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The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science

Alexey Stakhov
World Scientific
Publication Date: 
Number of Pages: 
Series on Knots and Everything 22
[Reviewed by
Mark Bollman
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At Mathfest in 2007, I attended a talk on “Puzzling Probabilities Featuring the Street Game of Craps” by Jack Alexander of Miami Dade College. The mathematical level of the talk was fairly elementary, but the talk was extremely engaging, and I walked away thinking what a pleasure it was to encounter very familiar mathematics approached in a new way. I had much the same reaction to Stakhov’s book, which begins with the golden section (the source for what is termed “harmony mathematics”) and ranges widely throughout many areas of mathematics.

As is often the case with works involving the golden section, there are a number of sections which overestimate the significance of the ratio, but these do not detract from the otherwise fine exposition. The lively treatment of such topics as hyperbolic functions, Fibonacci codes, and non-Euclidean geometry is a welcome collection of diverse topics in a single book.

Where the book goes a bit too far is evident only in the epilogue, where, in a list of conclusions about the significance of “harmony mathematics”, we find the following:

Thus, the neglect of the “golden section” and its associated idea of mathematical harmony is one more “strategic mistake” in not only mathematics and mathematics education, but also theoretical physics. (p. 625)


We affirm that the Mathematics of Harmony should become a base for the reform of modern mathematical education on the base of the ancient idea of Harmony and golden section. (p. 660)

I’m not convinced that these conclusions are valid, and even less certain that they’ve been adequately justified here. These crankish pronouncements aside, this is a worthwhile collection of elementary and not-so-elementary results.

Mark Bollman ( is associate professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.

  • Classical Golden Mean, Fibonacci Numbers, and Platonic Solids:
    • The Golden Section
    • Fibonacci and Lucas Numbers
    • Regular Polyhedrons
  • Mathematics of Harmony:
    • Generalizations of Fibonacci Numbers and the Golden Mean
    • Hyperbolic Fibonacci and Lucas Functions
    • Fibonacci and Golden Matrices
  • Application in Computer Science:
    • Algorithmic Measurement Theory; Fibonacci Computers
    • Codes of the Golden Proportion
    • Ternary Mirror-Symmetrical Arithmetic
    • A New Coding Theory Based on a Matrix Approach