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Radomes & Soccer balls

Regular and semi-regular polyhedra have long interested paperfolders - they have a fascination which appeals to the "Origami Mind". This fascination has increased for paperfolders with the development of modular folding as many folders, from beginners to creative giants have struggled to fit their chosen modules of folded paper together to construct ever larger and more complex species of polyhedra.

The connection between modular folding and ordinary origami has always seemed to me somewhat tenuous, because the only aspect of real paperfolding involved is the folding of countless identical modules of minimal paperfolding interest and, as even enthusiastic modular folders admit, it very boring. (It can, too, have a transcendental effect for those who are prepared to explore techniques of meditation.) The real interest has been the fitting of the modules together, and that is something quite different from paperfolding. Nevertheless I, too, have been fascinated and I have followed the work of Tomoko Fuse devotedly as she has produced book after book crammed with unexpected ideas for modular origami. My own interests, outside paperfolding, have extended to the geodesic domes designed by or invented by or conceived by or discovered by the American architect Richard Buckminster Fuller. There is a great fascination in the radomes which litter the world's landscape and in the larger geodesic domes such as the one built for Expo 67 at Montreal.

My bedtime reading this week is "The Most Beautiful Molecule" by Hugh Aldersey-Williams about the discovery of the third molecular form of carbon after graphite and diamond, (which have themselves intrigued mankind ever since we began to take an interest in the world about us). The discovery by Rick Smalley of the United States and Harry Kroto of Great Britain and their assistants, among them Bob Curl, was so unexpected that it has captured the imaginations of millions who were never remotely interested in chemistry and has helped to rescue chemistry from the eclipse it had suffered in the shadow of the great discoveries during this century, of nuclear physics and astronomy. The research started in September, 1985 when Harry Kroto, of the University of Sussex near Brighton on the south coast of England paid a visit to Rice University in Houston Texas. There, a complicated piece of apparatus had been developed, which Kroto though could help him in his research into the formation of carbon molecules he had found to exist in space. Research quickly led to a wholly unexpected result, a new form of carbon, which had not only not been known before, but which had not existed even in the wildest imaginations of speculative scientists.

The new form of carbon appeared to be a molecule of sixty atoms, and once this was realised the conundrum was how a model should be constructed to give a picture of the structure and chemical bonding of the molecule. Graphite is in the form of flat "nets" of carbon atoms and It was thought that the "nets" might wrap round and link up to form spheres of sixty atoms. What geometry would they take? A pattern of hexagons was suggested, but hexagons could not in any way be curved to make a sphere It was thought that geodesic domes might be an analogy and someone seemed to recall that they also contained pentagons. Different members of the team experimented with cocktail sticks and jelly sweets to build models. Rick Smalley went home and cut out first cardboard hexagons, which would not work. He then cut out some pentagons and found that by surrounding a pentagon with hexagons, he could make a curved dish. He persisted and before long, he could see how he could make a quasi-sphere using twenty hexagons separated by twelve pentagons. Then the startling revelation came to him that it would have exactly sixty apexes or points! The apexes must be where the sixty carbon atoms lay in relationship to each other. The puzzle was solved.

When the team met the next day, it was quickly discovered that this model with sixty apexes was the simplest of Buckminster Fuller's geodesic domes from which all the more complex domes were derived. In his larger, more complex domes, Fuller merely divided the hexagons and pentagons into triangles, and then subdivided the triangles into smaller triangles, to created the thousands of facets of a large geodesic dome. Look carefully among the subdivided hexagons of a geodesic dome and you will always find just twelve pentagons (allowing for the theoretical ones which are obscured by the base of the dome). The researchers also discovered that their quasi-sphere had a name and that it was officially known as a Truncated Icosahedron. More prosaically, it was found to be in the form of a common soccer ball, very familiar to most of the children on the Earth and some of their fathers, too!

And my origami sighting? It comes on page 75 of "The Most Beautiful Molecule":

"The group duly assembled in Rick's office. Rick came in and tossed his paper model onto the table. They had all felt the solution, whatever it was would be novel. But this was beyond their wildest expectations.

"Bob [Curl] kept his cool. He inspected the model. Then he said that he would only believe it if the bonds worked out in the conventional fashion of other aromatic molecules....... Rick had called it a day in the early hours without going this final distance. He had done enough origami for one night. In the event, it was readily shown with more sticky tape that the double and single bonds went round the sphere in alternating pattern. It passed the Curl Test. It really was special."

The new form of carbon was cumbersomely named "Buckminsterfullerene" in honour of the inventor of the geodesic dome, or "buckyballs" for short. (I cannot help thinking, however, that simply "Fullerene" would have been more manageable and elegant.)

Since then the chemists of the world have been busily engaged in working out related forms of carbon having the shapes of other polyhedra with different numbers of atoms or constructed as long unclosed tubes. What practical use will all be? We have to admit that we do not yet know for certain. Many exciting ideas have been put forward; an immense amount of research is being carried out in numerous laboratories throughout the world, and it will remarkable if nothing comes of this wholly unexpected extension of organic chemistry, the chemistry of life itself.

Despite Rick Smalley's weariness with "origami", it would be stretching the point and completely unrealistic to claim that origami had much to do with the discovery of Carbon 60. Yet the incident reminds us that the most fundamental and important discoveries come from very simple concepts. Modular paperfolders have long been familiar with the structures of polyhedra of every kind. Mainstream origami, too, has its own peculiar geometry. Perhaps one day, that geometry itself will throw light on a scientific problem of the greatest importance and value for mankind, which at present nobody has ever dreamed about.

Later Rick Smally and Harry Kroto compared notes and found that in 1967 they had both made private visits with their wives to EXPO 67 at Montreal and had both been fascinated by the huge geodesic dome that formed the United States pavillion. They had not then the slightest inkling that one day the most elementary of geodesic domes would lead them to a fundamental discovery of incalculable importance Now they are united again. The two of them together with Bob Curl, have jointly been awarded the Nobel Prize for Chemistry. It is to be presented to them in Stockholm next Tuesday, 10th December.

David Lister Grimsby, England.

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