The Zen of Venn
Cogwheels of the Mind: The Story of Venn Diagrams. A. W. F.
Edwards. xvi + 110 pp. The Johns Hopkins University Press. $25.
Every high school graduate has been exposed to Venn diagrams. Few,
however, know anything about their originator, John Venn, or about
the interesting and beautiful mathematics that arises from the
consideration of Venn diagrams with more than a few sets. There is
no better place to start than with Cogwheels of the Mind,
by A. W. F. Edwards.
The author, who was the last undergraduate student of the famous
statistician R. A. Fisher at the University of Cambridge, proposed
in 1988 that the centenary of Fisher's birth be commemorated with a
pair of stained-glass windows—one for Fisher and one for
Venn—in the Hall of Gonville and Caius College, where both men
had been Fellows (a title Edwards also holds). This project
stimulated Edwards's interest in Venn diagrams. He then went on to
discover an important class of Venn diagrams as well as several
rotationally symmetric Venn diagrams.
But what exactly is a Venn diagram? Pick up a pencil and draw a
curve that meets itself only where the pencil tip first touched the
paper; this is curve number 1. This curve divides the paper into two
parts, the region inside the curve and the region outside it. Repeat
the process, for curves 2, 3 and up to n, and label each
region according to the identifying numbers of the curves that
enclose it. Now imagine taking a sharp knife and cutting along each
of the curves. If 2 n pieces of paper result and
each one has a unique label, then what you had before the cutting
started was a Venn diagram. Most readers will be familiar with the
classic Venn diagram formed of three interlocking circles (as shown
in the photograph below of the stained-glass window dedicated to
Venn), which create eight distinct regions by the process just described.
Edwards begins by presenting a most interesting history of Venn
diagrams and then discusses some of his own discoveries, such as the
fact that a very nice sequence of Venn diagrams results from
successively applying the rule of subdividing the regions adjacent
to the first drawn curve. Venn's own general rule was that one
subdivides the regions adjacent to the most recently added curve.
Edwards presents an extremely personal view of Venn diagrams. He
devotes the majority of the book to explaining how he became
intrigued by the diagrams and discovered some of their fascinating
properties. The volume is lavishly illustrated in color and even
reproduces Edwards's "scratch work" pertaining to his
research. However, the book contains only one diagram that is
attributed to a living researcher other than Edwards: Branko
Grünbaum's symmetric diagram of five ellipses. So do not expect
a balanced overview of recent research on Venn diagrams.
I have a few minor quibbles with some of Edwards's choices. For
example, he talks in the second chapter of drawing "an endless
line on a piece of paper so that it cuts itself any number of
times." It would be better to call it a curve and mention that
it is closed. Figure 3.2 shows a "corrected redrawing" of
C. S. Peirce's attempt to draw a seven-set Venn diagram. It would
have been more helpful to the reader to show Venn's construction for
seven curves and thus illustrate what Venn referred to as its
"comb-like" structure. There are several places where
Edwards seems to be implying incorrectly that Venn did not have a
general method for constructing Venn diagrams.
In the third chapter, Edwards states that "every n-set
Venn diagram constructed by the sequential addition of sets has the
same structure when considered as a mathematical graph." I am
not sure what he intended by this statement, but it seems incorrect,
because it is certainly the case that for both Venn and Edwards the
general method of constructing diagrams is "by the sequential
addition of sets," and yet their diagrams are distinct as
mathematical graphs whenever the number of curves is greater than four.
This book is aimed at readers interested in recreational mathematics,
who will particularly enjoy Edwards's discussion of the connections with
such classic topics as binomial coefficients and Gray codes and his
references to personalities such as Lewis Carroll. In spite of my few
reservations about the book, I heartily recommend it for readers
interested in knowing more about John Venn and the geometric properties
of Venn diagrams. It will also be appreciated by those interested in the
process of mathematical discovery.—Frank Ruskey, Computer
Science, University of Victoria, British Columbia, Canada