Papercrafts and other fun things: Exploring the Five-Color Torus: A Mathematical Journey

Exploring the Five-Color Torus: A Mathematical Journey

This summer I attended the ICERM Illustrating Mathematics Reunion/Expansion and had the pleasure of meeting Alba Málaga Sabogal from the Université de Lorraine. Alba gave a fascinating presentation on a structure called the five coloring of a torus. You can watch her talk in the ICERM video archive; at about 4:02 she begins discussing the piece.

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.

That experiment sparked a new challenge for me: given that the four-color theorem guarantees any map can be colored with just four colors, I wondered if I could create a four-color torus using only four folded modules. The theorem shows that you need at most four colors to color any flat map so that adjacent regions—those sharing a border, not just touching at a point—have different colors. Since Francis Guthrie first proposed this concept in 1852 (earning it the alternate name "Guthrie's problem"), I was curious whether this mathematical principle could translate into a physical origami construction.

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.

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.