Falling Down a Hole through the Earth
Andrew J. Simoson
Drop a magic pebble into the rotating Earth, allowing it to fall, drilling its own hole without resistance. With respect to a stationary reference frame, the pebble's path is an ellipse whose center is the center of the Earth (a result proved by Newton) and whose nearest approach to the Earth's center is over 300 km when the pebble is dropped at the Equator. With respect to the rotating reference frame of the Earth, the pebble's hole is a star-shaped figure, an analysis of which leads naturally into determining the pebble's dynamic distance from straight down and into a discussion of the early pebble-drop experiments of Galileo and Hooke. When hypothetically changing the Earth's gravity field, the resultant paths twist as precessing ellipses and raise some natural questions such as why the Spirograph Nebula looks like a tangle of pebble holes falling through the gravitational field of the dust cloud.
PROOF WITHOUT WORDS
Euler’s Arctangent Identity
Rex H. Wu
The identity arctan(1/x) = arctan(1/(x+y)) + arctan(y/(x(x+y)+1) is one of many elegant arctangent identities discovered by Leonard Euler. He employed them in the computation of π.
Upper Bounds on the Sum of Principal Divisors of an Integer
Roger B. Eggleton and William P. Galvin
By the Fundamental Theorem of Arithmetic, a positive integer is uniquely the product of distinct prime-powers, which we call its principal divisors. For instance, the principal divisors of 90 are 2, 5, and 9. Brian Alspach recently asked for a nice proof that, apart from prime-powers, every odd integer greater than 15 is more than twice the sum of its principal divisors. Thus 21 is more than twice 3 plus 7, while 15 is almost, but not quite, twice 3 plus 5. How can Alspach’s observation be proved? To what extent is it true for even integers? For example, 20 is more than twice 4 plus 5, but 22 is less than twice 2 plus 11. When an integer has more than two principal divisors, are there stronger bounds on their sum? Part of the challenge is to find elegant bounds, the rest is to find elementary proofs.
PROOF WITHOUT WORDS
Every Octagonal Number Is the Difference between Two Squares
Roger B. Nelsen
A proof without words that the nth octagonal number is (2n-1)2-(n-1) 2. NOTES
Centroids Constructed Graphically
Tom M. Apostol and Mamikon A. Mnatsakanian
Two different methods are described for locating the centroid of a finite number of points in 1-space, 2-space, or 3-space by graphical construction, without using coordinates or numerical calculations. The first involves making a guess and forming a closed polygon. The second is an inductive procedure that combines centroids of two disjoint sets to determine the centroid of their union. For 1-space and 2-space, both methods can be applied in practice with drawing instruments or with computer graphic programs.
There Are Only Nine Finite Groups of Fractional Linear Transforms with Integer Coefficients
Gregory P. Dresden
Many of us recall fractional linear transforms (also known as Mobius transforms) from complex analysis, where we learned that these conformal functions map lines and circles to lines and circles (in the complex plane). In this article, we show that under composition, and using only integer coefficients, there are exactly nine distinct finite groups of fractional linear transforms. We establish the identical result for projective integer matrices of dimension two. The proof requires only a small amount of abstract algebra, and is entirely appropriate for upper-division math majors.
The One-dimensional Random Walk Problem in Path
Path representations are very useful in physics and mathematics to transform hard algebraic problems into easier combinatorial ones, and this is true even of combinatorial problems themselves! We put the elementary problem of the random walk of a particle on the real line into a path representation form and use the geometry of the lattice paths to solve for the probability that the particle ends up at the origin given that it starts anywhere.
Why Some Elementary Functions Are Not Rational
Gabriela Chaves and José Carlos Santos
We prove, in an elementary way, that the restrictions to an open interval of certain elementary functions, such as the sine function or the exponential function, are not rational functions, and we do it by using the concept of degree of a rational function.
Another Look at Sylow's Third Theorem
Sylow's third theorem tells us that the number of p-Sylow subgroups in a finite group is congruent to 1 modulo p, and, in particular, there must exist p-Sylow subgroups of the group. But, to obtain Sylow's third theorem, it has been necessary to first show the existence of a p-Sylow subgroup. In this note we show how Mobius inversion can be used to obtain a generalized Sylow third theorem directly.