Paper flat tori

Date started: November 2019
Leads: Alba Málaga, Samuel Lelievre, Edmund Harriss, Pierre Arnoux


A flat torus is usually described as the result of gluing together the pairs of parallel sides of a square – actually the same construction works fine for any parallelogram. This abstract gluing can be described mathematically very precisely using equivalence relations. However, to actually do the gluing while keeping a surface enclosing a non-degenerate volume require some thought. We learned one way to do it thanks to Henry Segerman’s 3d-printed hinged torus. We produced many paper flat tori and one transparent flat torus in acetate and we plan to keep going. Glen Whitney showed us a green mylar paper flat torus he made. One question is can one make a “square torus” using such a model. Each such model is based on two regular n-gons, an amount of twist between them, a height, and two choices of how much to shift the pairings of top edges to bottom vertices (one for the outer and one for the inner part), so there are three discrete and two continuous parameters. We plan to visualize the curves that the moduli of these families of tori draw on the modular surface.


Flat tori, prism version.
Flat tori, antiprism version.

Acknowledgements: The many paper tori we produced during the “Illustrating Mathematics” semester would have not been possible without Glen Whitney who kindly shared his paper cutter with all the semester participants.


  • Hinged flat torus on Henry Segerman’s youtube channel
  • Tore plat on
  • Zalgaller, V. A. Some bendings of a long cylinder. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 246 (1997), Geom. i Topol. 2, 66–83, 197; translation in J. Math. Sci. (New York) 100 (2000), no. 3, 2228–2238
    Article available at (in russian)
  • Prisms and Antiprisms at George Hart’s website