Exploring the Five-Color Torus: A Mathematical Journey
Because I’ve long been intrigued by the torus, I was immediately drawn to this project. The model was created by undergraduate students Alix Kremer and Djatil Krichenane, under the mentorship of Alba and Samuel Lelièvre. Their work combines geometry, design, and craftsmanship into a tangible structure: five folded modules, each a different color, joined together by hinges. Alba included their work in her ICERM presentation on “Tangible Mathematics Research Internships.” The project itself grew out of two prompts: Samuel Lelièvre’s 7-colored diplotorus and Saul Stahl’s paper “The Other Map Coloring Theorem.”
The result is a pentagonal torus — a surface of genus 1 (topologically equivalent to a doughnut) with pentagonal symmetry. Instead of the familiar smooth torus generated by rotating a circle, this version is polygonal, assembled from five wedge-like modules. What makes it so elegant is that each colored region touches all the others, perfectly embodying the idea of a five-coloring in three dimensions.
At the conference, Alba wanted to build a version to show how the coloring works. She was prepared to cut it by hand, but since I had brought my Silhouette Cameo cutting machine, I offered to help produce a cleaner, more precise version.
When I got home, I couldn’t stop thinking about the model. I decided to build one myself, this time experimenting with hinged paper tabs instead of Scotch tape. It was my first attempt at this joining method, and I quickly discovered the net could be assembled in two ways: clockwise or counterclockwise, depending on how the pieces were arranged.
This turned out to be much trickier than expected. The geometry demanded precise control of the diagonal angles and the heights of the outer triangles. Even the slightest mismatch caused the paper to warp or wrinkle as shown in the above photos.After many trials, I settled on a two-piece approach: one piece for the central diagonal, and another for the two outer triangles. I also scaled the outer triangles to 101% to account for cardstock thickness. At exactly 100%, they were just a bit too small to close smoothly into a ring.
In the end, the process gave me a new appreciation for Alix and Djatil’s original five-color model. What began as a photograph shared at ICERM grew into an exploration of geometry, color, and structure — and left me eager to keep experimenting with new variations.
In the end, the process gave me a new appreciation for Alix and Djatil’s original five-color model. What began as a photograph shared at ICERM grew into an exploration of geometry, color, and structure — and left me eager to keep experimenting with new variations.
I have since tried a six color version. Again, there is warping. I tried to ask ChatGPT to design a program to determine the correct calculations. This program does need further modification but it is interesting to see what AI can do with this mathematical color journey. Here is the html to run the program, Dynamic Twisting Polygon https://drive.google.com/file/d/1hJY8uP0mBRd8wXsxd0-RfgXi6y9Klyex/view?usp=sharing Click on the file and then download it. Once downloaded, open the file to run the html code, toggle on the twisted edges and the diagonals to see the internal structure of the torus. After further investigation, the measurements are wrong! Interesting to see that AI is not always right. I will correct its mistake in the next blog posting.
Here are the files for the colored torus:
The PDF for a five colored torus. https://drive.google.com/file/d/13C4T0GLoQzU01qpDz258_CiddF5CMQ_J/view?usp=sharing
The .Studio file for a four and five colored torus.
The SVG file for a four and five colored torus.
