Preprints and Papers for Thomas C. Hull

I'll be trying to make as complete a list of my papers here as I can. (i.e., as much as the publishers will let me!)  All are in PDF format.  I apologize if any of the graphics in these files look yucky.

See also my papers on the arXiv (some of which might not be on the below list yet).

• Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean (arXiv link, but the published paper can be found here), Riccardo Foschi, Thomas C. Hull, Jason S. Ku, Physical Review E, Vol. 106, No. 5 (2022), 055001-055011.
This paper started as merely trying to prove that a new folding table that architect professor and origami artist Riccardo Foschi designed has only one folding mode. That is, proving that it can fold in only one way, meaning that it is a good design for practical use. But the vertices in Foschi's design are non-developable, meaning that they are not made from normal, flat paper. Instead, at each vertex there is < 2π of paper around each vertex. In order to prove our result, I decided to create better folding angle equations for non-developable degree-4 origami vertices, and that is what the focus of this paper is about. The equations (in Theorem 2 of the paper) look ugly at first, but they have some amazing symmetry to them. We include some applications of these equations, including proving that Foschi's table has only one way it can fold without the material self-intersecting. Mathematica code that simulates the folding table seen in Figure 8 of this paper can be found here.

• Maximal origami flip graphs of flat-foldable vertices: properties and algorithms Manuel Morales, Thomas C. Hull, Sarah Nash, and Natalya Ter-Saakov, Journal of Graph Algorithms and Applications, Vol. 26, No. 4 (2022), 503-517.
This was research done at the MathILy-EST 2019 REU with undergraduate students from Arizona State University, New College of Florida, and MIT. It is the first paper to explore the configuration space of face flips for single-vertex flat-foldable crease patterns where all the sector angles around the vertex are equal.

• Rigid folding equations of degree-6 origami vertices Johnna Farnham, Thomas C. Hull, and Aubrey Rumbolt, Proceedings of the Royal Society A, Vol. 478, No. 2260 (2022), 20220051 (19 pages).
This paper contains work in collaboration with undergraduate students at Western New England University. We classified the symmetrically-different ways in which the all-60-degree angles degree-6 origami vertex can rigidly fold and then computed the folding angle equations in these symmetric cases. Surprisingly, we discovered that the tangent half-angle parameterization of the folding angles leads to fairly elegant folding angle equations, as in the degree-4 case.

• Counting locally flat-foldable origami configurations via 3-coloring graphs, Alvin Chiu, William Hoganson, Thomas C. Hull, and Sylvia Wu, Graphs and Combinatorics, Vol. 37, No. 1 (2021), 241-261.
This was research done at the MathILy-EST 2019 REU with undergraduate students from Clemson, Georgia Tech, and Swarthmore College. We extended the idea from my 2014 paper with Jessica Ginepro on the equivalence between locally-valid MV assignments of the Miura-ori and proper 3-vertex colorings of grid graphs (with one vertex pre-colored). In this new work, we found graphs (which we call SAW graphs, for Sylvia, Alvin, and William) whose 3-colorings are in bijection with locally-valid MV assignments for a wide range of flat-foldable crease patterns.

• Face flips in origami tessellations, Hugo Akitaya, Vida Dujmovic, David Eppstein, Thomas C. Hull, Kshitji Jain, and Anna Lubiw, Journal of Computational Geometry, Vol. 11, No. 1 (2020), 397-417.
This paper investigates face flips in valid MV assignments of flat-foldable origami crease patterns. Flipping a face means to reverse the MV parity of all creases surrounding the face. We explore the configuration space of face flips for a given crease pattern. In particular, we look at the problem of whether two flat-foldable MV assignments are connected by a path of face flips, finding polynomial-time algorithms for this in the case of square grid and Miura-ori crease patterns, and show that this problem is NP-hard for triangle lattice crease patterns.

• Rigid foldability is NP-hard, Hugo Akitaya, Erik Demaine, Takashi Horiyama, Thomas C. Hull, Jason Ku, and Tomohiro Tachi, Journal of Computational Geometry, Vol. 11, No. 1 (2020), 93-124.
We prove that deciding the rigid foldability of a given crease pattern is weakly NP-hard, and that rigid foldability with optional creases is NP-hard. This captures, in a sense, the difficulty that engineers have when using origami for mechanical purposes and find it difficult to make a material rigidly fold into a desired configuration from the flat, unfolded state.

