Buckminsterfullerene Triangulated

by Bill Lauritzen

Abstract

Visual models and verbal phrases to describe the Buckminsterfullerne molecule such as "truncated icosahedron," "soccer-ball", and "pentagons and hexagons" are discussed in reference to the stability of the molecule and are shown to be inadequate to explain such stability in light of principles discovered by Buckminster Fuller and experiential evidence presented by the author using the low-technology materials of string and straws and methods devised by Fuller. A new triangulated visual model is presented that accounts for the stability of the molecule. Reasons are offered for the fact that chemists and others have failed to discover and use the truer model.

It is fitting that a molecule be named in honor of R. Buckminster Fuller. And Richard E. Smalley and Harry Kroto should be congratulated for discovering it. However, it has been described as being made up of "12 pentagons ... 20 hexagons."(1) Or "soccer-ball shaped".(2) Or labeled a "truncated icosahedron."(3) Many other authors and editors have referred to it using these terms and images.(4) All of these terms and images are misleading because none of them show the triangulation which gives the molecule its stability. Buckminster Fuller's geodesic domes, although made up of pentagons and hexagons, are all triangulated. The Buckminsterfullerene is also triangulated, but this is never mentioned in articles, (except in references to geodesic domes) and it is not clearly evident from stick models or computer images. It is only seen when we go directly to a photograph of the molecule: a picture generated from x-ray diffraction.

It can also be seen using proper models made from straws and string.

Fuller says, "The triangle is the only flex-cornered polygon that holds its shape: ergo, it alone accounts for all structural shaping in the Universe."(6)

He suggests demonstrating this by making a necklace.(7) Take three straws and run a string through them and tie it off to form a triangle. Now do the same with four straws to form a square. The triangle is stable. The square collapses.

A pentagon and hexagon also collapse. Only a triangle, of all polygons, is stable. And only structures made from triangles are inherently stable. (Even in Fuller's tensional integrity, or tensegrity, structures, in which the triangulation is not always so apparent, the stability results from tensional triangles.) A string necklace can also make polyhedrons. A tetrahedron, composed of four triangular faces, is a stable structure. A cube collapses.

A pentagonal dodecahedron also collapses. Hexagons and above cannot be made into closed, regular shapes.However, if we continue to build using only triangles, we can progress beyond the tetrahedron. For example, the next shape beyond a tetrahedron is an octahedron.

After that we can make an icosahedron.

The "truncated icosahedron" is the untriangulated structure that is shown in popular articles as well as scientific journals as being a Buckminsterfullerene. It is formed by taking a icosahedron and slicing off each vertex one third of the way down the edge. I applied the "necklace test" to this structure. It collapsed upon itself. I then triangulated all the hexagons and pentagons, and I now had a stable structure. Comparing the two shows the surprising difference.

Curl and Smalley say that it is the symmetrical, closed, structure of C-60, the "truncated icosahedron" shape, that creates its extraordinary stability.8 But, according to principles uncovered by R. Buckminster Fuller, and by my verification, it is the triangulation of the carbon molecules that causes the stability.Chemists use clay and sticks to form models of molecules. The resiliency of clay tends to mask the triangle's unique property.

Chemists also use computers to form molecules. The artificiality of the computer again masks the triangle's property. In addition, the computer images tend show the carbon bonds as lines, whereas in the x-ray diffraction generated image we see the more realistic, carbon-triangulated version. Perhaps these reason help explain why the triangulation of the Buckyball has not been pointed out and properly emphasized.

Another well-known carbon substance that is made from triangles (formed into tetrahedrons) is diamond. Although textbooks acknowledge the tetrahedral (and thus triangular) shape of diamonds, graphite is not shown to be triangulated although I would suspect that it is. This failure to show the proper triangulation of graphite would also help explain the misconception regarding the Fullerene.

On a more fundamental level, it is probably the lack of knowledge of the stability of the triangle versus the square that causes the misconception.

We should do away with "truncated icosahedron," "twelve pentagons and twenty hexagons," and "soccerball" as words to describe the Buckminsterfullerene and as images to model it. "Triangulated soccerball" would be OK. "Twelve pentagons and twenty hexagons that have been triangulated" would be OK. A "geodesic sphere" is OK because geodesic structures are triangulated.

Remember, the triangle is the only structure. But don't believe me. Tri it.

NOTES:

Robert F. Curl and Richard E. Smalley, "Fullerenes," Scientific American, Oct. 1991, p. 54.
Ibid., p. 54.
Ibid., p. 58.
Just one example out of dozens is Volume 140 of Science News, August 24, 1991, p.120, on which we see a truncated icosahedron.
R. Buckminster Fuller, Synergetics Vol. 1, (New York, MacMillian, 1975) p. 319.
R. Buckminster Fuller, Comsmography, (New York, MacMillian, 1992) p. 47.
R. Buckminster Fuller, Synergetics Vol. 1, (New York, MacMillian, 1975) p. 317.
Curl and Smalley, op. cit., p.58.