Max-Planck-Institut für Festkörperforschung Andersen Group LMTO C60 GW Clusters

# History of the truncated icosahedron

## Archimedes(287(?) - 212 B.C.)

Archimedes is believed to have conceived the thirteen "Archimedean solids", among which the truncated icosahedron is found.

To find out more about Archimedes and the Archimedean solids

## Piero della Francesca(1420-1492)

This is oldest known picture of a truncated icosahedron (the shape of C60. It is from the book Libellus de quinque corpibus regularibus ("On the Five Regular Bodies"), by the Italian renaissance painter and mathematician Piero della Francesca.

A copy of the volume can be found in the Vatican library.

## Leonardo da Vinci (1452-1519)

This rendition of the truncated icosahedron is from the book De Divina Proportione ("On Divine Proportion") by Fra' Luca Pacioli, published around 1498. The drawing is ascribed to Leonardo da Vinci. He seems to have invented this type of "skeleton model", which lets us view both the outside and the inside of the polyhedron.

## Johannes Kepler(1571-1630)

The concept of truncating a polyhedron was introduced by Johannes Kepler. Polyhedra were actually quite central to his research. In his work Prodromus Dissertationum Mathematicarum Continens Mysterium Cosmographicum ("The Cosmographic Mystery") he argued that the distances of the planets from the sun were determined by the Platonic solids. (see e.g. his Biography from the Galileo Project at Rice University)

## Fine Arts

An example of the appearance of the truncated icosahedron in art is shown in this picture from an Italian cathedral. At the top we see an icosahedron. It is bounded by twenty equilateral triangles. At each of the 12 vertices of the icosahedron, five of the triangles meet. Cutting off ('truncating') these vertices thus replaces each of them by a pentagonal face; it also converts each of the twenty former triangular faces into a hexagon. We can see the resulting truncated icosahedron at the bottom of the picture. This is the shape of the C60 molecule.

## Vatican Library

To get a glimpse of the treasures housed in the Vatican library, take a look at the exhibit Rome Reborn: The Vatican Library & Renaissance Culture at the Library of Congress in Washington. Especially the page on mathematics shows many works by Euclid, Archimedes, Pierro della Francesca, and many others.

## Biographies

You can find the biographies of many mathematicians on the History of Mathematics site at the University of St Andrews, e.g.
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