Math Goes to the Movies

Chapter 5 Nitpicking in Mathmagic Land

Chapter 5 Nitpicking in Mathmagic Land In Donald in Mathmagic Land (1959), we accompany Donald Duck (played by himself) as the True Spirit of Adventure introduces him to the wonders of mathematics. This gem of a movie is to mathematics what Disney’s Fantasia is to classical music. Donald in Mathmagic Land is half documentary, half duck action movie. For decades, Mathmagic Land has been an invaluable asset for teachers seeking to inspire their students. However, much of Donald’s math is unclear, and some is incorrect. Our intention in this chapter is to sort out what is really going on. We hope it will prove a helpful resource for teachers, and parents, with budding mathematicians to inspire. Most of the mathematics presented is similar to that in other introductions to the beautiful side of mathematics: the golden section (or golden ratio), the Pythagorean theory of music, conic sections, infinity, and so on. One notable exception is the diamond rule, a mathematical rule of thumb for playing three-cushion billiards. In the following, we’ll spend less time on the obvious parts of the movie, with which most readers of this book are probably familiar. Instead, we’ll focus upon aspects that are easily overlooked, and we’ll do some nitpicking of mathematical claims that are not quite right (or just plain wrong). In particular, we’ll draw attention to some of the popular myths about the golden section, and we’ll try to make better sense of the diamond rule than is done in the movie. 5.1 Tiny Nitpick: What Is Pi? Mathmagic Land is populated by plants with number-shaped branches and leaves, trees with square roots (!), and other intriguing mathematical creatures . Particularly memorable are the pencil and pi creatures. The pencil creature has a pencil head and a right-triangled body. It writes random digits on the ground and challenges Donald to a game of tic-tac-toe (which of course the pencil creature wins). 71 72 5 Nitpicking in Mathmagic Land The pi creature recites “π is equal to 3.14159265389747 etcetera, etcetera, etcetera.” Surprisingly, he goofs the last two digits: in fact, the decimal expansion of π begins 3.141592653589793 . . . However, the pi creature can at least take comfort in the fact that most other movies featuring a chunk of the decimal expansion of π do even worse; see chapter 18 (“Money-Back Bloopers”). 5.2 Historical Nitpick: Pythagoras and the Pythagoreans To convince Donald that mathematics is “not just for eggheads,” the Spirit whisks Donald off to ancient Greece to meet “Pythagoras, the master egghead of them all.” The background for the first scene is the painting shown in figure 5.1. Fig. 5.1 The Pythagoreans: the theorem, some music, and their pentagram logo. Time for some math spotting! The pentagram at the top was the logo of the Pythagoreans. Underneath, we see what is clearly supposed to be the name “Pythagoras” written in Greek, but there are mistakes: it should be spelled Πνθαγóρας. Then comes the famous diagram that accompanies the first proof of Pythagoras’s theorem in Euclid’s Elements. This raises an important historical point: while watching the charming Pythagoras, one should keep in mind the uncertainty as to which of “his” discoveries should rather be attributed anonymously to the Pythagorean cult, 5.3 Small Nitpick: Pythagorean Music 73 and which have nothing to do with the Pythagoreans whatsoever.1 As a telling example, Pythagoras’s theorem probably belongs in the latter category !2 5.3 Small Nitpick: Pythagorean Music In the painting we also see a number of musical instruments and, in the scenes that follow, the Pythagorean contribution to music theory is sketched. Donald is very impressed that mathematics can be found in music and we witness a musical jam session, with Donald on bongos and Pythagoras on lyre. Then Donald is initiated into the Pythagorean society: Pythagoras stamps a regular pentagram onto Donald’s palm. All great fun! However, this fun also disguises some clumsiness and confusion : SPIRIT: Pythagoras discovered the octave had a ratio of two to one. With simple fractions he got this [a chord is played upon the lyre]. And from this harmony in numbers developed the musical scale of today. Fig. 5.2 Donald demonstrating the Pythagorean theory of music. In figure 5.2 we’ve highlighted the numbers, indicating the string ratios, to which the Spirit refers; the shortest string is...