»  Solutions to puzzles in my VDARE.com monthly Diary

  February 2017


My February diary had some notes on sphere-packing problems. Those notes concluded thus.

Suppose your identical spheres (we're back in three dimensions here) are made of some material — one of the workable metals, perhaps — that are rigid up to a point but can be deformed with sufficient pressure. Fill up a very large container with zillions of spheres arranged at random, not in any regular way. Now apply pressure from all directions, squishing the spheres together until there is no space left between them.

Each sphere is now an irregular polyhedron, a solid figure with plane faces. What is the average number of those faces?

I read the answer many years ago in some math journal and remember it as thirteen point something. I thought it was in Hilbert and Cohn-Vossen, but it isn't, and googling hasn't turned anything up. Anybody got a reference?


• Follow-up

Well, I remembered the number correctly, but to my shame and embarrassment forgot the source. It was not "some math journal": It was the late great H.S.M. Coxeter's fine classic Introduction to Geometry.

For penance I shall copy out the relevant passages from pp. 409-410 of my 1961 edition of that marvelous book.

Since hexagonal close packing has the same density as cubic close packing, namely 0.74048 …, it is natural to ask whether some still less systematic packing (without any straight rows of spheres) may have a greater density. This remains an open question. The best theoretical approach to an answer is the proof by Rogers that, if such a packing exists, its density must be less than 0.7797 …

Experiments in this direction began as long ago as 1727, when Stephen Hales stated, in his Vegetable Staticks,
I compressed several fresh parcels of Pease in the same pot, with a force equal to 1600, 800, and 400 pounds; in which Experiments, tho' the Pease dilated, yet they did not raise the lever, because what they increased in bulk was, by the great incumbent weight, pressed into the interstices of the Pease, which they adequately filled up, being thereby formed into pretty regular Dodecahedrons.
Hales presumably reached his conclusion by observing some pentagonal faces on his dilated peas. They could not all have been regular dodecahedra. For, since the dihedral angle of the regular dodecahedron is less than 120° [it's 116° 34′ — JD], three such solids with a common edge will leave an angular gap of about 10° 19′. In fact, dodecahedra {5, 3} are the cells of the configuration {5, 3, 3}, which is not an infinite three-dimensional honeycomb but a finite four-dimensional polytope.

In 1939, the botanists J.W. Marvin and E.B. Matzke repeated Hales's experiment, replacing his peas by lead shot, "carefully selected under a microscope for uniformity of size and shape," in a steel cylinder, compressed with a steel plunger at sufficient pressure (40,000 pounds) to eliminate all interstices. When the shot were stacked in cannon-ball fashion and compressed, they became nearly perfect rhombic dodecahedra. But "if the shot were just poured into the cylinder the way Hales presumably put his peas into the iron pot, irregular 14-faced bodies were formed." Almost all the faces were either quadrangles, pentagons, or hexagons, with pentagons predominating. Another botanist examined cells in undifferentiated vegetable tissues, and concluded that the internal cells have an average of approximately 14 faces, though the most prevalent shape (occurring 32 times among the 650 cells examined) had 13 faces: 3 quadrangles, 6 pentagons, and 4 hexagons. The few cells that had only 12 faces were neither rhombic dodecahedra nor regular dodecahedra.

Matzke also made a microscopic examination of a froth of 1900 measured bubbles. "For 600 central bubbles examined, the average number of contacts was 13.7." The commonest shape had again 13 faces: 1 quadrangle, 10 pentagons, and 2 hexagons.

In 1959, Professor Bernal confirmed the prevalence of pentagonal faces by a remarkably simple experiment in which equal balls of "Plasticene" (oily modeling clay) were rolled in powdered chalk, packed together irregularly, and pressed into one solid lump. The resulting polyhedra were found to have an average of 13.3 faces. [Ha! — JD]

A couple of footnotes to that.

• C.A. Rogers was one of my own instructors at University College, London, 1963-66. I think he was in fact head of the Math Department. I attended his lectures on convexity, a topic in which he was a leading researcher.

• Look at the odd concidence here. The first name in Coxeter's account of what we might call "the soft-sphere problem" is Stephen Hales, whose dates are 1677-1761. The Kepler Conjecture was proved at last by Thomas Hales, born 1958 and still with us. Genetic transmission?