Jean Pedersen

Catalan Numbers, Their Generalization, and Their Uses

Probably the most prominent among the special integers that arise in combinatorial contexts are the binomial coefficients (~). These have many uses and, often, fascinating interpretations [9]. We would like to stress one particular interpretation in terms of paths on the integral lattice in the coordinate plane, and discuss the celebrated ballot problem using this interpretation. A path is a sequence of points Po P1 9 9 9 Pro, m >I O, where each P, is a lattice point (that is, a point with integer coordinates) and Pz+l, i 1> 0, is obtained by stepping one unit east or one unit north of P,. We say that this is a path from P to Q if Po = P, Pm= Q. It is now easy to count the number of paths.

Geometry Turned On: Dynamic Software in Learning, Teaching and Research

Part I. Personal Reflections On Investigation, Discovery, and Proof: 1. Discovery and dissection of a geometric gem Douglas R. Hofstadter 2. The role and function of proof in dynamic geometry: some personal reflections Michael de Villiers 3. Dynamic geometry renews interest in an old problem Dan Bennett 4. Dynamic geometry as a bridge from Euclidean geometry to analysis Albert A. Cuoco and E. Paul Goldenberg Part II. Dynamic Geometry In The Classroom: 5. Dynamic visualization from middle school through College James Morrow 6. Geometers sketchpad in the classroom Tim Garry 7. Students discovering geometry using dynamic geometry software Michael Keyton 8. Moving triangles 9. Experiences with geometers sketchpad in the classroom Kathryn W. Boehm 10. Beyond elementary constructions: selected exercises Arnold Perham and Bernadette H. Perham 11. Interactive generation, manipulation, and application of loci Heinz Schumann 12. Calculus with dynamic geometry Catherine A. Gorini 13. Beginning geometry at college Tony Hampson 14. Identifying transformations by their orbits James M. Parks 15. Dynamic proofs that use similarities James King 16. Visualization of group theory concepts through dynamic geometry Doris Schattschneider 17. Using the geometers sketchpad with preservice teachers Zhonghong Jian and Edwin McClintock 18. Fish in the pond: inquiry with dynamic geometry Fadia Harik Part III. Dynamic Visualization In History, Perception, Optics and Aerodynamics: 19. Drawing logarithmic curves with geometers sketchpad: a method inspired by historical sources David Dennis and Jere Confrey 20. Lost in the funhouse: an application of dynamic projective geometry Susan Addington and Stuart Levy 21. The use of dynamic geometry software in teaching and research in optometry and vision science Benjamin T. Backus 22. Creating airfoils from circles: the Joukowski transformation John Olive Part IV. The Worlds Of Dynamic Geometry: Issues In Design And Use: 23. Drawing worlds: scripted exploration environments in the geometers sketchpad R. Nicholas Jackiw 24. Dynamic geometry and declarative geometric programming Richard J. Allen and Laurent Trilling.

EXTENDING THE BINOMIAL COEFFICIENTS TO PRESERVE SYMMETRY AND PATTERN

Abstract We show how to extend the domain of thee binomial coefficients (rn) so that n and r may take any integer value. We argue from two directions; on the one hand we wish to preserve symmetry and pattern within Pascals triangle (thus, creating Pascals hexagon), and on the other hand we wish the binomial coefficients to preserve their algebraic role in terms of the Taylor series and Laurent series expansions of (1 + x)n, valid when |x| 1, respectively. A geometric configuration within the Pascal triangle, called the Pascal flower, has some extraordinary properties—these properties persist into the hexagon. Moreover, the binomial coefficients may, by the use of the Γ-function, even be extended to all real (or complex) values of n and r, with the conservation of their principal properties.

Euler's Theorem for Polyhedra: A Topologist and Geometer Respond

In their stimulating paper [1], to which we here make a friendly and constructive response the authors, Branko Grunbaum and Geoffrey Shephard, introduce the interesting geometric concept of polyhedral set, generalizing the familiar notion of polyhedron, but confining themselves, for their present purposes, to subsets of 1R3. They discuss dissections of such sets, especially relatively open convex dissections of bounded polyhedral sets and show, by easily accessible arguments, the nice properties of the Euler characteristic X relative to such dissections. They are thereby led to a formula for X(P), namely, Theorem 4 of [1],