• Self-foldability of monohedral quadrilateral origami tessellations, T. C. Hull and T. Tachi, Origami7: Proceedings of the 7th International Meeting on Origami Science, Mathematics, and Education, Tarquin (2018), 521-532.
Joint work with Tomohiro Tachi. We use our theoretical model of self-foldability to study monohedral origami tessellations made from quadrilateral tiles, investigating when they will and won't be uniquely self-foldable.

• Rigid foldability of the augmented square twist, T. C. Hull and M. T. Urbanski, Origami7: Proceedings of the 7th International Meeting of Origami in Science, Mathematics, and Education, Tarquin (2018), 533-543.
This is work done with a Western New England University undergraduate math major. We fully describe the rigid foldability of a modified square twist crease pattern. See this web page for the paper that includes fun animated GIFs of the rigid foldings, as well as Mathematica code for making them!

• Sculpting the vertex: manipulating the configuration space topography and topology of origami vertices to design mechanical robustness, B. Liu, J. L. Silverberg, A. A. Evans, C. D. Santangelo, R. J. Lang, T. C. Hull, and I. Cohen, Nature Physics, Vol. 14 (2018), 811-815. (And here is the arxiv version)
We look at a way to combine high DOF (degree of freedom) vertices into larger crease patterns in order to pare down the range of motion of the system. In particular, we look at ways to modify the Miura-ori crease pattern by (a) first breaking the mirror symmetry of the vertices, but otherwise keeping the crease pattern the same and (b) second by again breaking the mirror symmetry of the vertices, but this time using this to wrap the Miura-ori into a ring, so that the crease pattern is on an annulus instead of a rectangle. We study the configuration space of the kinematics of these systems and use this to explain some of the interesting mechanics of these crease patterns.

• Double-line rigid origami, T. Hull and T. Tachi, Proceedings of the 11th Asian Forum on Graphic Science, August 6-10, 2017, Tokyo, Japan, H. Suzuki (ed.)
In this conference paper we examine a way to take an origami crease pattern C and "fatten" all the crease lines by converting them into double-line pleats and at the same time turing all the vertices into polygons, to get a new crease pattern DL(C). Why? Because while C might be very complicated, with vertices of all sorts of degrees, the double-line version DL(C) will have only flat-foldable vertices of degree 4. Thus the kinematics of the vertices in DL(C) behave very nicely, and this can tell us things about the kinematics of C. Yeah!

• Zero-area reciprocal diagram of origami, E. Demaine, M. Demaine, D. Huffman, T. Hull, D. Koschitz, T. Tachi, Proceedings of the IASS Annual Symposium 2016 "Spatial Structures in the 21st Century," September 26-30, 2016, Tokyo, Japan, K. Kawaguchi, M. Ohsaki, T. Takeuchi (eds.)
We describe a graphical tool, called the zero-area reciprocal diagram, to describe and design the rigid foldability of origami crease patterns. This diagram describes the second-order rigid motion of the crease pattern, making it a useful tool for rigid folding kinematic analysis and design. Nifty examples if rigidly-foldable crease patterns are included.

• Self-foldability of rigid origami, T. Tachi and T. Hull, ASME Journal of Mechanisms & Robotics, Vol. 9, No. 2 (2017), 021008-021017.
In this paper, we present a mathematical model for self-foldability. Specifically, we devise a geometric definition for what it would mean for a given crease pattern to be able to self-fold, rigidly, from a starting folded state to a target folded state, using a set of actuators (forces) on the creases. (This paper won the 2016 A. T. Yang Memorial Award in Theoretical Kinematics from the ASME.)

• Box Pleating is Hard, H. Akitaya, K. Cheung, E. Demaine, T. Horiyama, T. Hull, J. Ku, T. Tachi, R. Uehara, in Akiyama J., Ito H., Sakai T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science, Vol. 9943, Springer (2016), 167-179.
We give a proof that determining whether or not a box-pleated origami crease pattern (i.e., one where the plane angles between adjacent creases is always a multiple of 45 degrees) is flat-foldable is NP-hard, even if the mountain-valley assignment is given.

• Rigid origami vertices: conditions and forcing sets, Z. Abel, J. Cantarella, E. Demaine, D. Eppstein, T. Hull., J. Ku, R. Lang, and T. Tachi, Journal of Computational Geometry, Vol. 7, No. 1 (2016), 171-184.
This paper presents a necessary and sufficient condition for a single vertex to fold rigidly, kind of like a Kawasaki condition but for vertex rigid-foldability instead of flat-foldability. We then use this condition to find forcing sets for rigidly-foldable vertices.

