*Beyond Measure: A Guided Tour Through Nature, Myth, and
Number*

By Jay Kappraff

World Scientific, 2002

US $38.00, 582 pp.

ISBN 981-02-4702-8

REVIEWED BY THOMAS HULL

It would be very interesting to read a survey, from a mathematician's perspective, of the variety of popular math books that have emerged over the years. Of course, books of mathematical problems and puzzles have been popular for over a century, and there have been a bevy of historical, sometimes evangelical books on developing trends in the mathematical sciences since James Gleick's Chaos: the Making of a New Science. Martin Gardner's unique blend of history, puzzles, and honest-to-goodness math practically define their own category in this field.

Perhaps
Jay Kappraff does as well. In the
two books he has written, Kappraff describes interrelated fields of mathematics
with some common theme or purpose in mind. His first book, *Connections: The Geometric Bridge Between
Art and Science*, tied together subjects
like graph theory, transformational geometry, the golden ratio, and polyhedra
to exhibit the field of "design science," which lives in the intersection of
art, architecture, math, and several other sciences. The result was a delightful journey, appealing to both
artistic and mathematical minds while informing each about the other's
discipline. This seems to be a
common thread in Kappraff's work: making interdisciplinary connections,
especially between math, science, and art in hopes of fostering more
interactions across disciplines.

His second book, the one under
review, is *Beyond Measure: a Guided Tour Through Nature, Myth, and Number*. This
time the connection between the various subjects is not as direct, but they nonetheless
intertwine for Kappraff's main thesis.
In his own words,

A theme of this book is that our efforts to understand natural phenomena may be enhanced by broadening our approach to science and mathematics to include ideas from art and architecture, both ancient and modern. (p. xviii)

In
fact, Kappraff further claims that, "science has come to realize the limits
inherent in its ability to model reality," (p. xix) citing chaos theory as an example. The idea is that some mathematical
models are so sensitive that small changes, or measurement errors, in the
initial conditions will result in radically different behavior. Thus, if such models are indicative of
reality, then this "calls into question previously formulated models which have
assumed that the results of an experiment were intrinsically reproducible." (p.
xix) Is Kappraff attacking the
scientific method? He only
comments on this directly in the book's introduction, giving the argument that
the scientific method is inherently based on measurement, where any amount of
error can sometimes have a profound effect on the outcome. Perhaps, he goes on to suggest,
scientists could overcome these limitations to go *beyond measure* (hence the book's title) and gain understanding
about the world by taking cues from ancient cultures. Some cultures, as Kappraff attempts to demonstrate via
numerous examples throughout the book, seemed to overcome their limited
measurement tools to gain an advanced scientific understanding of their
world. He implies that modern
science lacks a key tool that ancient cultures had, that modern scientists are
aware of this lack, and that it could be regained via interdisciplinary work
between scientists and folks in other disciplines, especially artists.

This is a rather romantic outlook on things, but Kappraff's purpose is not to provide a detailed philosophical argument for his thesis. Rather, he aims to convince via examples. In this effort, the book certainly presents a wide variety of mathematical topics juxtaposed and related to just as many discussions about art and science from ancient cultures. For example, several chapters are devoted to mathematical patterns found in music and how they may have influenced different cultures around the world. Right triangles, the Brunes star, and other Euclidean geometry elements are connected to historical findings from ancient Egypt, Mesopotamia, and Megalithic Britain. The Farey sequence is shown to appear time and time again in a variety of settings.

However, Kappraff is treading some dangerous ground here. Any such discussion, with its rather explicit conclusion that scientists and mathematicians would better serve the pursuit of knowledge if they did things differently, runs the risk of offending any mathematically sophisticated reader. Furthermore, readers with less mathematical background might be convinced of things that do not ring true, like the book's implication that mathematicians actually pay attention to Biblical numerology. The fact is that Kappraff's own personal beliefs on the role art, symbolism, and myth should play in the math and science community come through quite strongly, and with only examples (sometimes flawed) and rhetoric (mostly empty) to prove his points, he doesn't serve either audience very well.