Some Isonemal Fabrics on Polyhedral Surfaces

The motivation for the mathematics presented here should really be viewed as originating with the practitioners of the weaver’s craft. The catalyst that resulted in this particular effort, however, was some recent work of Branko Grunbaum and G. C. Shephard [4,5]. They have carefully analyzed certain geometric objects which represent an idealization of woven fabrics in the plane and their investigations lead, among other things, to remarkable theorems concerning the number and nature of the different kinds of what they call “isonemal”1 fabrics in the plane. They have posed many open problems. The models described and pictured here (see Plates A-E, following page 120) were the result of my investigating one of their problems. The resulting models were a joy to discover and are truly beautiful to behold, but as so frequently happens in mathematics, as the existence of the answer to the original question was unveiled other similar questions seemed to spring forth. And herein lies the major difficulty involved with presenting such embryonic material. It is tempting (and, of course, desirable in the long run) to attack the problem with a great deal of mathematical rigor and preciseness (a) because it will certainly yield to that kind of discussion and (b) because there are beautiful and psychologically satisfying results. I will choose not to do that here because I believe that it is beneficial for the reader to observe first some of the natural beauty and surprise that is felt when viewing these models for the first time (unencumbered by technical detail). My second reason is that I wish, right now, to write an article—not a book.

On the complementary factor in a new congruence algorithm

In an earlier paper the authors described an algorithm for determining the quasi-order, Q t ( b ) , of t mod b , where t and b are mutually prime. Here Q t ( b ) is the smallest positive integer n such that t n = ± 1 mod b , and the algorithm determined the sign ( − 1 )  ϵ  ,  ϵ  = 0 , 1 , on the right of the congruence. In this sequel we determine the complementary factor F such that t n − ( − 1 )  ϵ  = b F , using the algorithm rather that b itself. Thus the algorithm yields, from knowledge of b and t , a rectangular array a 1 a 2 … a r k 1 k 2 … k r  ϵ  1  ϵ  2 …  ϵ  r q 1 q 2 … q r The second and third rows of this array determine Q t ( b ) and  ϵ  ; and the last 3 rows of the array determine F . If the first row of the array is multiplied by F , we obtain a canonical array, which also depends only on the last 3 rows of the given array; and we study its arithmetical properties.

The general quasi-order algorithm in number theory

This paper deals with a generalization of the Binary Quasi-Order Theorem. This generalization involves a more complicated algorithm than (0.2)t. Some remarks are made on relative merits of two dual algorithms called the ψ-algorithm and the ϕ-algorithm. Some illustrative examples are given.

Fibonacci and Lucas Numbers

Consider the following number trick–try it out on your friends. You ask them to write down the numbers from 0 to 9. Against 0 and 1 they write any two numbers (we suggest two fairly small positive integers just to avoid tedious arithmetic, but all participants should write the same pair of numbers). Then against 2 they write the sum of the entries against 0 and 1; against 3 they write the sum of the entries against 1 and 2; and so on. Once they have completed the process, producing entries against each number from 0 to 9, you suggest that, as a check, they call out the entry against the number 6. Thus their table (which, of course, you do not see) might look like the table in the margin. You now ask them to add all the entries in the second column, while you write 341 quickly on a slip of paper.

SYMMETRY IN MATHEMATICS

Abstract The role of symmetry in geometry in universally recognized. The principal purpose of this article, on the other hand, is to show how it plays many significant, but varied, roles throughout the whole of mathematics. We illustrate this fact through characteristics examples; in most of these examples the mathematics is well known, but the symmetry aspects of the arguments have not been rendered explicit as a guiding principle. In one example, however, we do look at an unfamiliar geometrical construction of regular star polygons, which we relate to number-theoretical properties. In this example we draw attention to the presence of symmetry of an unexpected and untraditional nature, not obviously related to the regularity of the polygons. We have attempted to identify specific principles illustrated by our examples of symmetry in mathematics. We draw attention to the Halfway Principle (Sec. 3) and the principle of the symmetric definition of symmetric concepts (Sec. 4).

On Problems with Solutions Attainable in More Than One Way.

Jean Pedersen, born later than George Pdlya, is a member of the Mathematics Department at the University of Santa Clara. Her principal research interests are in polyhedral geometry, combinatorics, and mathematics education, in which fields she has published several articles and books. Her most recent book is Fear No More: An Adult Approach to Mathematics (Addison-Wesley, 1983), co-authored with Peter Hilton. This book is the first of a series of 3 volumes