» Vol. 9, Issue 1, 2008 «

Title: Classification of Quasplatonic Abelian Groups and Signatures Author: Robert Benim, University of Portland Author Bio     Abstract: A quasiplatonic surface is a compact Riemann surface, X, which admits a group of automorphisms, G, (called a quasiplatonic group) such that the quotient space, X/G, has genus 0 and the map pG:X-->X/G is branched over three points. For a given genus, by using computational methods we can determine all quasiplatonic groups which act on a quasiplatonic surface of that genus. Though in principal this method can be used to calculate all quasiplatonic groups, in practice it is unrealistic. Another approach is to fix a group and see what genera this group can act upon as a quasiplatonic group. In this paper we classify all Abelian groups which can act on a quasiplatonic surface. A partial classification has previously been supplied. This partial classification provides necessary but not sufficient conditions for a group to act upon a quasipatonic surface. The classification shown in the following provides both. We accomplish this by first looking at cyclic groups with orders of a single prime power. All other Abelian groups can be built up from this case, and so can the classification of the quasiplatonic surfaces upon which they act upon. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Natural Families of Triangles I:  Parametrizing Triangle Space      Author: Adam Carr, Carleton College Julia Fisher, Carleton College Andrew Roberts, Carleton College Peng (David) Xu, Carleton College Author Bio     Author Bio     Author Bio     Author Bio     Abstract: We group triangles into families based on three parameters: the distance between the circumcenter O and the centroid G, the circumradius, and the measure of angle Ð GOA where A is one vertex. Using these parameters, we present triangle space, a subset of R3 in which every triangle is represented by exactly one point. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Natural Families of Triangles II:  A Locus of Symmedian Points Authors: Julia Fisher, Carleton College Adam Carr, Carleton College Andrew Roberts, Carleton College Peng (David) Xu, Carleton College Author Bio     Author Bio     Author Bio     Author Bio     Abstract: We group triangles into families based on three parameters: the distance between the circumcenter O and the centroid G, the circumradius, and the measure of angle Ð GOA where A is one vertex. We focus on the family of triangles which allows Ð GOA to vary and fixes the other two parameters. By construction, this grouping produces triangles which share the same Euler line. Perhaps unexpectedly, if we examine the family's locus of a triangle center known as the symmedian point, we find that it always forms an arc of a circle centered at a specified point on the Euler line. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Fundamental Reflection Domains for Hyperbolic Tesselations Authors: Jacob Grosek, University of Utah Author Bio     Abstract: This paper summarizes Vinberg's algorithm for finding the subgroup generated by reflections of the group of integral matrices that preserve particular quadratic forms of signature (n,1). Also, many fundamental reflection domains of different hyperboloids, found by the author using Vinberg's algorithm, are listed in this paper. Plus, Matlab code, written by the author, is included, which serves to help one discover potential perpendicular vectors to the hyperplanes (mirrors) that enclose the fundamental domain. . Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Unique Properties of the Fibonacci and Lucas Sequences Authors: Stephen Parry, Elmira College Author Bio     Abstract: The algebraic structure of the set of all Fibonacci-like sequences, which includes the Fibonacci and Lucas sequences, is developed, utilizing an isomorphism between this set and a subset of the 2 by 2 integer matrices. Using this isomorphism, determinants of sequences, and Fibonacci-like matrices, can be defined. The following results are then obtained: (1) the Fibonacci sequence is the only such sequence with determinant equal to 1, (2) the set of all Fibonacci-like sequences forms an integral domain, (3) even powers of Lucas matrices are multiples of a Fibonacci matrix, and (4) only powers of multiples of Fibonacci matrices or Lucas matrices are multiples of Fibonacci matrices. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Chaotic Dynamics, Fractals, and Billiards Author: Aaron Peterson, Luther College Sarah Rozner, University of Wisconsin at La Crosse Emily Sutter, Cornell College in Mount Vernon, Iowa Author Bio     Author Bio     Author Bio     Abstract: Chaotic dynamics occur in deterministic systems which display extreme sensitivity on initial conditions. These systems often have attractors which are geometric figures exhibiting affine self-similarity that have non-integer dimension, otherwise knows as fractals. We investigated the link between chaos and the eventual fate of a ball on a frictionless elliptical billiards table with one pocket. The result is a fractal generated by these dynamics. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: The Box Problem Authors: Marissa Predmore, California Lutheran University Author Bio     Abstract: The box problem is taken from Calculus 1, where a student is asked to maximize the volume of a box constructed from a rectangular piece of cardboard with squares removed at the corners. We are interested in what the width and length need to be in order to have at least a rational answer for the optimum height. In 2000, Cuoco used Eisenstein triples to find the dimensions. Hotchkiss expanded on Cuoco's work in 2002 and used an elliptical equation to find the dimensions needed for the box. This paper answers two open questions posed by Hotchkiss: proving that the smallest possible distinct dimensions that produce an integral solution are 5 and 8. Also the minimum distnct dimensions are examined in general. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: On Involutions With Many Fixed Points in Gassmann Triples Author: Jim Stark, University of Virginia Author Bio     Abstract: We show that in a non-trivial Gassmann triple (G,H,H') of index n there does not exist an involution t in G such that the value of the permutation charactger on t is n-2. In addition we describe a GAP program designed to search for examples of Gassmann triples and give a brief summary of the results of this search. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP