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

Jay Kappraff
World Scientific
Publication Date: 
Number of Pages: 
Series on Knots and Everything 28
[Reviewed by
Charles Ashbacher
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Applications of mathematics can be found just about anywhere, and in this collection of essays, Kappraff shows us many different places where mathematics is embedded in the structure. From the patterns of Amish quilts, to the mathematics of musical scales, the number theory behind the optimal spacing of leaves to collect the greatest amount of sunlight to the symmetry of the designs in the floor of the Laurentian Library in Florence, mathematics appears everywhere.

Not all of the topics can be considered mathematical, by far; my favorite chapter was number 12, "The Flame-Hand Letters of the Hebrew Alphabet." The early writing of the Egyptians and Chinese was based on pictographs, where a simplified picture of the object became the symbol for the object. The Semitic tribes developed the first truly alphabetic written language and the point raised in this chapter deals with the origin of those letters. It was argued, although it is extremely controversial, that the letters were developed by projecting a vortex onto a two-dimensional surface. This vortex could be the swirling of a fire and the projection could be that of a hand. Hence, the title of the chapter. I am not qualified to pass judgment on the validity of the argument, but it was certainly interesting to think about.

What I really liked about the book was that while some of the standard material of popular math books appeared, topics such as chaos, fractals, and spiral symmetry are covered, there were many examples that I had not seen before. The magic square formed from DNA codons was pretty cool. As the co-editor of Journal of Recreational Mathematics, I have seen many magic squares. To see one with applications to an understanding of genetics was "worth the price of admission." This is one of those books where everyone with an interest in math, from the professor to the interested public, will be able to understand it and come away enriched.

Charlie Ashbacher ( is the principal of Charles Ashbacher Technologies, a company that offers state of the art computer training. He is also an adjunct instructor at Mount Mercy College in Cedar Rapids, Iowa, and at the end of this academic year, he will be three courses short of having taught every class in the math and computer science majors. A co-editor of the Journal of Recreational Mathematics , he is the author of four books in mathematics and one in computer programming.


* Essays in Geometry and Number as They Arise in Nature, Music, Architecture and Design:
* The Spiral in Nature and Myth
* The Vortex of Life
* Harmonic Law
* The Projective Nature of the Musical Scale
* The Music of the Spheres
* Tangrams and Amish Quilts
* Linking Proportions, Architecture, and Music
* A Secret of Ancient Geometry
* The Hyperbolic Brunes Star
* The Hidden Pavements of the Laurentian Library
* Measure in Megalithic Britain
* The Flame-hand Letters of the Hebrew Alphabet
* Concepts Described in Part I Reappear in the Context of Fractals, Chaos, Plant Growth and Other Dynamical Systems:
* Self-Referential Systems
* Nature's Number System
* Number: Gray Code and the Towers of Hanoi
* Gray Code, Sets, and Logic
* Chaos Theory: A Challenge to Predictability
* Fractals
* Chaos and Fractals
* The Golden Mean
* Generalizations of the Golden Mean - I
* Generalizations of the Golden Mean - II
* Polygons and Chaos
* Growth of Plants: A Study in Number
* Dynamical Systems