April 23, 1968

Dr. R. E. Williams

Douglas Advanced Research Laboratories

5251 Bolsa Avenue

Huntington Beach, California 92646

Dear Dr. Williams:

I have been interested to read your paper (Douglas Paper 5016, April 1968) on a new space-filling polyhedron.

This is a subject that I have thought about from time to time. I shall not discuss all aspects of the subject in this letter, but shall attempt mainly the question that you ask in the last sentence of your paper, why the composition of faces exemplified in your polyhedron seems to be preferred in nature to the more regular composition of faces of the Kelvin polyhedron.

A number of years ago I prepared a paper on this subject, which has, however, remained unpublished. The argument that I used involved the values of the six angles about a tetrahedral point, common to four polyhedra. The maximum stability would be associated with equality of the six angles, each of which would then have the value of the tetrahedral angle, 109° 28 minutes. Of the regular polygons, the pentagon, with angle 108°, comes closest to this value, the hexagon, 120°, next, and the square, 90°, next, with deviation nearly twice as great as for the hexagon and 13 times as great as for the pentagon. The deviation for the hexagon is 7 times that for the pentagon, and in the opposite direction.

For small distortions from the regular tetrahedral directions, the sum of the six angles is close to 6 times the tetrahedral angle. It is not possible, accordingly, to construct a space-filling polyhedron with faces that are regular pentagons.

We see from the above values that the minimal amount of strain would be associated with use of a polyhedron with mainly pentagonal faces, and about one hexagonal face to every 8.4 pentagonal faces.

An approximation to this expected optimal structure is provided by the 46-H.O Clathrate crystals. The principal polyhedron is the tetrakaidecahedron with two hexagonal faces and 12 pentagonal faces. The ratio of pentagonal faces to hexagonal faces is 6 to one, somewhat smaller than the calculated 8.421. The structure is completed by inclusion of one third as many pentagonal dodecaheora as tetrakaidecahedra, leading to 9 pentagons per hexagon. The average bond angle is 109.33°, quite close to the tetrahedral angle.

The 136-H20 hydrate framework consists of two dodecahedra for each hexa-kaidecahedron, the latter having four hexagonal faces and 12 pentagonal faces. The ratio of pentagonal faces to hexagonal faces for this structure is 9 to one, and the average value of the angle is 109.2°, again close to the tetrahedral value.

I may point out that for the Kelvin polyhedron the average value (for six squares and eight hexagons) is 110°, half a degree larger than the tetrahedral value. The mean deviation from the tetrrhedral angle is 13.3° for the Kelvin polyhedron, 2.5° for the 46-H20 structure, and 2.4° for the 136-H20 structure. I think that it is not surprising that the two latter structures are such common ones for Clathrate hydrates.

I know many other structures involving polyhedra with pentagonal and hexagonal faces, coming together at tetrahedral points, but I have not yet found one in which all of the polyhedra are equivalent. I think that the best chance of finding one is by studying the two other tetrahaidecahedra with two hexagonal faces and 12 pentagonal faces, which may be described as the ortho and meta tetrakaidecahedro of this class, with the one in the 46-HjO structure called the para polyhedron.

I may mention that I do not think that there is any important reason for the polyhedra in nature to be all equivalent to one another, and that accordingly in my opinion the 46-H-O structure may be the best solution to the problem.

Sincerely,

Linus Pauling

cc: Dr. Art Robinson