Diplotori - flat polyhedral tori


Diplotori - flat polyhedral tori
Files for producing various diplotori, either to 3d-print or to paper-fold. The zip archive contains a readme file in English and in French.


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The original idea by Ulrich Brehm was later considered, propagated and/or studied by various people, including Branko Grünbaum, Geoffrey C. Shephard, Jörg M. Wills, Guy Valette, Robert Ferréol, Henry Segerman, Vincent Borrelli, Francis Lazarus, Florent Tallerie, Takashi Tsuboi... and the present authors: Alba Málaga, Pierre Arnoux and Samuel Lelièvre. The patterns we designed allow to fold up paper tori without glue or tape.
Pierre Arnoux
Samuel Lelièvre
Alba Málaga
Supported by: 
By reporting results from the Illustrating Mathematics program, this material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 and the Alfred P. Sloan Foundation award G-2019-11406 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, USA. Also supported by - Institut de mathématiques de Marseille (I2M), Aix-Marseille Université - Laboratoire de mathématique d'Orsay (LMO), Université Paris-Saclay - Laboratoire lorrain de recherche en informatique et ses applications (Loria) - IUT de Saint-Dié, Université de Lorraine - CENTURI Multi-Engineering Platform, Luminy, Marseille - Centre international de rencontres mathématiques (CIRM), Marseille

Diplotori are polyhedra “of genus one” (that is, “with one hole”, like a buoy or a donut) which are “flat” in the sense that the total angle at each vertex is exactly 360 degrees.

These tori are flat in the sense that the total angle at each vertex is 360 degrees.

We provide files to produce patterns that can fold into these tori in several ways.

One way is to print the pattern as an image, then cut with scissors or a paper knife and crease and fold by hand.

Another way is to send an image to an automatic paper cutter, which, by controlling a blade or a laser beam, can cut and pre-crease the paper very precisely.

We also provide shape files which can be used to 3d-print diplotori.

These flat tori, apparently introduced by Ulrich Brehm in 1978, were not thoroughly studied until around 2020.

They have since been discovered many times, and called “Genus one polyhedra whose Gaussian curvature is zero at all vertices”, “everywhere flat polyhedral tori in E3”, “toroidal isogonal polyhedra”, “basis-free isogonal prismatoids”, “acoptic isogonal prismatoids”, “polyhedral isometric embeddings of flat tori”, “origami tori”, “iris toroids”, and possibly other names, and we call them “diplotori”.

Any flat torus (abstractly obtained by gluing opposite sides of a parallelogram) can be realized as a diplotorus.