Mentees: Danielle Amoateng, Yipeng Cao, Minu Choi, Doğa Eker, Temuulen Enkhtuvshin, Keegan Jewell, Otso Karali, Enoch Liu, Yashraj Vishwakarma.
Description: 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.
Description: 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.
Zome and polyhedra
Mentors: Brandon Shapiro
Mentees: Yunjie Tong Annie, Rachel Earle, Ari Escandon, Grayson Gemmell, Kate Peevey, Pierce Tan.
Description: 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.
Fall 2024
Computer graphics and Robotics
Mentor: Bakhyt Aitzhanova
Mentees: David Cao, Zachary Forino, Josh Gao, Alex Wei, Alexander Yao.
Description: We learned a standard method for solving the forward kinematic problem 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 end-effector (or the tip) of a robot arm based on its joint parameters.
Billiards and Translation Surfaces
Mentors: Lam Nguyen, Oliver Wang
Mentees: Mariam Grigorian, Brian Lai, Emily Moore, Alessia Natalucci, Ismoil Nosirov, Dominic Tran, Jiaming Zhang.
Description: If a billiards player hits a ball in a particular direction, will it eventually drop in a pocket, and if so, after how many bounces? Questions like this one are the subject of mathematical billiards, and answering them surprisingly requires us to connect multiple branches of math including geometry, number theory, and linear algebra. In this project, we explored how examining billiards naturally leads to the geometric notion of translation surfaces, and we investigated the properties of flows on these surfaces.
Coding Theory and Finite Fields
Mentors: J.D. Quigley, Michael Wills
Mentees: Nate Bryerton, Arun Jannupreddy, Malik Kurtz, Dailin Li, Rohan Radadiya, Ridge Redding, Eva Simpson
Description: We learned about how one can mathematically harden information against random errors in transmission by the strategic addition of redundancy. We wrote functioning code implementing various systems for achieving this, and learned how to compare their performance. Many familiar objects from geometry can be described using polynomial equations. For instance, the unit circle is the set of points in the xy-plane such that x^2+y^2-1=0, while the equation x^2-y^2-1=0 describes a hyperbola. In this project, we studied polynomials in exotic contexts for arithmetic called “finite fields”. One motivation to investigate this kind of geometry comes from coding theory, which is the mathematics of how to make digital messages resistant to random noise.
Mentees: Maya Clifford, Kyla Fish, Justin Gu, Wenwan Xu.
Description: 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 explored the geometry of shapes we can build using Zome, such as angles, counting faces of solid shapes, symmetries, knots, and/or many other possibilities.
Spring 2024
Knots: Spirographs, Wire, Clay
Mentors: Sarah Blackwell, Jiajun Yan
Mentees: Sharini Rahman, Ethan Wu
Description: We explored mathematical knots through various different media such as spirographs, wire, clay, and yarn. We investigated the types of knots that can arise from spirograph curves, and also used wire, clay, and yarn to make 3D models of the knots we discussed.
Mentees: Cassie Buxbaum, Jai Khosla, Alice Zhang, Melissa Zheng
Description: We explored 3-dimensional polyhedra and 4-dimensional polytopes using Zome to build models of many of these shapes, compute various angles within them, and reason about their existence. In particular, we used Zome to see why there are only 5 3-d platonic solids and 6 in 4-d, and worked towards figuring out how to count the number of faces of the 4-d solids.
Paper
Mentors: David Chasteen-Boyd, Mason Hart
Mentees: Anthony Doll, Lucia Dwyer, Wynn Smith
Description: We explored 2-dimensional geometry and topology through (2-dimensional) paper. Topics included tilings of the plane, platonic solids, Euler characteristic, and gluing diagrams for surfaces. Our main project was exploring the classification of surfaces by cutting and manipulating paper gluing diagrams. In the end, we described an algorithm for this cutting and gluing.
Computational Geometry
Mentors: Lam Nguyen, Alec Traaseth
Mentees: Kai Garcia-Lorincz, Mitansh Kagalwala, Ridge Redding, Matthew Schlueter, Anna Tu
Description: We explored algorithms in computational geometry including convex hulls, Voronoi diagrams, and Delauney triangulations and coded them in Python. Students undertook individual capstone projects that included topics such as orthogonal range searching, line-segment intersection, Page rank, and hyperbolic convex hulls. We also had one final project to explore Djistraka’s algorithm.
Fall 2023
Spirographs
Mentors: Sarah Blackwell, Zivile Puospekaite
Mentees: Kayla Denoo, Erol Guleyupoglu, George Hammer, Yuhong Liu
Description: We explored the mathematics of spirographs, which involved investigating the properties of the curves produced in our drawings and deriving parametric equations for these curves.
