In this month's issue you'll learn about a geometric method of generating road-wheel pairs. You'll also find out about the golden string (a recursively defined string of letters that is intimately connected to Fibonacci numbers and the golden ratio), sums of subsequences of a sequence and their connection with Cantor sets, and much more. —Dan Velleman, Editor
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Roads and Wheels, Roulettes and Pedals
We revisit the idea of road-wheel pairs, first introduced 50 years ago by Gerson Robison and later popularized by Stan Wagon and his square-wheeled tricycle. We show how to generate such pairs geometrically: the road as a roulette curve and the wheel as a pedal curve. Along the way we gain geometric insight into two theorems proved by Jakob Steiner relating the area and arc length of a roulette to those of a corresponding pedal. Finally, we use our results to generate parabolas, ellipses, and sine curves as roulettes.
The Golden String, Zeckendorf Representations, and the Sum of a Series
In this article we explore some wonderfully intricate relationships between three mathematical objects, each of which is associated in some way with the Fibonacci sequence. One of these objects is a particular finite series comprising n terms, and, via the other two, our mathematical journey culminates in the derivation of an expression for the sum of this series in terms of sums and products of certain Fibonacci numbers.
Achievement Sets of Sequences
Given a real sequence (xn), we examine the set of all sums of the form , as I varies over subsets of the positive integers. We call this the achievement set of (xn), and write it AS(xn). For instance, AS(1/2n) = [0, 1] by the existence of binary expansions, and AS(2/3n) is the Cantor middle-third set. We explore the properties of these two sequences that account for their very different achievement sets. We give a sufficient condition for a sequence to have an achievement set that is an interval, and another sufficient condition for the achievement set to be a Cantor set. We also examine what sets can occur as achievement sets, and give results on the topology of achievement sets.
Polynomials, Ellipses, and Matrices: Two Questions, One Answer
Pamela Gorkin and Elizabeth Skubak
Consider the following questions for points a1, a2 in the unit disc, . If q(z) = (z – a1)(z – a2), when is q the derivative of a polynomial with all of its zeros on the unit circle, δ ? If an ellipse E with foci a1, a2 is inscribed in a triangle with vertices on δ, when is E tangent at the midpoints to a triangle with vertices on δ? We show that these problems are essentially the same. In fact, the answer to both is a very simple: if and only if 2| a1a2| = | a1+a2|. We also discuss generalizations of these problems and their solutions.
A Lost Counterexample and a Problem on Illuminated Polytopes
Ronald F. Wotzlaw and Günter M. Ziegler
Daniel A. Marcus claimed in a “Note added in proof” to his 1984 paper on positively k-spanning vector configurations that some minimal positively k-spanning vector configurations in m-dimensional space have more than 2km elements, but the example he found (for k = 2 and m = 12) seems to be lost.
We produce such an example by applying Gale duality, a linear algebra technique developed by Micha Perles in the sixties, to a result on illuminated polytopes by Peter Mani from 1974. Our example has exactly the parameters claimed by Marcus.
Conversely, we show how results on positively k-spanning vector configurations, again via Gale duality, can be used to solve a problem by Mani on nonsimplicial illuminated polytopes.
Conway’s Conjecture for Monotone Thrackles
János Pach and Ethan Sterling
A drawing of a graph in the plane is called a thrackle if every pair of edges meet precisely once, either at a common vertex or at a proper crossing. According to Conway’s conjecture, every thrackle has at most as many edges as vertices. We prove this conjecture for x-monotone thrackles, that is, in the case when every edge meets every vertical line in at most one point.
Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal’s Identity
Kieren MacMillan and Jonathan Sondow
A well-known and frequently cited congruence for power sums is
where n ≥ 1 and p is prime. We survey the main ingredients in several known proofs. Then we give an elementary proof, using an identity for power sums proven by Pascal in 1654. An application is a simple proof of a congruence for certain sums of binomial coefficients, due to Hermite and Bachmann.
Meromorphic Continuation of Dirichlet Series via Derivations
We show how to use derivations on the ring of Dirichlet series to achieve meromorphic continuations of completely multiplicative Dirichlet series. In particular we present infinitely many new representations of the Riemann zeta function as a quotient of series converging on Re(s) > 0.
PROBLEMS AND SOLUTIONS
Continuous Symmetry: From Euclid to Klein.
William Barker and Roger Howe
Reviewed by Robin Hartshorne
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