• Critical transition to bistability arising from hidden degrees of freedom in origami structures, J. L. Silverberg, J. Na, A. A. Evans, T. Hull, C. D. Santangelo, R. J. Lang, R. C. Hayward, and I. Cohen, Nature Materials, Vol. 14, (2015), 389-393.
We look at the family of square twist crease patterns and demonstrate how they show bistability after a critical value of the rotation angle parameter. We then argue how this same mechanical transition behavior exists in other folded patterns as well.

• Minimum forcing sets for Miura folding patterns, B. Ballinger, M. Damian, D. Eppstein, R. Flatland, J. Ginepro, and T. Hull, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA15), P. Indyk ed., SIAM (2015), 136-147.
We find minimum forcing sets in the Miura-ori crease pattern, with an efficient algorithm and everything.

• Coloring connections with counting mountain0valley assignments, Origami6: Proceedings of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 3-11.
We survey more recent attempts at enumerating the number of mountain-valley assignments that allow a given crease pattern to locally fold flat. In particular, we solve this problem for square twist tessellations and generalize the method used to a broader family of crease patterns. We also describe the more difficult case of the Miura-ori and a recently-discovered bijection with 3-vertex colorings of grid graphs.

• Locked rigid origami with multiple degrees of freedom, Z. Abel, T. Hull, and T. Tachi, Origami6: Proceedings of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 131-138.
We devise a rigidly-foldable origami crease pattern that is (a) composed only of triangle faces and (b) has disconnected configuration space. This means that the origami model could be put it into a rigid folded state (if we, say, cut along the creases, fold it, and then tape them back together when done) from which the mdoel could not be unfolded via rigid motions.

• Rigid flattening of polyhedra with slits, Z. Abel, R. Connelly, E. D. Demaine, M. L. Demaine, T. Hull, A. Lubiw, and T. Tachi, Origami6: Proceedings of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 109-117.
By the Bellow's Theorem, it is impossible to collapse or flex a tetrahedron along creases made on its faces or its edges in such a way that (a) leaves all faces of the creased tetrahedron rigid (i.e., flat) and (b) changes the volume of the tetrahedron. We show how we can do this if we make a small slit in one of the faces of the tetrahedron, thus getting around the requirements of the Bellow's Theorem. We are able to get away with a very small slit, one that is only 0.046 units in length if an edge of the original tetrahedron is length 1.

• Programming reversibly self-folding origami with micro-patterned photo-crosslinkable polymer trilayers, J.-H. Na, A. A. Evans, J. Bae, M. C. Chiappelli, C. D. Santangelo, R. J. Lang, T. C. Hull, and R. C. Hayward, Advanced Materials, Vol. 27 (2015), 79-85.
This paper describes a really cool way to make micro-scale polymer hydrogels swell in such a way as to fold the gel into various objects. Our method was successful enough to allow us to properly fold an origami tessellation crease pattern with over 200 creases.

• Using origami design principles to fold reprogrammable mechanical metamaterials, J. L. Silverberg, A. A. Evans, L. McLeod, R. Hayward, T. Hull, C. D. Santangelo, and I. Cohen, Science, Vol. 345, No. 6197 (2014), 647-650.
We describe, both experimentally and via analysis, how one can use pop-through defects to program the stiffness and mechanical behavior of a Miura-ori folded structure. This paper has gotten a lot of citations.

• Counting Miura-ori foldings with Jessica Ginepro, Journal of Integer Sequences, Vol. 17 (2014), Article 14.10.8.
My student Jessica and I found a bijection between the number of locally flat-foldable MV assignments of an m x n Miura-ori crease pattern and the number of ways to properly 3-color the vertices of an m x n grid graph with one vertex pre-colored.

• Minimal forcing sets for 1D origami, M. Damian, E. Demaine, M. Dulieu, R. Flatland, H. Hoffman, T. Hull, J. Lynch, and S. Ramaswami, preprint (2013).
We demonstrate an algorithm for finding a minimal forcing set in an arbitrary 1D crease pattern (i.e., a linkage).

• The flat vertex fold sequences, E. Chang, T. Hull, in Origami5: Fifth International Meeting of Origami Science, Mathematics, and Education, P. Wang-Iverson, R.J. Lang, M. Yim editors, A K Peters/CRC Press, Natick, MA (2011), 599-607.
With my Master's student Eric Chang, we examined the sequence of numbers one gets when counting the number of ways a degree-2n vertex can fold flat. (E.g., a degree-4 vertex can fold in 4, 6, or 8 different ways depending on the specific angles between the creases.)