Mentees: Sarah Child, Paris Phan, Ili Rong, Nahli Zzaman
Description: We explored 3-dimensional polyhedra and 4-dimensional polytopes using Zome to build models of many of these shapes, compute various angles within them, and reason about their existence. In particular, we used Zome to see why there are only 5 3-d platonic solids and 6 in 4-d, and pursued questions about duality and symmetry of these solids.
Paper
Mentors: David Chasteen-Boyd, Biying Wang
Mentees: Baz DeVaul, Justin Gu, Jae Wan Jung, Wynn Smith
Description: We started the semester exploring various constructions that can be done with origami, including visual representations of cyclic groups and their subgroups, trisecting an angle, and approximating a parabola. The latter half of the semester was focused on studying properties of surfaces by constructing models of them with modular origami.
Description: We learned some basic concepts of hyperbolic geometry and models to represent the hyperbolic plane using math. We applied this knowledge to create our own models using crochet and varying the curvature of our models. We also learned how to do 3D shapes like spheres and figured out a math algorithm for the number of stitches each row had to have. Additionally, we had a joint project with the crochet group of MEGL (Mason Experimental Geometry Lab) at George Mason University where we created hyperbolic flowers which we presented at the farmer’s market at IX park. We also visited GMU and joined them for their final presentation of all the MEGL projects.
Computer Visualization
Mentors: Gennady Uraltsev
Mentees: Lincoln Curtis, Diego Dimattina, Michelle Giulajan, Yejun Kim, Liran Li, Katherine Moore, Mandy Unterhalter, Zerui Wang
Description: This project explores the perception of a world with a geometry unlike our own, with a special focus on vision. We examine the standard computer graphics infrastructure and discuss challenges associated with visualizing non-Euclidean geometries. We reference various media that depict non-Euclidean worlds; the project aims to understand the mechanics of these models beyond their visually striking effects: Optics, Mirages, and Curved Space - Steve Trettel; Living on Surface of 4D Sphere - Gijs Bellaard; Hyperbolica - A whimsical Non-Euclidean adventure; HyperRogue - A rougelike on the hyperbolic plane; 4D Toys. An interactive toy for 4D children.; Rotation Tesseract (4D cube). First, we constructed platonic solids and discussed how a two-dimensional being on these surfaces would interpret geometric concepts and vision. We considered how a two-dimensional being would perceive and communicate concepts such as ‘straight line’ and ‘angle. A significant part of the project was understanding the concept of “dimension”. We used the surface of a 4D cube as a model of a non-Euclidean three-dimensional space.” Discussions revolved around how a three-dimensional being would experience living on the surface of a 4D cube. We reformulated the basics of vision, including three-point perspective through the language of linear algebra and projective geometry. This knowledge was applied to understand how the standard computer graphics pipeline works. The project also involved learning the basics of programming with Python notebooks, as well as 3D modeling with the A-frame web library. The team successfully projected a 4D cube wireframe to 3D space and applied 3D standard visualization. Ultimately, the project resulted in a 3D Virtual Reality depiction of a non-Euclidean world, created by integrating combinatorial logic coded by students with a standard Euclidean graphics pipeline. This exploration provided insights into visualizing dimensions beyond our own and the potential for computer-generated visualizations of complex geometries.
3D Printing
Mentors: David Chasteen-Boyd, Alec Traaseth
Mentees: Marco Baessler, Clara Grimmelbein, Nadara Hudson, John Layne, Sam Lichtman, Charlie Wang, Alice Wanner
Description: We used specialized software to design models for 3D printing. This included several platonic solids, fractal-like objects, and some objects from hyperbolic geometry such as pleated surfaces and hyperbolic polyhedra.
Description: We first started learning about the basic concepts of crochet, mainly doing chains and creating squares with single, half-double and double crochet stitches. We then explored the math behind creating euclidean shapes (circles, triangles and other polygons) and “hyperbolic” shapes (adding some ruffles). We explored the math behind counting stitches in rows and adding new ones, and translated it to creating our desired shapes. We created rectangular and circular models for hyperbolic spaces and used them to understand geodesics using thread to visualize them. In our final project, we each contributed a lot of equilateral triangles of the same size and put them together to create a hyperbolic plane model.
3D Printing
Mentors: David Chasteen-Boyd, Filippo Mazzoli
Mentees: Jade Gregoire, Owen Guttilla, Sam Lichtman, Alice Wanner
Description: We studied how to classify isometries of the Euclidean and hyperbolic planes, and designed and 3D-printed models of each type of isometry.
Computer Visualization
Mentors: Gennady Uraltsev
Mentees: Liran Li, Zihan Li, Anna Mack, Maria Vizcaino