Beyond Measure: A Guided Tour Through Nature, Myth, and Number
By Jay Kappraff
World Scientific, 2002
US $38.00, 582 pp.
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 R3 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 – 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 (). 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 ) 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 – in Mesopotamian base 60 arithmetic – 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, "… the Riemann mapping theorem which states that a unit circle of the complex plane … 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 perpendicular…" (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.
 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.
 Kappraff, J., "Hidden Pavements in Michelangelo's Laurentian Library" The Mathematical Intelligencer, Vol. 21, No. 3 (1999), 24-29.
Department of Mathematics
North Andover, MA 01845 USA