# Project: Visualizing Fractals Using 3D Printing

**Project supervisor:** Olga Lukina

**Student researchers:** Divye Bhagwani, Harsha Podapati, Theodore Spiro

**Description:** Fractals are self-similar patterns that are created by repeating a geometric construction over and over at smaller and smaller scales. Patterns with fractal characteristics occur naturally in nature, for example, a snowflake or a leaf of a plant may have a structure resembling fractals. In mathematics, we can construct a variety of fractals using iterated function systems, or using substitutions. During this project we will use 3D printing to visualize simple fractals in three dimensions.

# Project: Destructive Topology: Slicing Surfaces in VR

**Project supervisor:** David Dumas

**Student researchers:** Alexander Adrahtas, Alexander Guo, Gregory Schamberger

**Description:** Closed surfaces (like the torus shown above) can be constructed by stitching together the sides of a polygon according to a certain pattern. However, it can be difficult to visualize what a pattern drawn inside the polygon would look like when wrapped around the surface, or conversely, what a design on the surface would look like when unwrapped to the flat polygon.

The purpose of this project is to develop a virtual reality mathematical visualization and teaching tool that will allow users to explore this correspondence (between flat polygons and curved surfaces). Building on a prototype developed in Unity 3D by Professor Dumas, the student researchers in this project will create a virtual surface laboratory where the user can slice, stitch, paint, and manipulate surfaces in space while seeing the same operations performed on the corresponding flat polygons.

**Prerequisites:** Some 3D graphics programming experience is required. The ideal candidate would have experience with the Unity 3D engine, would be familiar with software version control systems (e.g. git), and would have taken a linear algebra course.

# Project: Two-radius Planar Disk Packings

**Project supervisor:** Ali Mohajer

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

**Description:** The densest packing of disks of the same radius in the plane is the hexagonal lattice. If disks of two different radii are allowed, it is sometimes possible to pack disks more densely than the hexagonal lattice, however this is only possible if the ratio of the radii is not very close to one. Exactly how close is “very close” is an area of active research, and much remains to be discovered about the behavior of two-radius packings. In this research project we will study the densities of randomly generated two-radius disk packings with various ratios of radii, with a goal of obtaining experimental insights into the variation of the optimal density as the ratio of radii changes.

This project will build on the results of the Spring 2018 MCL semester project *Efficiency of Planar Disk Packings*.

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

# Project: Robot Kinematics

**Project supervisor:**Julius Ross

**Student researchers:**Ativ Aggarwal, Leticia Garcia, Sara Sayeed

**Description:** Robot kinematics considers the following kinds of problems. Imagine a robot arm made up of a number of straight segments joined together by hinges. One end of the arm is fixed on a table and at the other end is a tip (or finger) of some kind. Suppose the robot wants to move its finger to a give point, then what angles must it choose for each of the joints? Is it the case that the robot can move its finger to any position, or are there some that it cannot reach? What about if it wants to move its finger to a given position and make it point in a given direction? One can ask this question in either two dimensions or three dimensions (or even more!).

This project will consider these questions by understanding the problem as finding solutions to a system of simultaneous polynomial equations, which then becomes a problem in numerical algebraic geometry.

**Prerequisites:** Math 310 or Math 320 (Linear Algebra).