The age-old gum ball conundrum

Originally published 25 March 1991

How many gum balls can you fit into a gum ball machine? And does it matter?

It prob­a­bly does­n’t mat­ter to gum-ball machine own­ers, who aren’t par­tic­u­lar­ly con­cerned about squeez­ing a few more balls into their machines, espe­cial­ly if it means pay­ing some­one to pack them in a par­tic­u­lar way. But it mat­ters to math­e­mati­cians. Ques­tions like this are the math­e­mati­cian’s stock in trade.

So how do you fill a space most com­plete­ly with equal-sized spheres? Renais­sance astronomer Johannes Kepler guessed the answer almost 400 years ago: Arrange a lay­er of balls in a tri­an­gle (like bil­liard balls in the stan­dard tri­an­gu­lar rack) and then pile more balls on top (like a stack of can­non balls). Extend the arrange­ment in every direc­tion. Each ball will touch twelve neigh­bors, six in the same lay­er, three above, and three below. There’s no more com­pact way of stack­ing spheres.

The trou­ble is, Kepler could­n’t prove his guess and math­e­mati­cians like iron-clad cer­tain­ty. Now along comes a math­e­mati­cian who thinks he has found a rig­or­ous proof of Kepler’s guess. Wu-Yi Hsiang of the Uni­ver­si­ty of Cal­i­for­nia at Berke­ley is putting the fin­ish­ing touch­es on a 100-page proof of the 400 year-old unsolved prob­lem. If Hsiang’s fel­low math­e­mati­cians can’t find a flaw in his proof, the sphere-pack­ing prob­lem will be laid to rest forever.

Why care?

Did I hear a sti­fled yawn? Are you won­der­ing why any­one should care how many gum balls can be packed into a gum-ball machine? Well, math­e­mati­cians care because that’s the sort of thing that turns math­e­mati­cians on (real­ly!). And cer­tain engi­neers care because sphere pack­ing does have appli­ca­tion to such prac­ti­cal mat­ters as radar and CAT scans. But for most of us the real inter­est in gum-ball pack­ing lies else­where, and to find it we go back to the orig­i­nal sphere-pack­ing man, Johannes Kepler.

Kepler dis­cussed the prob­lem in a lit­tle book called The Six-Cor­nered Snowflake, pub­lished in 1611. The book is not near­ly as well known as it should be. Kepler is best known for fig­ur­ing out that plan­ets move in ellip­ti­cal paths. In The Six-Cor­nered Snowflake he asks an even more fun­da­men­tal ques­tion: Why does nature take the shapes it does?

All around him Kepler observed beau­ti­ful shapes in nature: six-point­ed snowflakes, the ellip­ti­cal orbits of the plan­ets, the hexag­o­nal hon­ey­combs of bees, the 12-sided shape of pome­gran­ate seeds. Why? he asks. Why does nature dis­play such math­e­mat­i­cal perfection?

Why does the stuff of the uni­verse arrange itself into five-petaled flow­ers, spi­ral galax­ies, dou­ble-helix DNA, rhom­boid crys­tals, the rain­bow’s arc? Why the five-fin­gered, five-toed, bilat­er­al­ly sym­met­ric beau­ty of the new-born child? Why?

Kepler strug­gles with the prob­lem, and along the way he stum­bles onto sphere-pack­ing. Why do pome­gran­ate seeds have twelve flat sides? Because in the grow­ing pome­gran­ate fruit the seeds are squeezed into the small­est pos­si­ble space. Start with spher­i­cal seeds (the shape of gum balls). Pack them as effi­cient­ly as pos­si­ble (with each sphere touch­ing twelve neigh­bors). Then squeeze. Voila!

And so he goes, con­vinc­ing us, for exam­ple, that the bee’s hon­ey­comb has six sides because that’s the way to make hon­ey cells with the least amount of wax. His book is a tour-de-force of play­ful mathematics.

The shape of nature

In the end, Kepler admits defeat in under­stand­ing the snowflake’s six points, but he thinks he knows what’s behind it all, behind all of the beau­ti­ful forms of nature: A uni­ver­sal spir­it per­vad­ing and shap­ing every­thing that exists. He calls it nature’s “for­ma­tive capacity.”

We would be inclined to say that Kepler was just giv­ing a fan­cy name to some­thing he could­n’t explain. To the mod­ern mind, for­ma­tive capac­i­ty sounds like emp­ty words.

We can do bet­ter. For exam­ple, we explain the shape of snowflakes by the shape of water mol­e­cules, and we explain the shape of water mol­e­cules with the math­e­mat­i­cal laws of quan­tum physics.

Since Kepler’s time, we have made impres­sive progress towards under­stand­ing the vis­i­ble forms of snowflakes, crys­tals, rain­bows, and new­born babes, for starters, by prob­ing ever deep­er into the heart of matter.

But we are prob­a­bly no clos­er than Kepler to answer­ing the ulti­mate ques­tions: What is the rea­son for the curi­ous con­nec­tion between nature and math­e­mat­ics? Why are the math­e­mat­i­cal laws of nature one thing rather than anoth­er? Why do nat­ur­al forms exist at all?

It was think­ing about such prob­lems that led Kepler to dis­cov­er the sphere-pack­ing the­o­rem that Wu-Yi Hsiang now claims to have proved.

And that’s why sphere-pack­ing mat­ters to the rest of us. Math­e­mat­ics and nat­ur­al forms are inti­mate­ly relat­ed. Think­ing about gum balls in gum ball machines can be a pro­found­ly philo­soph­i­cal activity.

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Hsiang’s 1990 proof was lat­er found to be incom­plete. It would­n’t be until 2017 when a for­mal proof to Kepler’s sphere-pack­ing con­jec­ture would be accept­ed by the math­e­mat­i­cal com­mu­ni­ty. ‑Ed.