Stan Wagon

Rhombic Penrose tilings can be 3-colored

The four-color theorem tells us that any tiling in the plane can be 4-colored, where adjacent tiles are to get different colors. Anyone who tries to color a Penrose tiling will wonder whether they are all 3-colorable; this question was first proposed by John H. Conway [2, p. 27]. We show, by a very simple argument, that any tiling by Penrose rhombs is 3-colorable; our method yields an algorithm, the results of which are illustrated in Figure 1. We were led to this result by a Mathematica implementation of a 4-coloring algorithm for plane maps based on Kempe chains; see [5] and [7]. It had no difficulty in 3-coloring rhombus tilings. For background on Penrose tilings see [2], [4], and [8]. Note that there is no simple characterization of the 3-colorable planar maps, and the problem of recognizing 3-colorable planar maps is NP-complete [3].

Roads and Wheels

San Franciscos Exploratorium contains an intriguing exhibit of a square wheel that rolls smoothly on a road made up of linked, inverted catenaries (see FIGURE 4). That exhibit inspired us to generate a computer animation of a rolling square and further explore the relationship between the shapes of wheels and roads on which they roll. In a sense, we are bringing up to date the paper by G. Robison [4], showing how much more can be done, both numerically and graphically, with modern computer hardware and software. The problem of the square wheel has been rediscovered and solved several times; see [5, 7]. All the diagrams and animations were prepared in Mathematica. Our package that generated the diagrams and the associated animations (see Section 5) can be obtained by sending a Macintosh disk to one of the authors. It is noteworthy that some of the results of this paper, in particular the discovery of a cycloidal locus generated by a noncircular wheel, were discovered only after viewing certain graphics. Mathematica was also used to do all the symbolic integrations that occur. For further applications of symbolic and graphic computation to wheel/road problems, in particular, a complete discussion of the cycloid, see [6, Chapter 2]. The paper is organized as follows. Section 1 discusses the theory and the fundamental differential equation. Section 2 contains many closed-form examples. Section 3 shows how numerically approximating the solution to the differential equation is an excellent approach to diverse examples, even those solvable in closed form. Section 4 squares the circle by considering Fourier approximations to the catenary. And Section 5 discusses the Mathematica package that we built.

How to pick out the integers in the rationals: an application of number theory to logic

STAN WAGON received his Ph.D. from Dartmouth College under the direction of James E. Baumgartner and taught at Smith College from 1975 to 1990, with visits to Carleton College, the University of Colorado, the University of California at Berkeley, and the University of Washington. His introduction to the area of Diophantine definitions came in a memorable course taught by Julia Robinson at the University of California, Berkeley, in 1975.

Venn Symmetry and Prime Numbers: A Seductive Proof Revisited

An n-Venn diagram is a Venn diagram on n sets, which is defined to be a collection of n simple closed curves (Jordan curves) C1,C2, . . . ,Cn in the plane such that any two intersect in finitely many points and each of the 2n sets of the form ∩C i i is nonempty and connected, where i is one of “interior” or “exterior.” Thus the Venn regions are all bounded except for the region exterior to all curves; each bounded region is the interior of a Jordan curve. See [6] for much more information on Venn diagrams. An n-Venn diagram is symmetric if each curve Ci is ρ i (C1), where ρ is a rotation of order n about some center (we use O for the fixed point of rotation ρ). We use Boolean notation for combinations of sets, with the 0-1 string e1e2 . . . en representing ∩C i i , where i is interior (respectively, exterior) if ei = 1 (respectively, 0). Thus 111 . . . 1 represents F , the full intersection of all the interiors, 000 . . . 0 is the intersection of all the exteriors (the unbounded region), and 100 . . . 0 represents the set of points interior to C1 and exterior to the others. In a symmetric Venn diagram, rotation of a region by ρ corresponds to a rightward cyclic shift of the Boolean string. The universally familiar three-circle Venn diagram is symmetric, as is the one on two sets using two circles. For about 40 years a major open question was whether symmetric n-Venn diagrams exist for all prime n. Henderson found one for n = 5 and also (unpublished) for n = 7. Much later, Hamburger [3] settled the case of 11, which was quite complicated, and then in 2004 Griggs, Killian, and Savage [1] found an approach that works for all primes. So we now have the strikingly beautiful theorem that a symmetric n-Venn diagram exists if and only if n is prime. But there is a small problem: Henderson’s proof, which appears to be very simple, has a gap. Here is the proof from [4]. Suppose 1 ≤ k ≤ n − 1. Since a symmetric n-Venn diagram is symmetric with respect to a rotation of 2π/n, the regions corresponding to the Boolean strings with k 1s must come in groups of size n, each group consisting of one such region and its images under repeated rotation by 2π/n. Therefore n divides (n k ) . This concludes the proof because the only n for which this is true for the specified k-values are the primes (an easy-to-prove fact of number theory; see [5]). This is a very seductive argument. The primeness arises in such a cute way that one wants it to be true. Thus the proof has been repeated in many papers in the decades since it was first published. Yet there are problems. The proof does not call upon the connectedness of the Venn regions. Without connectedness the result is false; see Figure 1 (due to Grunbaum [2]), which shows a diagram satisfying all of the conditions

