Curves for the Mathematically Curious by Julian Havil
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t is often fun to reflect on the top ten items of a favorite thing. For example, in the hit song ‘‘My Favorite Things,’’ from the 1959 Broadway musical The Sound of Music, by Richard Rogers and Oscar Hammerstein II, the top ten items (out of fourteen altogether) range from (1) raindrops on roses and (2) whiskers on kittens to (10) schnitzel with noodles. As a second example, who were the top ten people to shape world history and culture from the last millennium? One answer was given in the book 1000 Years, 1000 People [1], which ranked the top 1000 people from year 1000 to 2000; the top ten of mathematical note, along with their ranking in this list of 1000 names, are Galileo (4), Newton (6), Leonardo da Vinci (9), Einstein (17), Copernicus (18), Descartes (25), Kepler (33), Kant (40), Francis Bacon (84), and Leibniz (88). As can be surmised, such rankings may be somewhat arbitrary. Julian Havil’s book is no exception. His book highlights, in great detail, ten curves from the early Greeks to our time that are unpredictable, historical, beautiful, and romantic. Just like David Perkins’s book on the top four complex numbers [2], which showcases p, e, the golden mean /, and the imaginary number i but omits 0 and 1 because of their familiarity, Havil likewise skips lines and circles. Curves for the Mathematically Curious is not for the casual reader. Each of Havil’s curves when fully revealed to the mathematical community was a veritable bombshell, shaking and reformulating mathematical foundations. As Havil points out, each of these curve constructions ‘‘was meant to embrace rigour, and rigour is what we must embrace.’’ Besides a lively historical narrative of people and events, Havil gives line after line, page after page, of derivation for the algebraic representation of each curve. Figure 1 is a collage of Havil’s curves. The background curve within the square labeled ABCD is an early iteration toward David Hilbert’s space-filling curve, generated recursively, wherein each successive iteration increases overall curve length and reduces the freedom of motion of, say, a two-dimensional ant crawling in the spaces between the continuous curve, so that ultimately the ant, no matter how tiny, gets crushed. The roots of this elegant construction go back to Georg Cantor, who had dreamt of wild one-dimensional continuous curves that wound themselves completely onto two-dimensional surfaces—somewhat like how immensely long strands of DNA wind themselves up so as to fit within tiny cells. Fortunately, as Havil points out,
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Cantor had a seasoned mentor in Richard Dedekind, who listened and responded patiently to such nightmares, and affirmed that Cantor’s counterintuitive examples were sound. The curve in Figure 1 looking like a lightning bolt, labeled W, is the Weierstrass curve, an object that had been once thought not to exist. Given a continuous function of one variable, just how bad could its derivative be? Surprisingly, its derivative might not exist anywhere! This particular curve is the graph of 1 n X 1 cosð3n px
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