November 2012 Contents
In the November issue, Mark Frantz gives us A Different Angle on Perspective, Timothy Jones proves the irrationality of π, David Seppala-Holtzman finds An Optimal Basketball Free Throw, and Tom Brown and Brian Pasko find the probability of Winning a Racketball Match. There are also four Proofs Without Words, two Classroom Capsules, a Student Research Project, Media Highlights and Problems and Solutions.
A Different Angle on Perspective
When a plane figure is photographed from different viewpoints, lengths and angles appear distorted. Hence it is often assumed that lengths, angles, protractors, and compasses have no place in projective geometry. Here we describe a sense in which certain angles are preserved by projective transformations. These angles can be constructed with compass and straightedge on existing projective images, giving insights into photography and perspective drawing.
Euler’s Identity, Leibniz Tables and the Irrationality of Pi
Timothy W. Jones
Using techniques that show that e and Π are transcendental, we give a short, elementary proof that
Sometimes Newton’s Method Always Cycles
Joe Latulippe and Jennifer Switkes
Are there functions for which Newton’s method cycles for all non-trivial initial guesses? We construct and solve a differential equation whose solution is a real-valued function that two-cycles under Newton iteration. Higher-order cycles of Newton’s method iterates are explored in the complex plane using complex powers of x. We find a class of complex powers that cycle for all non-trivial initial guesses and present the results analytically and graphically.
Proof Without Words: An Alternating series
A visual proof that 1 – (1/2) + (1/4) – (1/8) + . . . converges to 2/3.
The numerical range of the Luoshu is a piece of cake—almost
Dietrich Trenkler and Götz Trenkler
The numerical range, easy to understand but often tedious to compute, provides useful information about a matrix. Here we describe the numerical range of a 3 x 3 magic square, in particular, one of the most famous of those squares, the Luoshu: 4 9 2/3 5 7/8 1 6, whose numerical range is a piece of cake—almost.
Proof Without Words: The Sine is Subadditive on [0, Π ]
A visual proof that the sine is subadditive on [0,Π].
A Fifth Way to Skin a Definite Integral
We use a novel approach to evaluate the indefinite integral of 1/(1+x4) and use this to evaluate the improper integral of this integrand from 0 to ∞. Our method has advantages over other methods in ease of implementation and accessibility.
Better than Optimal by Taking a Limit?
Designing an optimal Norman window is a standard calculus exercise. How much more difficult is its generalization to deploying multiple semicircles along the head (or along head and sill, or head and jambs)? What if we use shapes beside semi-circles? As the number of copies of the shape increases and the optimal Norman windows approach a rectangle, what proportions arise? How does the perimeter of the limiting rectangle compare to the limit of the perimeters? These questions provide challenging optimization problems for students and the graphical depiction of these window sequences illustrates the concept of limit more vividly than sequences of numbers.
Proof Without Words: Ptolemy’s Theorem
William Derrick and James Hirstein
A visual proof of Ptolemy’s theorem.
An Optimal Basketball Free Throw
A basketball player attempting a free throw has two parameters under his or her control: the angle of elevation and the force with which the ball is thrown. We compute upper and lower bounds for the initial velocity for suitable values of the angle of elevation, generating a subset of the configuration space of all successful free throws. A computer-assisted search of this configuration space yields a free throw shot most forgiving of error hence optimal.
Winning a Racketball Match
Tom Brown and Brian Pasko
We find the probability of winning a best-of-three racquetball match given the probabilities that each player wins a point while serving.
STUDENT RESEARCH PROJECT
Idempotents à la Mod
Thomas Q. Sibley
An idempotent satisfies the equation x2 = x. In ordinary arithmetic, this is so easy to solve it’s boring. We delight the mathematical palette here, topping idempotents off with modular arithmetic and a series of exercises determining for which n there are more than two idempotents (mod n) and exactly how many there are.
Rational Exponentials and Continued Fractions
Using continued fraction expansions, we can approximate constants, such as Π and e, with 2 raised to the power x1/x, x a suitable rational. We review continued fractions and an give an algorithm for producing these approximations.
Geometry of Sum-Difference Numbers
We relate the factorization of an integer N in two ways as N = xy = wz with x + y = w - z to the inscribed and escribed circles of a Pythagorean triangle.
PROBLEMS AND SOLUTIONS
REFEREES IN 2012
ADDITIONS, CORRECTIONS, EMENDATIONS, AND REVISIONS
GEORGE PÓLYA AWARDS FOR 2011