One
admittedly extreme example that illustrates this is Chapter 12 in the book,
titled "The Flame-Hand Letters of the Hebrew Alphabet." Kappraff describes the work of Stan
Tenen, who has made a very intriguing conjecture about the origin of the Hebrew
alphabet. Tenen claims that if a
certain spiral curve in **R**^{3}
is projected onto planes at various angles, one can generate startlingly close
approximations to Hebrew characters.
Of course, as Kappraff points out, the Hebrew alphabet is several
thousands of years old, and the exact shape of ancient Hebrew letters is
something of debate for Judaic scholars.
Kappraff even admits that while Tenen's work "has begun to be presented
for peer review, it would be premature to comment on its authenticity." (p.
265) Still, at the very least it's
an interesting idea. However,
Kappraff reproduces several of Tenen's illustrations which serve to undermine
the legitimacy of Tenen's work.
For example, Figure 12.17 (page 282) provides a diagram from one of
Tenen's publications where the spiral projecting curve is drawn as a gradually
thickening ribbon on a dimpled sphere.
This is fine, but part of the curve is drawn to make an enclosure-like
shape on the surface, where the words "Plato's Cave" are written. Further, surrounding the surface are
dozens of other alluring catch-phrases, like "4 rivers around Eden," "cosmic
egg," "the 'great sea' surrounds the Earth Plane" (presumably, the x-y plane),
and "Aladdin's Lamp." Yes, this is
balderdash, or at the very least we could say that Tenen seems to take great
liberties with his symbolic interpretations of shapes. Yet Kappraff actually seems to indicate
that this adds more value to Tenen's work. He states, "As one would do for poetry, Tenen's proposal of
flame-hand letters should be evaluated not only on its literal meanings but on
its ability to ring true at the level of metaphor." (p. 286) This seems to be one of the main
thrusts of this book Ð that myth and metaphor can provide understanding of our
world at a level that modern scientific methods cannot. However, examples such as this, where
the metaphors seem to be strewn about with random abandon, only make the thesis
seem to border on the ludicrous.

Not
all of Kappraff's examples are as flamboyant as this. In fact, some of them are quite fascinating, like the
investigation in Chapter 10 of "hidden" floor designs in the Laurentian Library
in Florence, Italy, about which Kappraff has previously written in the *Intelligencer* ([2]).
This library was designed by Michelangelo around 1523, and the pavements
in question were hidden under wooden desks for several centuries until their
accidental re-discovery in 1774.
Researchers conjecture that Michelangelo himself designed these pavements,
each of which illustrate a different study in geometry and design. Further, it is not hard to find such
classic gems as the golden ratio, the Brunes star, or the "sacred cut" in these
designs, making Kappraff (and others) view them as a synthesized synopses of
geometric "tricks of the trade" developed by various cultures up to that point
in time. However, since Kappraff
believes each of these geometric forms and methods carry symbolic weight from
the cultures in which they originated, he forms a very fanciful conclusion:

The extensive occurrence in the pavement panels of particular geometries and the numbers they spawn suggest that the pavement designers were cognizant of the bond between number and myth that modern scholarship is once again making available for us. (p. 231)

By "modern scholarship" he clearly means people like himself and those he credits in his book. But the implication here is also clear: that the rest of the scholarly community supports this view. Nothing could be further from the truth.

In
fact, one of the most egregious omissions of *Beyond Measure* is its failure to mention other, some would say more
rigorous research in the history of science. As one example, Kappraff presents briefly (p. 236) the
classic Chinese diagram from the *Zhou bi*, one of the earliest works of the classical Chinese mathematical
canon, that is often offered as proof that Chinese scholars had proven the
Pythagorean Theorem independently of the Greeks. Kappraff repeats this claim, ignoring completely the work of
Christopher Cullen (recently summarized in [1]) who tries to understand the
culture and methods of ancient Chinese mathematicians and arrives at the
conclusion that to hold the past work of the Chinese to the same proof
standards of Euclid is a big mistake.
On one hand, Cullen argues, the Chinese do not seem to have proven the
Pythagorean Theorem, but on the other hand a study of their methods has a lot
to tell us about different modes of mathematical understanding. Cullen even goes so far to say that
modern mathematicians could learn a thing or two from the ancient Chinese. This seems right up Kappraff's alley,
and what's more, it has the advantage of being carefully reasoned and
documented research with no hint of starry-eyed romanticism of past
cultures. The fact that Kappraff
not only omits but seems ignorant of such work makes one wonder what other
history of science research is missing from his treatise. The result is a huge loss of
credibility for the book in the eyes of any mathematician even marginally
familiar with history of science scholarship.

