Ball Arches – Mathematical Connections

The story

Two “infinite” ball arches burst forth from the Great Doodle Quilt and encircle the exhibit. One reaches skyward and disappears from view, representing a converging series. The other arch plunges deep into the bay, representing a diverging series. If the arches truly had an infinite number of balls, the length of the converging series would be finite, while the length of the diverging series would be infinite.

The two arches demonstrate the difference between a converging and a diverging series. The shorter of the two is the Converging Arch; it corresponds to a geometric series, in which each ball has a diameter that is a fixed fraction of the preceding (larger) diameter. The sum of all the diameters (the length of the arch) is finite; the Converging Arch stops at a finite distance from its start even though there are infinitely many balls. (The balls in its tail become too small to be seen.)

In the Diverging Arch, the ratio of the (n+1)st diameter to the nth is (n/n+1)^{^}{⅔}; although each ball has a diameter that is strictly smaller than the immediately preceding one, the decrease is so slow that the series diverges.

This arch, if it were continued ad infinitum, would go through the floors underneath, through the Earth, leave the solar system and eventually even our galaxy — it would go on forever. In practice we lose sight of it after it dives below the waters of the Bay.

Not every ball is adorned with embroidery. In each of the Arches, one can number the balls from the start; the largest (shared) ball is 1, followed by 2 (which is the same size in both arches, even though the series have such different behavior), 3 (already a bit smaller in the Converging than in the Diverging Arch), etc. With this numbering, one easily checks that only the balls with numbers  3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73 are embroidered.  We can only see the first 100 balls in the diverging arch (in the converging arch, we can distinguish much fewer; the 100th ball has diameter less than 0.0004” and is not visible); if we could see below the water surface, then it would turn out that balls 101 and 103 are embroidered too. The labels of embroidered balls are all the twin primes as far as we can see, that is, each label of such a ball is one of a pair of prime numbers that are only 2 apart, as illustrated in the figure.

(Note that spiral number arrangements of this type, often called Ulam spirals, bring out very interesting and as yet poorly understood properties of the distribution of prime numbers. At scales much larger than in this figure, lines and curves emerge on which primes occur much more often than elsewhere.) It is believed by most mathematicians, but not yet proved as of 2021, that there are infinitely many pairs of twin primes. (A breakthrough result from 2014 by Zhang proves that for some finite number N, infinitely many pairs of primes exist that are at most some finite number N apart. At first, values of N that allowed for a watertight proof were very large; a large collaborational mathematical effort brought down the value to N=246. As of January 2022, the twin prime conjecture, corresponding to N=2, is still open.)

The embroidery pattern for the adorned balls was inspired by the Catalan solids, yet another family of special polyhedra than the platonic, archimedean and Johnson solids illustrated elsewhere in the installation. (There is a relationship, though: the Catalan solids are duals of the archimedean solids.) As in those three other families, the Catalan polyhedra are convex; unlike them, they don’t have faces that are themselves regular polygons. What makes them special is that in each Catalan solid all the faces are identical polygons (which is not the case for Johnson or even archimedean polyhedra). There are 13 Catalan solids; each of them is illustrated by one (or two) of the embroidery patterns chosen for the 15 embroidered balls visible in the diverging arch; their smaller siblings in the Converging Arch sport the same pattern with a permutation of the embroidery colors.

This table illustrates the correspondence between the Catalan solids and the temari embroidery; the last two embroidered balls (71 and 73) show repeats of previously illustrated Catalan solids.