Tom's Origami Research Corner

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Configuration spaces of rigid foldings

This long-term project concerns rigid origami, where we imagine that we are folding stiff (but mathematically thin) metal and have (mathematically thin) hinges at the creases. Will we be able to smoothly fold and unfold such origami? If we can, then tag each crease with a number pi = the folding angle, the amount that the crease bends from the unfolded state. If the crease pattern has n creases, then the set of all points (p1, p2, ..., pn) made by the folding angles gives us the configuration space of the rigidly-foldable crease pattern. Understanding the configuration space gives us valuable information for using rigid origami crease patterns in applications.

Publications:

Counting valid mountain-valley assignments

A fundamental open problem in origami is: Given an origami crease pattern, in how many different ways can we fold it? There are many ways to answer this question, but one way I'm particularly interested in is counting the number of mountain-valley (MV) assignments that make the crease pattern fold flat (can be pressed in a book without crumpling). Each MV assignment represents a distinct way in which the crease pattern can fold. The combinatorial structure of MV assignments seems very rich, even for single-vertex crease patterns. In many cases MV assignments are related to vertex-colorings of graphs, but the jury is still out on classifying the general case.

Publications:

Self-folding

In 2015 Tomohiro Tachi and I began an investigation into modeling how and if a given rigidly-foldable crease pattern will be able to fold from the unfolded state to a target state reliably, given simple activation forces (like springs) on the creases. In other words, if we let the springs on the creases do their thing, will the crease pattern self-fold in the way we want, or will it mis-fold into some undesired shape? Our model is very theoretical (we assume mathematically thin paper, for one thing), but it had led to other studies of self-folding in the physics literature (such as in Nature Physics and Physical Review X). Our work in self-folding also won the 2016 A. T. Yang Memorial Award in Theoretical Kinematics.

Tachi and I are continuing to expand and refine our model by applying it to new developments in rigid origami configuration spaces and making it more realistic for actual materials.

Publications:

Videos:

My 2021-2022 research students, Gianna Biolo, Megan McGuinness, Taryn Padilla, and Hannah Zieminski at Western New England University created a series of TikTok videos documenting their work on fabricating self-folding origami models made of wood, plexiglass, and spring hinges. (Their attempts at using mousetrap spring hinges did not work, as they were far too strong. Fortunately, we were able to find more gentle and accurate spring hinges.) The videos are short, but they do demonstrate their work in action!

Work supported by NSF grants DMS-1906202, RUI: Configuration Spaces of Rigid Origami and EFRI-1240441 Mechanical Meta-Materials from Self-Folding Polymer Sheets.

Contact: Feel free to email me at thomas.hull and then "at" and then "fandm" dot "edu".

Or vist me at my professional web page or find me at Franklin & Marshall College.

Last updated 11/27/22