BOOK REVIEW
Numbering Nature
Will Wilson
Mathematics in Nature: Modeling Patterns in the Natural
World. John A. Adam. xxvi + 360 pp. Princeton University Press,
2003. $39.50.
I've always loved math, and as a child I especially loved word
problems about everyday things. The idea that the real world can be
described mathematically was, to me, simply wonderful. Today I get
the same enjoyment from mathematical descriptions of nature, which
are just more complicated versions of the word problems I adored in
my youth. A truly breathtaking range of such problems can be found
in John A. Adam's Mathematics in Nature, which tackles
quite a broad assortment of nature's patterns, going beyond the
typical ones with which we might all be familiar. Most, however, are
within nearly everyone's realm of experience. For example, the
book's 24 color plates present fascinating cloud patterns, a double
rainbow, sand dunes, ocean waves, plant and floral forms, patterns
on animals and cracks in asphalt. Adam builds the reader's
mathematical intuition as he discusses these phenomena.
The word problems I was raised on dealt with a more select subset of
experiences: bobs on strings, baseballs and missiles flying through
the air, and cars accelerating on highways. Later the problem fodder
got more diverse—but not by much. Even when I got to graduate
school in physics, it was pretty limited. I recall once having a
contest in a bar with some fellow grad students in which the goal
was to imagine the most implausible scientific discipline. I came up
with "theoretical biology." (Ironically, I'm now a
professor of theoretical ecology.) Certainly my contest entry was
the result of my own lack of imagination, but it was also a product
of the limited scope of word problems I had faced during my
education. A more diverse set would have given me, and other
mathematically oriented students, a better sense of how broadly
mathematics can be applied.
Mathematics in Nature is an excellent resource for bringing
a greater variety of patterns into the mathematical study of nature,
as well as for teaching students to think about describing natural
phenomena mathematically. However, the book is not a light read for
anyone who is not already well versed in the solution of partial
differential equations. It could be used for an upper–level
undergraduate course, or more likely a graduate–level seminar.
It may be perfect for biology types who like math well enough but
were immensely bored or put off by studying blocks sliding down
inclined planes.
The initial chapters of the book define some terms, present
guidelines and principles, and provide a warm–up set of two
dozen estimation problems. These include such challenges as
estimating "how fast human hair grows (on average) in miles per
hour" and "the speed of descent of a cloud droplet in
still air." I can't recall ever seeing these types of problems
used so explicitly in any book; they're employed to overall good
effect here.
I'm happy to report that dimensional analysis is given full coverage
in a separate chapter on the problem of scale, with useful word
problems to get the point across. I find that estimation and
dimensional analysis are quite lacking in mathematical biology,
particularly population biology and the dimensionally suspect
concept of "population density." More attention needs to
be paid to these features, but I've never known how to bring them
up. That problem is solved here.
Next Adam examines various categories of patterns. Two chapters on
meteorological optics deal with such subjects as shadows, rainbows,
halos and the sun's reflection off rippled water. (I hadn't realized
that a math problem lurked behind that last item.) These problems
morph into questions regarding clouds ("How heavy is a
cloud?" "Why do we see farther in rain than in fog?")
and sand dunes, and there is an excellent discussion of hurricanes.
Another chapter deals with all sorts of wave phenomena, bravely
presenting a clear mathematical discussion of dispersion relations.
This is followed by an excellent treatment of spatial stability
analysis. These concepts, which involve some heavy math, are covered
remarkably well.
A chapter on the Fibonacci sequence and the Golden Ratio represents
a departure from the differential–equations mindset. A really
sensible argument, based on efficient packing, is put forth for the
striking seed patterns seen in sunflower heads—patterns that
are based on the Fibonacci numbers. Adam then moves on to describe
applications of packing to honeycombs, bubbles and mud cracks. After
that, he brings differential equations back to discuss river
meanders. Certainly many people have been fascinated with river
meanders and the concomitantly formed canyons that can be observed
from the sky. This work is then extended to tell you everything
you've always wanted to know about trees—how they grow, bend,
and filter light.
There is a short chapter on bird flight, and then the final chapter
touches on biological pattern formation. The latter chapter is no
match for what can be found on that topic in James D. Murray's book
Mathematical Biology, now in its third edition. Murray
provides excellent coverage of the application of
reaction–diffusion models to biological pattern formation, and
probably nothing will ever beat his presentation.
Adam could have included more patterns in his book. Ecological
patterns are essentially absent (except for the implicit natural
selection motivation for maximal seed packing in sunflower heads).
Population dynamics, behavior, mating systems and evolutionary
patterns—phenomena that constitute a huge part of theoretical
biology—are not considered. However, the first three chapters
alone make Mathematics in Nature an incredibly useful
source for lecture material, and the breadth of patterns studied is
phenomenal.—Will Wilson, Biology, Duke University