• Solving cubics with creases: the work of Beloch and Lill,T. Hull, The American Mathematical Monthly, Vol. 118, No. 4 (2011), 307-315.
This paper is the result of my translating the original papers by the Italian mathematician Margherita Piazzolla Beloch (circa 1930s) into English and discovering that the mathematics they contain is really very pretty! Of special note is her use of Lill's Method, which seems to have been largely forgotten over the decades.

• Configuration spaces for flat vertex folds, T. Hull, in Origami4: Proceedings of the Fourth International Meeting of Origami Science, Mathematics, and Education, R. Lang, ed., A.K. Peters, Ltd., Natick, MA (2009), 361-370.
In this paper the configuration space is computed of the set of all flat-foldable vertex crease patterns of degree 2n.

• Folding regular heptagons, T. Hull, in Homage to a Pied Puzzler, E. Pegg Jr., et al., eds., A.K. Peters, Ltd., Natick, MA (2009), 181-191.
If you're interested in how to construct a regular heptagon out of origami, look no further!

• Constructing pi via origami, T. Hull, preprint (2007).
This is more of a philisophical musing on what the consequences would be for origami geometric constructions if we allowed curved creases. It turns out that if we allow them, constructing pi becomes fairly easy. Hmmm...

• Origami Quiz, T. Hull, in The Mathematical Intelligencer, Vol. 26, No. 4 (Fall 2004), 38-39, 61-63.
This is a fun quiz that I was invited to write for the Intelligencer's Mathematical Entertainments column.

• The Combinatorics of Flat Folds: a Survey, T. Hull, in Origami3: Proceedings of the Third International Meeting of Origami Science, Mathematics, and Education, A.K. Peters, Ltd., Natick, MA (2002).
A survey of flat-folding combinatorics results.

• Counting Mountain-Valley Assignments for Flat Folds, T. Hull, Ars Combinatoria, Vol. 67 (2003), 175-188.
This paper covers the problem of counting the number of valid MV assignments for a flat-foldable origami vertex, resulting in recurrence equations for the answer (as well as some other useful tools in the process).

• Modeling the Folding of Paper into Three Dimensions using Affine Transformations, s.-m. belcastro, T. Hull, Linear Algebra and its Applications, Vol. 348 (2002), 273-282.
Another one of my most heavily-cited papers, because it gave the first proof of the matrix product necessary condition for rigidly-foldable crease patterns, which is very useful for engineers using folding mechanics in their designs.

• Classifying Frieze Patterns Without Using Groups with s.-m. belcastro, The College Mathematics Journal, Vol. 33, No. 2, (2002), 93-94.

• In Search of a Practical Map Fold, T. Hull, Math Horizons, February (2002), 22-24.
This is a fun paper describing a letter I received from a military veteran describing an interesting way to fold a map.

• Defective List Colorings of Planar Graphs, N. Eaton, T. Hull, Bulletin of the Institute of Combinatorics and its Applications, Vol. 25 (1999), 79-87.
This paper came from my Ph.D. dissertation. It was the first paper on defective list colorings, and as a result it's one of my most cited papers, making me wonder if I should have stayed in this area!

• A Note on "Impossible" Paper Folding, T. Hull, The American Mathematical Monthly, Vol. 103, No. 3 (1996), 242-243.
I also wrote this one while in grad school, so please forgive the rather flippant tone.

• Unit Origami as Graph Theory, in COET95: Proceedings of the 2nd International Conference on Origami in Education and Therapy, V'Ann Cornelius, ed., Origami USA, New York (1995), 39-47.
This was another grad school paper where I tried to describe some of the connections between (and the usefulness of) modular origami and graph theory. A shorter version of this paper appeared as "Planar Graphs and Modular Origami" in Origami Science and Art: the 2OSM Proceedings, which was published in 1997.

• On the Mathematics of Flat Origamis, T. Hull, Congressus Numerantium, Vol 100 (1994), 215-224.
I wrote this one while in grad school. Please ignore my attempts to define "origami" rigorously, as it is flawed. The other proofs and such are fine though.

## Presentation Slides

• Counting and 3-Edge Coloring Spherical Buckyballs, presented at the Joint Meetings of the AMS and the MAA, San Antonio, TX, Jan. 12-15, 2006.
• Origami and Constructible Numbers talk, which I've given versions of at number of places, including Erik Demaine's Computational Origami class at MIT, some regional MAA Section meetings, and the 2008 Joint Meetings in San Diego. It talks about Margherita Beloch's proof that origami can solve cubic equations and Lill's method for finding real roots of polynomials.