An Easy Algorithm for Students to Generate Implicit N-Holed Tori in Maple and Mathematica
Joshua J. Leiter

Abstract
The purpose of this paper is to provide a pedagogical approach on how to make an n-holed torus in Maple and Mathematica for undergraduate students. It shows how to 3D-print out an N-holed torus from only a mathematical equation. This eliminates the hassle of fudging with parameters or drilling the holes yourself. An algorithm to generate N-Holed tori is provided with all parameters for reasonable bounds and parameters given. A specific application of this algorithm in Mathematica shows the print out of a seven-holed torus in figure 9. Furthermore, this method automatically adds in a discrete symmetry determining where the tori’s holes occur. Human beings like symmetry and the discrete symmetry in the algorithm makes the surface aesthetically pleasing. Hopefully the examples generated from this method will allow for: fun 3D printing, and provide simple examples useful for teaching 2D implicit geometry and topology. An explicit construction of a 64-holed torus is given.

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