A Fractal Made of Golden Sets
by Marc Frantz
Most discussions of the golden rectangle include a picture of it tiled by an infinite spiral of squares — a figure which also contains infinitely many golden rectangles. We refer this figure as the "golden set." While the golden set is often referred to as being self-similar, it falls short of this description, being the union of a smaller copy of itself and a square. To put things aright, we construct a self-similar fractal that is the closure of a countable union of golden sets—and therefore golden rectangles —in a way that makes the presence of these sets pleasingly obvious. In the process, we discover good exercises for students on conjecturing and proving whether a set is a subset of a given fractal.
A Survey of Euler’s Constant
by Thomas P. Dence and Joseph B. Dence
The mathematical constant, denoted by γ, and referred to as Euler’s constant, with value approximately 0.5772156, is not nearly as well known as the constants π and e, but is still important enough to warrant serious discussion. We present here a variety of instances in mathematics where γ occurs, and also a number of different expressions (typically involving either an integral, a series, or a limit) that all represent this fascinating number. An extensive reference serves to provide interested readers with alternate sources.
Keeping Dry: The Mathematics of Running in the Rain
by Dan Kalman and Bruce Torrence
The age-old question of whether one stays driest by walking or running in the rain is addressed. In the previous literature the traveler has always been modeled by a rectangular solid. Here we consider the consequences for more well-rounded individuals, in particular, for ellipsoids. We find that indeed, shape matters, and reach conclusions that differ from those utilizing rectangles. Indeed, we find that in the presence of a tail wind, the optimal pace for the ellipsoidal traveler is always slightly faster than the tail wind.
Starting with Two Matrices
by Gilbert Strang
Linear algebra begins with vectors, and their combinations, and the matrix that produces those combinations. We offer two examples that illustrate the central ideas (independent or dependent vectors, invertible, or singular matrix) before any general theory is attempted. We believe that students will learn to use this language by working with specific examples like these two.
To Buy or Not to Buy: The Screamin’ Demon Ticket Game
by Frederick Chen and Alexander Kays
Your school's football team is expected to do well in the upcoming season, and your school has implemented the following system for allocating football tickets to students. There is a fixed number of tickets available for each game. For a moderate sum of money, students can purchase season tickets. Alternatively, students can enter an online drawing every time there is a home game to try to win free tickets, where the number of tickets available for the drawing is equal to the total number of tickets available minus the number of tickets sold. Should you buy season tickets or not?
What Do You Get When You Cross a Power Sum with an Iraqi Bank Note?
by Gregory M. Boudreaux
After using the limit definition to evaluate the definite integral of a polynomial function, the calculus student is more apt to appreciate the power of the Fundamental Theorem of Calculus. However, this algebraically demanding evaluation depends on knowing the closed form of the sum of the k-th powers of the first n positive integers. A new treatment of a millennium old result by ibn al Haytham, who is featured on the new Iraqi 10,000 dinar bank note, yields arguably the best method of deriving these formulas using an intuitive and generalizable approach.
Q Is Not Complete
In this note we introduce a series of rational numbers that converge absolutely to a rational number and it converges to an irrational number. We use such series to prove that the rational numbers are not complete.
A Low-level Proof of Chebyshev's Pre-Prime Number Theorem
by M. Scott Osborne
It is not uncommon, in advanced calculus, to use the Euler product for the Riemann zeta function to establish the divergence of the sum of the reciprocals of all the primes. This paper takes a closer look at the character of that divergence, also using standard techniques from advanced calculus. The conclusion, originally observed by Chebyshev in 1852, is that the prime distribution given by the Prime Number Theorem is the most viable possibility.
Tile in a Corner
by Richard P. Jerrard and John E. Wetzel
When a rigid tile moves in the first octant with its corners on the coordinate planes (and at least one corner on each), its center traces out a surface whose shape depends on the dimensions of the tile. We investigate this surface by studying certain of its sections, and we illustrate the possibilities with drawings.