phi,\pm 1)$, $(\pm 1,0,\pm \phi)$, and $(\pm \phi,\pm 1,0)$, where $\phi$ is the Golden Ratio. These points divide the Edges of an Octahedron into segments with lengths in the ratio $\phi:1$.


The Dual Polyhedron of the icosahedron is the Dodecahedron. There are 59 distinct icosahedra when each Triangle is colored differently (Coxeter 1969).


\begin{figure}\begin{center}\BoxedEPSF{PentagonApothem.epsf}\end{center}\end{figure}

To derive the Volume of an icosahedron having edge length $a$, consider the orientation so that two Vertices are oriented on top and bottom. The vertical distance between the top and bottom Pentagonal Dipyramids is then given by

\begin{displaymath}
z=\sqrt{\ell^2-x^2},
\end{displaymath} (7)

where
\begin{displaymath}
\ell={\textstyle{1\over 2}}\sqrt{3}\,a
\end{displaymath} (8)

is the height of an Isosceles Triangle, and the Sagitta $x\equiv R'-r'$ of the pentagon is
\begin{displaymath}
x={\textstyle{1\over 2}}a{\textstyle{1\over 10}}\sqrt{25-10\sqrt{5}}\,a,
\end{displaymath} (9)

giving
\begin{displaymath}
x^2={\textstyle{1\over 20}}\sqrt{5-2\sqrt{5}}\,a^2.
\end{displaymath} (10)

Plugging (8) and (10) into (7) gives
$\displaystyle z$ $\textstyle =$ $\displaystyle a\sqrt{{\textstyle{3\over 4}}-{\textstyle{1\over 20}}(5-2\sqrt{5})} = a\sqrt{15-(5-2\sqrt{5})\over 20}$  
  $\textstyle =$ $\displaystyle a\sqrt{10+2\sqrt{5}\over 20}={\textstyle{1\over 2}}a\sqrt{10+2\sqrt{5}\over 5}$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 10}}\sqrt{50+10\sqrt{5}}\, a,$ (11)

which is identical to the radius of a Pentagon of side $a$. The Circumradius is then
\begin{displaymath}
R=h+{\textstyle{1\over 2}}z,
\end{displaymath} (12)

where
\begin{displaymath}
h={\textstyle{1\over 10}}\sqrt{50-10\sqrt{5}}\,a
\end{displaymath} (13)

is the height of a Pentagonal Dipyramid. Therefore,
$\displaystyle R^2$ $\textstyle =$ $\displaystyle (h+{\textstyle{1\over 2}}z)^2$  
  $\textstyle =$ $\displaystyle ({\textstyle{1\over 10}} \sqrt{50-10\sqrt{5}}+{\textstyle{1\over 20}}\sqrt{50+10\sqrt{5}}\,)^2a^2$  
  $\textstyle =$ $\displaystyle \left({{5\over 8}-{3\over 8\sqrt{5}}+{\sqrt{20}\over 10}}\right)a^2={\textstyle{1\over 8}} (5+\sqrt{5}\,) a.$ (14)

Taking the square root gives the Circumradius
\begin{displaymath}
R=\sqrt{{\textstyle{1\over 8}}(5+\sqrt{5})}\,a = {\textstyle{1\over 4}}\sqrt{10+2\sqrt{5}}\,a \approx 0.95105a.
\end{displaymath} (15)

The Inradius is
\begin{displaymath}
r={\textstyle{1\over 12}} (3\sqrt{3}+\sqrt{15}\,)a \approx 0.75576a.
\end{displaymath} (16)

The square of the Interradius is
$\displaystyle \rho^2$ $\textstyle =$ $\displaystyle ({\textstyle{1\over 2}}z)^2+{x_l}^2$  
  $\textstyle =$ $\displaystyle [({\textstyle{1\over 4}})({\textstyle{1\over 100}})(50+10\sqrt{5}\,)+{\textstyle{1\over 100}}(25+10\sqrt{5}\,)]a^2$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}(3+\sqrt{5})a^2,$ (17)

so
\begin{displaymath}
\rho=\sqrt{{\textstyle{1\over 8}}(3+\sqrt{5}\,)}\,a={\textstyle{1\over 4}}(1+\sqrt{5}\,)a \approx 0.80901a.
\end{displaymath} (18)


The Area of one face is the Area of an Equilateral Triangle

\begin{displaymath}
A={\textstyle{1\over 4}}a^2\sqrt{3}.
\end{displaymath} (19)

The volume can be computed by taking 20 pyramids of height $r$
$\displaystyle V$ $\textstyle =$ $\displaystyle 20[({\textstyle{1\over 3}} A)r] = 20 {\textstyle{1\over 3}} {\textstyle{1\over 4}}\sqrt{3}\,a^2 {\textstyle{1\over 12}}(3\sqrt{3}+\sqrt{15}\,)a$  
  $\textstyle =$ $\displaystyle {\textstyle{5\over 12}} (3+\sqrt{5}\,)a^3.$ (20)

Apollonius showed that
\begin{displaymath}
{V_{\rm icosahedron}\over V_{\rm dodecahedron}} = {A_{\rm icosahedron}\over A_{\rm dodecahedron}},
\end{displaymath} (21)

where $V$ is the volume and $A$ the surface area.

See also Augmented Tridiminished Icosahedron, Decagon, Dodecahedron, Great Icosahedron, Icosahedron Stellations, Metabidiminished Icosahedron, Tridiminished Icosahedron, Trigonometry Values Pi/5


References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.

Davie, T. ``The Icosahedron.'' http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/icosahedron.html.

Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.

Klein, F. Lectures on the Icosahedron. New York: Dover, 1956.

Pappas, T. ``The Icosahedron & the Golden Rectangle.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 115, 1989.



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© 1996-9 Eric W. Weisstein
1999-05-26