Introduction to Circle Packing: The Theory of Discrete Analytic
Functions. Kenneth Stephenson. xii + 356 pp. Cambridge
University Press, 2005. $60.
In the past, books about science and math were written in many
different styles. For example, Galileo published his most famous
books as dialogues between Simplicio (a rather dense traditional man
who is a proponent of Ptolemaic/Aristotelian theory) and two more
enlightened citizens—Sagredo (a modern Copernican) and
Salviati (an unbiased representative of the educated public). Today,
unfortunately, scientific exposition on the research level and
especially mathematics writing have fallen into a trap—a rigid
professional code dictates the structure and style in which new
results are to be communicated. In mathematics, this tradition
forces the author to write a strictly linear sequence of
definitions, propositions, theorems, corollaries and lemmas, giving
little hint of where they came from, where he or she is going with
them and above all why. This strangely limited but pure style traces
back to Euclid and to his modern-day reincarnation, the French
mathematics collective that wrote under the pseudonym "Bourbaki."
What is remarkable about Introduction to Circle Packing is
that its author, Kenneth Stephenson, a professor of mathematics at
the University of Tennessee in Knoxville, is working hard to break
out of this mold. I share his ambition, having written a book three
years ago that is not altogether unrelated to his.
The landscape of mathematics books may be roughly described as
consisting of three domains: (a) professional books written in the
style described above; (b) popular books that are intended to
present important mathematical ideas but which, being constrained to
have no equations, are limited to anecdotes and vague analogies; and
(c) a small smattering of things in between. Examples of works in
the "popular" category are Chaos, by James Gleick
(1987), and Stalking the Riemann Hypothesis, by Daniel N.
Rockmore (2005). Both have the laudable goal of conveying the
excitement of an active and important area of mathematical research,
but neither book, in my view, ever actually explains what the
research is really about. The problem is not that Gleick and
Rockmore do not try. The problem is that you can't explain real math
without asking a little bit more of the reader.
Stephenson's book deals with another truly beautiful new development
in math. Moreover, it is one in which the stunning first-level
results involve only elementary geometry. The simplest and most
basic result is this: Imagine packing many small circles inside one
big one. If you do this as efficiently as possible (producing what
is called a "maximal" packing), the small circles will
touch one another as much as possible, and the only gaps will be
regions that are bounded by three mutually tangent circles (and thus
look like triangles with curved sides). An example is shown in the
illustration on this page: On the left side, 19 circles are packed
in one big circle. Any such a "packing" of a big circle by
smaller ones has an associated configuration: It is a graph. The
configuration for this circle packing is shown to the right of it.
Note how the black vertex marked w corresponds to the dark
w , and the neighboring vertices
v 1, v 2, v
3 and v 4 correspond to the lightly
shaded circles touching c
w . In general, to make a configuration, you
take one vertex for each circle, join two with an edge when the
circles touch and fill in a triangle whenever three circles are
mutually tangent. The wonderful result is that if you start with any
such configuration (technically, a topological triangulation of the
disk), then there is a nearly unique way to assign radii to the
circles so that there is a packing with this configuration. (The
only nonuniqueness comes from the possibility of moving the circle
packing by the three-dimensional group of Möbius
transformations.) What is remarkable here is that a discrete object,
the configuration, is determining the whole geometry of the circles.
Paul Koebe discovered and proved this remarkable fact in 1936, but
it lay dormant until it was picked up first by E. M. Andreev, then
by William Thurston, Kenneth Stephenson, Zheng-Xu He and Oded
Schramm. What a wonderful example of how international mathematics
is: The ball bounced from Germany to Russia to the United States,
China and Israel. Circle packing is now a full-fledged area of
investigation, which this book seeks to popularize.
How successful is Stephenson in explaining his ideas to a larger
audience? The book has four parts, which get harder as you go along.
Chapters 1 through 7, which include all of Part I and most of Part
II, might be accessible to someone with nothing more than a good
high-school math background who is willing to read slowly and put in
some time to feel his or her way into the ideas. This material is
written formally, with proofs. Stephenson does use some technical
math terms, such as homotopic and homeomorphic.
But he explains and illustrates them. He says, excusing his style,
Let me share something with you. Mathematicians
write linearly—I am compelled to put this
stuff here! On the other hand, if I were reading this, I
would glance briefly, then skip ahead to the main show, returning
only when I found myself in a bind.
Fortunately, Stephenson is one of a new breed of pure
mathematicians, growing in number, who love to combine experiment
with theory. This means he has computer code to carry out these
packings and investigate their properties. And the book is
interlaced with experiments—some successful, some not, some
which worked one day but not the next when pushed further. His
immense enthusiasm for this subject comes through on every page.
After Chapter 7, things get more heavily mathematical in many
places. As a whole, the book is suitable for graduate students in
math. In addition to making these ideas accessible, Stephenson has a
second agenda: to make other mathematicians believe that circle
packings are the basis for a discrete version of complex analysis
that is every bit as rich and deep as traditional complex analysis,
which deals with continuous variables. To make this point clear, the
last part of the book, chapters 19 through 23, makes explicit the
links between discrete and continuous complex analysis.
At this point, the book is really for mathematicians only. But along
the way, there are many beautiful examples and discussions that are
not very technical, such as the discrete Belyi functions
covered in sections 16.3 and 23.1. And although it looks like heavy
math from the outside, the argument about hyperbolic and parabolic
uniqueness given in section 8.3 (which introduces geometric tools
based on winding numbers) is explained simply and slowly and is
quite breathtaking. With proper preparation, an honors high-school
math class could understand it.
Three years ago I wrote a book, Indra's Pearls, with
Caroline Series and Dave Wright, on a different
aspect of complex analysis—the fractal shapes generated by
infinite discrete groups of conformal maps. Like Stephenson, we
tried to start very slowly and explain the basic concepts on a
high-school level, using many pictures. But as we went on, our
excitement took over, and we wrote a book that only a graduate
student or an avid hobbyist or groupie is going to read all the way
through to the end.
I'm not sure what the best way is to find a middle ground between
standard research monographs and journalistic reports from the math
front. But it is, I feel, important to try to bridge the undeniable
gap between those who are math-literate (physical scientists,
economists, computer scientists and engineers) and the greater
educated public. I am glad to see a new genre of math books with
both pictures and formulae in them, written so that a reader who is
willing to put in a little hard thinking can come away really
understanding something wonderful in mathematics.