Mechanical Circle-Squaring

Barry Cox (barryc@uow.edu.au; www.uow.edu.au/-barryc) is a lecturer in applied mathematics at the University of Wollongong (Australia) and holds a postdoctoral fellowship from the Australian Research Council. The primary focus of his research is modelling nanoscaled devices. He became interested in the problem of drilling polygonal holes through the Macalester Problem of the Week, to which he has subscribed for many years. Barrys personal interests include recreational mathematics, public speaking, tennis, and chess.

The Gaussian Zoo

We find all the maximal admissible connected setsof Gaussian primes: there are 52 of them. Our catalog corrects some errors in the literature. We also describe a totally automated procedure to determine the heuristic estimates for how often various patterns, in either the integers or Gaussian integers, occur in the primes. This heuristic requires a generalization of a classical formula of Mertens to the Gaussian integers, which we derive from a formula of Uchiyama regarding an Euler product that involves only primes congruent to 1 (mod 4).

The Ultimate Flat Tire

The Problem Suppose a bicycle has square wheels. How can a roadbed be designed so that the ride is smooth, in the sense that the seat of the bike stays horizontal? History The problem was first considered and solved by G. B. Robison in 1960. In 1992 Leon Hall and Stan Wagon investigated other possibilities for shapes of wheels. The first square-wheel bike at Macalester was installed in 1998 and was featured in many television and newspaper pieces, including Ripley’s Believe It Or Not. The new, improved model was installed in 2004. What Is It Good For? Round pieces of wood (quarter-circles) have been found near the sites of Egyptian pyramids and it has been suggested that they were used as a road bed in roughly the shape as the one here, so that large blocks of stone could be moved efficiently. The Answer Each arch of the road is an inverted catenary, the curve one gets by just dropping a piece of chain or rope while holding its ends. Derivation of the Solution Intending to build a model of our solution (using a tricycle to improve stability), we first noted that a commercially available tricycle could be modified slightly so that the distance between its front and rear axles was exactly 42 inches. The roadbed would be a series of arches, and we decided to have three arches span the wheelbase, so that each arch would have a horizontal span of 14 inches.

Repeatedly Appending Any Digit to Generate Composite Numbers

Abstract We investigate the problem of finding integers k such that appending any number of copies of the base-ten digit d to k yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits such that repeatedly appending any digit yields a composite number.

Drilling for Polygons

Abstract We solve the problem of designing a simple device that uses rotary motion to drill a hole with a cross-section that is a regular polygon with an odd number of sides: the main idea is to use a polygonal trammel and a family of rotors. By using different rotors, one can produce a hole with a cross-section that is in any proportion of the trammel size from zero to exactly one. The key geometric idea is a result about the envelope of an edge of a triangle that rotates so that the other two edges maintain tangential contact with two fixed circles.

A Heuristic for the Prime Number Theorem

ConclusionMight there be a chance of proving in a simple way thatx/π(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linear functionL(x) is increasing becausex/π(x) → ∞ asx → ∞. Moreover, using PNT, we can give a proof thatL(x) is indeed asymptotic tox/π(x). But the point of our work in this article is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem.