Contents for January 1998
The World's Biggest Taco
David D. Bleecker and Lawrence J. Wallen
A taco is the solid formed by bending a circular tortilla around a cylinder and filling it to the border. A natural problem is to find the cylinder that yields the taco of largest volume for a tortilla of a unit radius. For circular cylinders the volume of the taco is a Bessel function of the cylinder's radius, and for cylinders with other familiar cross-sections the volume of the corresponding taco also involves special functions. But in each case, with the aid of a computer algebra system the methods of calculus can be applied to find the taco of maximal volume. However, the general case is a nontrivial problem in the calculus of variations. The existence of a taco of maximal volume for a suitably general class of cylinders can be proved, and numerical experiments are given to show how the shape and volume of this "world's largest taco" can be approximated.
The Brahmagupta Triangles
Raymond A. Beauregard and E. R. Suryanarayan
This short article commemorates the fourteenth centenary of the Indian astronomer and mathematician Brahmagupta, who analyzed the class of triangles having consecutive integer sides and integer area. The authors show that these "Brahmagupta triangles" can all be generated from the powers of a certain 2 by 2 matrix. At first glance this modern method might seem foreign to Brahmagupta's approach, but it turns out that he had discovered a way to combine two solutions to get a third one, and we see in retrospect that his method of composition of solutions amounts to the multiplicative property of 2 by 2 determinants, which is the key ingredient in the modern method.
A Class of Pleasing Periodic Designs
By superimposing a few sublattices of the lattice of all points in the plane with integer coordinates, each of which is a regular array of points with no artistic interest, surprisingly intricate and attractive patterns of points appear. This provides a semiautomatic method for generating pleasing geometric designs suitable for floor tiles or fabrics. The author, a retired architect, has spent many years exploring this and other methods for generating artistically pleasing designs. A mathematical classification for the sublattices of the integer lattice in the plane is sketched, and this classification then serves as a guide for choosing a set of sublattices that will produce an aesthetically pleasing design when superimposed.
Zeroing In on the Delta Function
Joan R. Hundhausen
By finding the response of an undamped harmonic oscillator to various forcing functions that each deliver a unit impulse over a time interval [0, tau], and then examining the limiting behavior as tau decreases to zero, students can gain an intuitive understanding of the properties of the Dirac delta function. This program is carried out for one family of forcing functions and two other families suitable for students to explore are given. The discussion combines techniques from calculus and graphical analysis with physical considerations to help students achieve a deeper understanding of the delta function in one of its simplest applications.
Eugene C. Boman and Margaret A. Misconish, A Diagonal Perspective on Matrices.
Multiplication of a column vector by an n by n matrix whose only nonzero entries are in the (k, k+i) positions, for some i, shifts the entries of the vector up or down i places and scales the entries of the shifted vector by the entries in the nonzero ith subdiagonal or superdiagonal of the matrix. Thus such matrices are called "shift matrices". Elementary linear algebra textbooks emphasize thinking of matrices as collections of row vectors or column vectors, but for some purposes it is useful to think of square matrices as sums of their constituent shift matrices. Three examples are given where this diagonal perspective on matrices is especially useful.
Yong-Zhuo Chen and Richard F. Melka, Finding a Determinant by Bordering.
For a particular family of n by n matrices the inverse of a typical matrix in the family is easily found in an especially attractive form if one thinks of the given matrix as embedded in an n+1 by n+1 matrix. One finds the inverse of the "bordered" matrix by row reduction, and by comparing the result with a general formula for the inverse of a matrix in block form, the inverse of the original matrix is apparent.
Bettina Richmond and Tom Richmond, Characterizing Power Functions by Volumes of Revolution.
The volume V(r) of the solid obtained by revolving about the y-axis the region under the graph of a power function f(x) over an interval [0, r] is easily shown to be proportional to the volume C(r) of the right circular cylinder with radius r and height f(r). By solving an appropriate differential equation the authors show that, conversely, the only twice differentiable increasing functions for which the ratio V(r)/C(r) is constant are power functions.
Richard Johnsonbaugh, A Discrete Intermediate Value Theorem.
A discrete analogue of the intermediate value theorem of calculus is this: if f is an integer-valued function defined on the integers in [m, n], and the absolute difference between the values of f at any two successive integers in this interval is never more than 1, then if f(m)f(n) x in (m, n) where f(x) = 0. As an application, this discrete intermediate value theorem is shown to provide an elegant proof of an existence theorem in discrete mathematics.
Duane W. DeTemple, Colored Polygon Triangulations.
A side-colored polygon P is a polygon whose sides are colored red, blue or green in such a way that no pair of adjacent edges have the same color. A side-colored triangulation T of P is a triangulation (possibly containing some new vertices on the edges of P) whose edges are colored so that all three colors do not appear on the edges incident to any vertex, and any side of T that is along a side of P retains the color of that side. By associating a graph with any side-colored triangulation, and applying the "handshaking lemma" from graph theory, it is shown that for a side- colored polygon with an odd number of sides, every side-colored triangulation contains an odd number of triangles whose three edges have different colors. Related results are left as exercises for students, and similar results for vertex-colored polygons and triangulations are sketched.
Richard B. Thompson, Designing a Baseball Cover.
The problem of designing the pattern for the two congruent pieces of leather that are stitched together to form the cover of a baseball can be approached in two ways. 1) Draw a pattern in the plane and then wrap copies around two perpendicular circular cylinders of equal radius and see if they fit together (form a "preball") with their seam lying on a sphere whose radius is that of a baseball. A radial expansion of the preball, which corresponds to a slight stretching of the leather, would then form the desired cover of the ball. 2) Find a parametrization of the seam of the ball, then unwrap the two regions that form the surface of the corresponding preball, to form the pattern pieces in the plane. Using a computer algebra system, the arc-length formula from calculus, and basic analytic geometry, the second of these approaches is carried through with little difficulty. The first method, a trial and error version of which was probably used to design the pattern for the modern baseball cover, leads to a rather complicated differential equation. However, with modern computer methods this equation can be numerically solved; thus the trial and error can be eliminated. Final remarks compare the pattern used for real baseball covers with the ideal pattern found by mathematical analysis.