Compass and Straightedge Constructions (Evangelia Gazaki, Michael Wills): This project will examine compass and straightedge constructions in the plane. We will use these tools to carry out explicit constructions (bisecting an angle, drawing a regular hexagon) as mathematicians have been doing for over 2000 years. Even as more and more constructions were discovered, some (such as constructing a square with the same area as a given circle) remained elusive. This project will also examine the seemingly distant branch of math known as Galois theory, and how it was used in the 1800’s to show the mathematical impossibility of these famous open problems.
Computer graphics and Robotics (Bakhyt Aitzhanova): We will learn a standard method for solving the forward and inverse kinematic problems for a given robot “arm”. The forward kinematic problem is a fundamental concept in robotics and computer graphics that involves determining the position and orientation of the hand of a robot arm based on its joint parameters. Unlike forward kinematics, where the hand position is computed from known joint parameters, inverse kinematics works in reverse: given the hand position, we find the joint parameters. The inverse kinematic problem is more challenging because it often requires solving a system of non-linear polynomial equations.
Zome and polyhedra (Brandon Shapiro): The Zome toolkit makes it easy to build toy models of polyhedra, 3 dimensional solid shapes like cubes and pyramids built out of polygons on the outside. Using the Zome pieces, we can show why there are only 5 polyhedra whose faces are all the same shape, and also why in the fourth dimension there are only 6 shapes with the same kind of property. We will explor the geometry of shapes we can build using Zome, such as angles, counting faces of solid shapes, symmetries, knots, and/or many other possibilities.