Gnomon: From Pharaohs to Fractals

In the words of the author "The theme of this book is self-similarity, which I call gnomonicity. ... The connotation of the term gnomon is that originally given by Hero of Alexandria, namely, 'A Gnomon is that form that, when added to some form, results in a new form similar to the original.'"

The most familiar form of a gnomon is the L-shaped object of that name (the gnomon of a square) that serves as the pointer on most sundials. This observation virtually completes the "Pharaohs" end of the story mentioned in the title.

Figurate numbers have familiar gnomons. The gnomon of the n-th triangular number, in its familiar rows-of-dots manifestation, is a row of n+1 dots. Other figures have analogous gnomons.

Continued fractions have obvious gnomonic tendencies, especially those with periodic regular representations. These lead naturally into physics, where ladder networks of resistors or capacitors are gnomonic, and their analysis leads back through continued fractions.

Most of us have seen the gnomonic properties relating the golden rectangle, the golden ratio and the Fibonacci numbers. In particular, the golden ratio, called phi in this book, is the limit of the ratios of consecutive Fibonacci numbers. If you construct a rectangle of sides 1 and phi, then the resulting rectangle is gnomonic and its gnomon is a square. Fewer of us know about what Gazalé calls the Silver Pentagon, a pentagon of sides 1, p, p², p³ and p⁴, where p is a root of p³ - p - 1 = 0. This pentagon is also gnomonic, and its gnomon is an equilateral triangle.

After these topics, Gazalé spends what seems to be an inordinate amount of time on spirals, especially logarithmic spirals. Bernoulli loved them, too, but the exposition has more rotation matrices than most readers will enjoy.

Finally, we get to the other end of the programme announced in the title, fractals. Gazalé takes a route to fractals that begins with the Kronecker product of matrices. He adapts this to form a Kronecker sum of vectors, and then adapts that to give sequences that can be read as instructions on drawing images of fractals. It is an interesting approach, but again at times he lets the ideas be hidden by the notation.

In sum, this reader found the book generally interesting, though at times over-burdened with notation. There is a considerable variety of topics here, all linked by the unifying concept of gnomonicity.

Ed Sandifer (sandifer@wcsu.ctstateu.edu) is professor of mathematics at Western Connecticut State University and has run the Boston Marathon 27 times.