# Mathematics Magazine - June 2003

### ARTICLES

The Fibonacci Numbers--Exposed
Dan Kalman and Robert Mena
The Fibonacci numbers are a popular topic for mathematical enrichment and popularization. They are famous for a host of interesting and surprising properties, and show up in text books, magazine articles, and web sites. From all of this attention, it would be easy to infer that they are a unique and singular phenomenon. Nothing could be further from the truth. Most if not all of the properties associated with the Fibonacci numbers are shared by a large class of number sequences--described by second order linear recurrences. This was known to Lucas and his nineteenth century contemporaries who discovered and described much of what we know about Fibonacci numbers today. But it seems to have been forgotten in the popular consciousness. This article reviews a sample of interesting Fibonacci properties, and shows that they arise simply and naturally for much more general classes of sequences.

The Fibonacci Numbers--Exposed More Discretely
Arthur T. Benjamin and Jennifer J. Quinn
Many identities for generalized Fibonacci numbers (or Lucas sequences) defined by the recurrence Fn = aFn-1 + bFn-2 can be proved by combinatorial methods. The paper was written to complement (and compliment) the paper "The Fibonacci Numbers Exposed!" by Dan Kalman and Robert Mena.

Gaspard Monge and the Monge Point of the Tetrahedron
Robert Alan Crabbs
We open this paper with some biographical background on Monge. In addition to noting the areas in which he made mathematical contributions (he made important contributions in three different branches), we describe how he got started as a mathematician, explain his importance as an educator, and discuss his close friendship with Napoleon Bonaparte. We then focus on a particular discovery of Monge: the Monge point of the tetrahedron. After defining the Monge point and proving it exits, we explore its relationship to other important points in the tetrahedron. In doing so we describe the contents of 1809 and 1811 papers by Monge and present some proofs Monge himself gave. Toward the end of the paper we go a bit beyond the work of Monge, considering the Monge point in a special-case tetrahedron and concluding with a theorem by Mannheim.

### MATH BITE

A Novel Proof of the Infinitude of Primes Revisited
Daniel Cass and Gerald Wildenberg
An old but little known proof of the infinitude of the primes is rephrased avoiding the topological terminology of the original.

### NOTES

Periodic Points of the Open-Tent Function
Danrun Huang and Daniel Scully
One of the simplest continuous functions with a point of period 3 is the open-tent function on the unit interval. According to Li-Yorke's famous paper "period three implies chaos," this function has points of all periods. When a curious student tries to iterate this function over and over again, many interesting nontrivial questions may arise. What numbers return home after n iterations? How many of them are there? Can we locate all of them? Even more peculiar questions could pop out such as, can we find a number that stays in the right-half interval for every single iteration except the one millionth? In this note, we answer all these questions and get a lot of other orbital information about this function. The idea is to digitally encode each number in the unit interval with a special infinite sequence of 0s and 1s that is different from its binary expansion, process the orbital information in terms of sequences, and then decode (translate) the digital information back for the open-tent function. Our approach is elementary and connects many "cool" topics in undergraduate mathematics like the golden mean, Fibonacci and Lucas numbers, directed graphs, matrices, binary expansions, coding, chaos, and others all for this simple function.

Fibonacci Numbers and the Arctangent Function
Ko Hayashi
This note provides several geometric illustrations of three identities linking the Fibonacci numbers to the arctangent function and invites interested readers to prove the identities using Cassini's identity.

Hypercubes and Pascal's Triangle: A Tale of Two Proofs
Michael L. Gargano, Marty Lewinter, and Joseph F. Malerba
The authors employ graph theory to prove the well-known combinatorial identity n2n-1 = C(n,1) + 2C(n,2) + 3C(n,3) + Â… + nC(n,n) by counting the number of edges of the hypercube, Qn, in two different ways. Traditional proofs involve calculus (it is included) and combinatorics. The authors' approach sheds light on why the identity is true, whereas other proofs may create the impression that it is merely an algebraic accident.

Two Simple Derivations of Taylor's Formula with Integral Remainder
Dimitri Kountourogiannis and Paul Loya
In this note we give a derivation of Taylor's formula with remainder based on interchanging integrals that avoids any unintuitive integration by parts substitutions.

A Theorem Involving the Denominators of Bernoulli Numbers
Pantelis Damianou and Peter Schumer
For what values of n and k are the average of the first n kth powers an integer? This paper completely answers that question by handling two cases. If k is odd, then the average of the first n kth powers is an integer if and only if n is incongruent to 2 modulo 4. If k is even, then the average of the first n kth powers is an integer if and only if n is not divisible by any prime p that divides the denominator of the n kth Bernoulli number. Equivalently, for k even, the average of the first n kth powers is an integer if and only if for every prime p that divides n, p - 1 does not divide k.

### PROOFS WITHOUT WORDS

Viviani's Theorem with Vectors
Hans Samelson
The proof is suggested by the fact that at a point at minimum total distance from three given points the three lines meet at 120 degrees.

Let Pi be 3.
Robert N. Andersen, Justin Stumpf, and Julie Tiller
According to folklore the Bible says that Π is equal to 3. One verse that suggests this value is First Kings, chapter 7, verse 23. Instead of discounting this value as a coarse approximation, we explore the possible values of the ratio of circumferences of circles to their diameters in noneuclidean geometries to see what conditions will suffice to make the Biblical ratio precise.