Configuration spaces of rigid foldings
This longterm 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 p_{i} = 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 (p_{1}, p_{2}, ..., p_{n}) made by the folding angles gives us the configuration space of the rigidlyfoldable crease pattern. Understanding the configuration space gives us valuable information for using rigid origami crease patterns in applications.
Publications:

Explicit kinematic equations for degree4 rigid origami vertices, Euclidean and nonEuclidean, Riccardo Foschi, Thomas C. Hull, and Jason S. Ku, Physical Review E, Vol. 106 (2022) 055001 (11 pages).
Mathematica code that simulates the folding table seen in Figure 8 of this paper, and shown in the blackandwhite animation to the left, can be found here.

Rigid folding equations of degree6 origami vertices, Johnna Farnham, Thomas C. Hull, and Aubrey Rumbolt, Proceedings of the Royal Society A, Vol. 478 (2022) 20220051 (19 pages).
 Rigid Foldability is NPHard, Hugo Akitaya, Erik D. Demaine, Takashi Horiyama, Thomas C. Hull, Jason S. Ku, and Tomohiro Tachi, Journal of Computational Geometry, Vol 11, No. 1 (2020), 93124.
 Sculpting the vertex: manipulating the configuration space topography and topology of origami vertices to design mechanical robustness, Bin Liu, Jesse L. Silverberg, Arthur A. Evans, Christian D. Santangelo, Robert J. Lang, Thomas C. Hull, and Itai Cohen, Nature Physics, Vol. 14 (2018), 811815.
 Rigid foldability of the augmented square twist, Thomas C. Hull and Michael T. Urbanski, Origami^{7}: Proceedings of the 7th International Meeting of Origami in Science, Mathematics, and Education, Tarquin (2018), 533543.
 Rigid origami vertices: conditions and forcing sets, Zachary Abel, Jason Cantarella, Erik D. Demaine, David Eppstein, Thomas C. Hull., Jason S. Ku, Robert J. Lang, and Tomohiro Tachi, Journal of Computational Geometry, Vol. 7, No. 1 (2016), 171184.
Counting valid mountainvalley 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 mountainvalley (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 singlevertex crease patterns. In many cases MV assignments are related to vertexcolorings of graphs, but the jury is still out on classifying the general case.
Publications:

Maximal origami flip graphs of flatfoldable vertices: properties and algorithms, Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya TerSaakov, Journal of Graphs, Algorithms, and Applications, Vol. 26, No. 4 (2021), 503517.
 Counting locally flatfoldable origami configurations via 3coloring graphs, Alvin Chiu, William Hoganson, Thomas C. Hull, and Sylvia Wu, Graphs and Combinatorics, Vol. 37, No. 1 (2021), 241261.
 Face flips in origami tessellations, Hugo Akitaya, Vida Dujmović, David Eppstein, Thomas C. Hull, Kshitij Jain, and Anna Lubiw, Journal of Computational Geometry, Vol. 11, No. 1 (2020), 397417.
 Counting Miuraori foldings, Jessica Ginepro and Thomas C. Hull, Journal of Integer Sequences, Vol. 17 (2014), Article 14.10.8.
 Counting MountainValley Assignments for Flat Folds, Thomas C. Hull, Ars Combinatoria, Vol. 67 (2003), 175188.
Selffolding
In 2015 Tomohiro Tachi and I began an investigation into modeling how and if a given rigidlyfoldable 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 selffold in the way we want, or will it misfold 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 selffolding in the physics literature (such as in Nature Physics and Physical Review X). Our work in selffolding 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:
 Selffoldability of monohedral quadrilateral origami tessellations, Thomas C. Hull and Tomohiro Tachi, Origami^{7}: Proceedings of the 7th International Meeting on Origami Science, Mathematics, and Education, Tarquin (2018), 521532.
 Selffoldability of rigid origami, Tomohiro Tachi and Thomas C. Hull, ASME Journal of Mechanisms & Robotics, Vol. 9, No. 2 (2017), 021008021017.
Videos:
My 20212022 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 selffolding 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!

Video 1, showing them building their first lasercut wood model.

Video 2, which is a montage of shots from their first handcut model with mousetrap springs, to their lasercut plexiglass and wood versions (much better). This video went semiviral, gaining 24K views in the first 24 hours it was posted!

Video 3, where Megan and Hannah explain a little bit of the math that makes selffolding origami work.

Q&A video 1, where Gianna answers a TikTok viewer's question.

Q&A video 2, where Megan answers a TikTok viewer's question.

Q&A video 3, where Hannah and Taryn answers a TikTok viewer's question.

Video 4, where the students recorded me demonstrating how the larger tessellation can successfully selffold to a desired state.