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Twisted Math and Beautiful Geometry

Four families of equations expose the hidden aesthetic of bicycle wheels, falling bodies, rhythmic planets, and mathematics itself.

Steiner's Porism

The first half of the 19th century saw a revival of interest in classical Euclidean geometry, in which figures are constructed with straight-edge and compass and theorems are proved from a given set of axioms. This “synthetic,” or “pure,” geometry had by and large been thrown by the wayside with the invention of analytic geometry by Pierre de Fermat and René Descartes in the first half of the 17th century.

Analytic geometry is based on the idea that every geometric problem could, at least in principle, be translated into the language of algebra as a set of equations, whose solution (or solutions) could then be translated back into geometry. This unification of algebra and geometry reached its high point with the invention of the differential and integral calculus by Newton and Leibniz between 1666 and 1676; it has remained one of the chief tools of mathematicians ever since. The renewed interest in synthetic geometry came, therefore, as a fresh breath of air to a subject that had by that time been considered out of fashion.

One of the chief protagonists in this revival was the Swiss geometer Jacob Steiner (1796–1863). Steiner did not learn how to read and write until he was 14, but after studying under the famous Swiss educator Heinrich Pestalozzi, he became completely dedicated to mathematics. Among his many beautiful theorems we bring here one that became known as Steiner’s porism (more on that odd name in a moment).

Steiner considered the following problem: Given two nonconcentric circles, one lying entirely inside the other, construct a series of secondary circles, each touching the circle preceding it in the sequence as well as the two original circles ( see the figure at right ). Will this chain close upon itself, so that the last circle in the chain coincides with the first? Steiner, in 1826, proved that if this happens for any particular choice of the initial circle of the chain, it will happen for every choice.

In view of the seeming absence of symmetry in the configuration, this result is rather unexpected. Steiner devised a clever way of exposing hidden symmetry by inverting the two original circles into a pair of concentric circles. As a result, the chain of secondary circles (now inverted) will occupy the space between the (inverted) given circles evenly, like the metal balls between the inner and outer rings of a ball-bearing wheel. These can be moved around in a cyclic manner without affecting the chain.

But that’s not all: It turns out that the centers of the circles of the Steiner chain always lie on an ellipse ( marked in red ) , and the points of contact of adjacent circles lie on yet another circle ( marked in green ) .

The images at left illustrate nine Steiner chains, each comprising five circles that touch an outer circle ( alternately colored in blue and orange ) and an inner black circle. The central panel shows this chain in its inverted, symmetric “ball-bearing” configuration. As happens occasionally, a theorem that has been known in the West for many years turned out to have already been discovered earlier in the East. In this case, a Japanese mathematician, Ajima Chokuyen (1732–1798), discovered Steiner’s porism in 1784, almost half a century before Steiner. An old Japanese tradition, going back to the 17th century, was to write a geometric problem on a wooden tablet, called sangaku, and hang it in a Buddhist temple or Shinto shrine for visitors to see. A fine example of Steiner’s—or Chokuyen’s—chain appeared on a sangaku at the Ushijima Chomeiji temple in Tokyo. The tablet no longer exists, unfortunately, but an image of it appeared in a book published about the same time as Steiner’s discovery.

It is somewhat of a mystery why this theorem became known as Steiner’s porism. You will not find the word porism in your usual college dictionary, but the online Oxford English Dictionary defines it as follows: “In Euclidean geometry: a proposition arising during the investigation of some other propositions by immediate deduction from it.“ Be that as it may, the theorem again reminds us that even good old Euclidean geometry can still hold some surprises within it.

Notes

• Steiner chains enjoy many additional properties. See http://en.wikipedia.org/wiki/Steiner_chain . For a proof of Steiner’s porism, see Coxeter, Introduction to Geometry, p. 87.
• See Hidetoshi and Rothman, Sacred Mathematics: Japanese Temple Geometry, p. 292.