Fair question - I can't answer it - but I can tell you how I produced them and hundreds like them. Some have similarities to the domes designed by the architect Buckminster-Fuller (and the C60 molecule named after him, Buckminsterfullerene) and to other geodesic structures.

One day while just sitting and thinking a question came into my mind. What would happen if some identical, weightless, electrically charged particles were put inside a glass sphere. Clearly two would repel each other and go to opposite ends of a diameter of the sphere. Three would probably form an equilateral triangle and four a tetrahedron. In each case the energy of the system is minimised. But what would 5, 6, 7 or any larger number do?

If you want to see now just click on the links below. Otherwise read on....

I've written a Visual Basic program to simulate the system. I choose a number of particles V (for vertices) and the program places them in random positions on the inner surface of the sphere.

The program now, for each particle in turn, sums the repulsive forces acting on it from all the other particles using the inverse square law. It then moves the particle along the surface of the sphere in the direction of the sum-of-force's tangential component much as would happen in real life. It repeats this for all the other particles.

The actions of the last paragraph are repeated until every particle is within one micro radian of the force acting on it. The energy of the system (En), now at a minimum, is calculated by summing the reciprocals of the distances between all particle-pairs. If this is a new value the data, including the xyz co-ordinates of the particles are saved to disc and a new set-up examined.

A second program reads this data back, finds the edges (E) and faces (F) and displays the results as shown here.

What are the results? Well, a tetrahedron does result from V=4. Two three-sided pyramids stuck base-to-base result from V=5 while V=6 makes an octahedron. Two five-sided pyramids base-to-base come from V=7 . For what I'm told is a skew square prism (think of a cube with the top face skewed by 45 degrees), set V=8. Just to continue the sequence see V=9.

Beyond this words fail me apart from V=12 which gives an icosahedron (pity Plato got there 4,000 years before me!). At V=16 an interesting effect starts. There are two quite distinct shapes that having different numbers of faces and edges for the one value of V, see V=16. This came as a surprise. A more complex example of this is V=68. I was also surprised to find that, at higher values of V, alternative configurations with identical V, F and E values occur with different energies (En) for example V=52.

One form of V=24 has six square faces and as these are slightly skewed it can exist in both right and left handed versions.

With larger values of V the polyhedra start to look quite similar to each other. Multiple forms of each becomes the norm with half a dozen or more shapes possible for each V value.
Now see some of the larger ones:-

V=44 ....V=64 ....V=242 ....V=257

At first I thought that sixty particles might take the shape of the C60 molecule with its hexagonal and pentagonal faces. That hasn't happened but I noticed that the were 60 faces on the lowest energy form of V=32. Vertices where 6 edges join are surrounded by vertices where 5 edges join so a quick faces to vertices transformation and bingo! Buckminsterfullerene!

If you'd like to know more or talk about this do please e-mail me,

martin@wmeadow.demon.co.uk

- I'd really like to hear from you.

Bye for now - Martin

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