Is there an Euler circuit in this graph shaped like a soccer ball? Why not?
Can you find a Hamilton circuit in this graph?
This is also the geometric structure of buckminsterfullerene, or "buckyball".
A molecule of buckminsterfullerene has 60 carbon atoms, one at each vertex.
Chemical bonds occur along each of the 90 edges.
Notice that this structure has two different kinds of shapes as faces.
What are they?
Diamond and graphite (the material in pencil "lead") are forms of pure carbon that
occur in nature, but the physical properties of the two materials are very different.
For example, diamond is one of the hardest materials known, while graphite is
one of the softest.
The physical properties of these two substances are different because their
geometric structures are so different. In diamond, the chemical bonds form a
tightly packed three-dimensional structure, while in graphite the chemical bonds
are arranged in sheets.
Buckminsterfullerene is also a form of pure carbon, but it has a very special
geometric structure and has unusual physical properties.