- 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.
- Desmos Torus Knot Generator

- Zome
- Mentors: Zivile Puospekaite, Brandon Shapiro
- 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.

- 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.
- Desmos Spirograph Generator (made by Erol and Sarah)

- Zome
- Mentors: Oscar Arévalo, Brandon Shapiro
- 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.

- Wire
- Mentors: Lam Nguyen, Alec Traaseth
- Mentees: Brandon Bennett, Samuel Bromberg, Grace Coleman, Casey Matney, Ridge Redding
- Description: We explored various models of the hyperbolic plane and isometries between them using wire constructions and string.

- Crochet
- Mentors: Beth Branman, Valentina Zapata Castro
- Mentees: Noah Amato, Ellie Berg, Blis Bodawala, Caroline Lingle, Michelle Monge
- 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.

- Crochet
- Mentors: Valentina Zapata Castro
- Mentees: Faith Ellis, Olivia O’Donnell, Emily Parnell, Lyla Zynn
- 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