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.
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
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) <0 there must be an integer 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.