# Project: 3D Printing Surfaces

**Project supervisor:** Daniel Groves

**Graduate mentor:** Paul Rapoport

**Student researchers:** Mario Aranda, Xuchen Wang, Dean Welby, Amal Yaghmour

**Description:** This project will explore surfaces in three-dimensional space by 3D printing models of them. The goal is to start by choosing appropriate equations and end with physical models which would be appropriate for use in multivariable calculus classes. This will involve learning about function graphing software, 3D modeling software and 3D printing. Time and student interest permitting, we can investigate such things as curvature, ruled surfaces and other things which touch upon ideas from topology, geometry and algebraic geometry.

**Prerequisites:** It is possible to succeed in this project whilst enrolled in Math 210 (Calculus III).

# Project: Efficiency of Planar Disk Packings

**Project Supervisor:** Ali Mohajer

**Student researchers:** Rohit Banerjee, Ivan Cruz, Jacob Krol

**Description:** A popular chemistry experiment shows that a mixture of certain pure liquids can be denser than each of the constituents. Mixing a liter of methanol with a liter of ethanol gives a solution with volume measurably less than 2 liters! A mathematical analog of this experiment is the fact that a packing of unequal disks in the plane can be denser than a packing of equal disks, as long as the radii of the disks aren’t very close. Exactly how close is “very close” is an area of active research. In fact, much remains to be discovered about the behavior of two-species packings, except at a handful of very special ratios of radii where everything fits together very nicely. In this research project we will study randomly generated two-species packings in order to gain insight into the shape of the density bounding function.

**Prerequisites:** Math 210 (Calculus III) and some computer programming experience (equivalent to MCS 260).