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American Mathematical Monthly -June/July 2009

June/July 2009

Musical Actions of Dihedral Groups
By: Alissa S. Crans, Thomas M. Fiore, and Ramon Satyendra,,
The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor triads. We illustrate both geometrically and algebraically how these two actions are dual. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.


Cutting a Pie Is Not a Piece of Cake
By: Julius B. Barbanel, Steven J. Brams, and Walter Stromquist,,
Is there a division among n players of a cake using n-1 parallel vertical cuts, or of a pie using n radial cuts, that is envy-free (each player thinks he or she receives a largest piece and so does not envy another player) and undominated (there is no other allocation as good for all players and better for at least one)? David Gale first asked this question for pies. We provide complete answers for both cakes and pies. The answers depend on the number of players (two versus three or more players) and whether the players' preferences satisfy certain continuity assumptions. We also give some simple algorithms for cutting a pie when there are two or more players, but these algorithms do not guarantee all the properties one might desire in a division. We conclude with two open questions.


Gaussian Integers and Arctangent Identities for Pi
By: Jack S. Calcut
John Machin, with his accurate and ready pen, first computed 100 decimal digits of pi in 1706 using his celebrated arctangent identity. Several others, including Newton, Euler, and Gauss, studied arctangent identities, and their utility was further displayed by the calculation of 100,000 digits of pi by Shanks and Wrench in 1961. In fact, the current record of more than 1.2 trillion digits of pi was computed by Kanada in 2002 using such identities. The existence of these identities is intimately related to equations over the Gaussian integers. One question, namely the existence of single-angle identities, has several applications to geometry. For instance, this leads to a very simple and intuitive proof that if the side lengths of a triangle form a Pythagorean triple, then its acute angles have irrational degree measures. Several other applications are presented as well. This article is self contained and includes an account of the relevant history for further reading.



Modifications of Thomae's Function and Differentiability
By: Kevin Beanland, James W. Roberts, and Craig Stevenson,,
In 1875, K. J. Thomae discovered the now-famous example of a real-valued function that is continuous on the irrationals and not continuous on the rationals. This function is presented in many undergraduate real analysis courses and is known, to some, as the 'popcorn' function. It is not difficult to see that Thomae's function is not differentiable anywhere. The question we examine here is whether it can be modified to be differentiable on some subset of the irrationals. The answer may surprise you—it certainly surprised us.

Over-Iterates of Bernstein's Operators - A Short and Elementary Proof
By: Ulrich Abel and Mircea Ivan,
In this note, we provide a short and elementary proof of the result by Kelisky and Rivlin (Pacific J. Math. 21 (1967) 511-520) concerning the iterates of the Bernstein operators.

Evaluating the Probability Integral Using Wallis's Product Formula for π
By: Paul Levrie and Walter Daems
... or how the square root of Wallis's product formula for π is the probability integral in disguise. To prove this only basic one-variable calculus is used.

Counting Fields of Complex Numbers
By: Gerald Kuba
We determine the (transfinite) maximal number of mutually non-isomorphic subfields K of the field C of complex numbers that have, respectively, one of the following properties: (1) K is arbitrary; (2) K is a subfield of the field R of all real numbers; (3) K contains only algebraic numbers; (4) K contains only real algebraic numbers; (5) K(i) = C; or (6) K is a proper subfield of R with countable codimension. We also determine the total number of all subfields of C that are isomorphic to C and R, respectively. Furthermore, we give two easy examples of a field of characteristic 0 such that its subfield lattice is a chain of infinite length containing only mutually non-isomorphic fields.


The Symmetries of Things.
By: John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss
Reviewed by: Branko Grünbaum