At
times one does see merits in *Beyond Measure*. Certainly a large number of
mathematical topics are presented, from projective geometry to grey codes to
fractals to basic number theory.
But time and time again the rhetoric is thick with wild
speculation. Another example (more
mild than the Hebrew alphabet projections) is Kappraff's presentation of the
work of Ernest McClain, who, despite Kappraff's association of him with
Biblical numerology (which will alarm any mathematician familiar with the
faux-statistics in books like *The Bible Code*) seems to have fascinating theories about how
ancient peoples probably discovered very early on the patterns found in musical
tones generated by plucking a taught string at various lengths, and that the
numbers in these patterns may have influenced other numerical modeling. For example, 12-tone musical scales are
found all over the world in ancient cultures, and this may have influenced the
creation of twelve signs of the Zodiac.
Now, McClain may or may not jump to fanciful conclusions in his work,
but Kappraff most certainly does:

McClain points out that Ð in Mesopotamian base 60 arithmetic Ð the 13 tones of the spiral of fifths require the monochord reference unit to be interpreted arithmetically as 60 to the fifth power, that is, as 777,600,000 base 10. And traditional Kabbalist interpretations of YHWH as 10-5-6-5 can be read as 10 to the fifth power (meaning 100,000) times 6 to the fifth power (meaning 7776)= 777,600,000. Is it possible that this reading is cleverly implanted in Genesis as the age of Noah's father (777) when the flood came in his 600th year, when his three sons were already 100? (pp. 78-79)

Kappraff then tries to provide further support of this nonsense. No, not all of Kappraff's exposition contains outlandish passages such as this, but one does not have to look very hard to find them either.

Additionally, the book does not seem to have been edited. In some parts of the book there seems to be at least one typographical error per page, including many mislabeled figures. Mathematical errors are also present. On page 283 one finds a picture labeled "Image of a hypersphere" which is definitely not a hypersphere; more likely it is representing the complement of a torus. Also, in Chapter 16: Chaos and Fractals, the Riemann mapping theorem is stated incorrectly. We quote, "É the Riemann mapping theorem which states that a unit circle of the complex plane É can be mapped onto any region of the complex plane bounded by a closed curve so that the radial lines and concentric circles are mapped to curves that continue to be mutually perpendicularÉ" (p. 432) Ignoring the fact that he meant to say "unit disk" and forgot to mention injectivity, this is an application of the Riemann mapping theorem to equipotential curves in the exterior of the Mandelbrot set and corresponding Julia sets, not the Riemann mapping theorem itself. Sure, leeway can be allowed in a general, expository text, but undergraduates and non-mathematicians are likely to walk away thinking that the Riemann mapping theorem is specifically a theorem about the Mandelbrot set. In any case, it's hard to imagine that the book was reviewed by any mathematician, given the authors penchant to juxtapose good mathematics with questionable research on the history of science and absurd conjectures. (Nonetheless, the publisher did manage to find three praiseworthy quotes for the back cover.)

In conclusion, mathematicians reading this book will no doubt find a plethora of mathematical topics to interest them, but will almost certainly become infuriated (or wildly amused) by the author's rhetoric. Non-mathematicians or students are advised to stay away. Mathematically inclined readers would be much better off with Martin Gardner or Keith Devlin than running the risk of being persuaded by Kappraff's hokum.

And it is sad, for Kappraff's initial intent is a noble one. Interdisciplinary communication does need to happen more often, and there is certainly a lot to be gained by all via conversations between mathematicians, artists, and designers. But this book is not the way. Rather, let each party spend time in the shoes of the other. Let the artist wrestle with mathematical inquiry and learn to appreciate the beauty of proof, and let the mathematician view the world, with all its social and aesthetic baggage, through the eyes of the artist. Then more fruitful collaborations or new creative insights might emerge. But trying to find common ground in wild speculations, as Kappraff suggests, is not a good idea.

[1] Cullen, C., "Learning from Lui Hui? A Different Way to Do Mathematics" *Notices
of the American Mathematical Society*, Vol.
49, No. 7 (2002), 783-790.

[2] Kappraff, J., "Hidden Pavements in Michelangelo's
Laurentian Library" *The Mathematical Intelligencer*, Vol. 21, No. 3 (1999), 24-29.

Thomas Hull

Department of Mathematics

Merrimack College

North Andover, MA 01845 USA

Thomas.Hull@merrimack.edu