Thomas C HullAssociate Professor of Applied Mathematics
Spotlight
F&M Math Professor Theorizes an Origami Computer
Recent research by Associate Professor of Applied Mathematics Thomas Hull was the subject of a feature article in Quanta Magazine. The story, titled covers a collaboration between Hull and mathematics professor Inna Zakharevich from Cornell University where they proved that origami, the historically Japanese art of paper folding, is "Turing complete." This means that, in theory, one could make a functioning modern computer by folding a piece of paper.As the article states, "An origami computer would be massively inefficient and impractical. But in principle, if you had a very large piece of paper and lots of time on your hands, you could use origami to calculate arbitrarily many digits of 蟺, determine the optimal way to route every delivery driver in the world, or run a program to predict the weather."
Quanta Magazine is an award-winning online scientific news publication run by the Simons Foundation that is highly read in math, science, and education circles.
In addition to being an art form practiced both seriously and recreationally all over the world, origami is being studied by scientists, mathematicians, and engineers for applications in robotics, materials, and architecture. Hull currently has an NSF grant to work on origami-math which he uses to fund research with F&M students. He states, "I'm honored to have our work featured in Quanta Magazine and appreciate the attention it brings to the excellent work done by the F&M Mathematics Department." (Image credit: Kristina Armitage/Quanta Magazine)
Biography
Associate Professor of Applied Mathematics, Department of Mathematics, Franklin & Marshall College, Lancaster, PA, 2023-present
Project Associate Professor, Graduate of Arts & Sciences, University of Tokyo, Tokyo, Japan, 2015
Associate Professor, Department of Mathematics, Western New England University, Springfield, MA, 2008-2023
Visiting Assistant Professor, Department of Mathematics, University of Cincinnati, Cincinnati, OH, 2002-2003
Assistant and Associate Professor, Department of Mathematics, Merrimack College, North Andover, MA 1997-2008
Education
Ph.D. in Mathematics, University of Rhode Island, Kingston, RI 1997
M.S. in Mathematics, University of Rhode Island, Kingston, RI 1992
B.A. in Mathematics and Philosophy, Hampshire College, MA 1991
More About Professor Hull
Books
Origametry: Mathematical Methods in Paper Folding, Cambridge University Press, Cambridge UK (2020).
Project Origami: Activities for Exploring Mathematics 2nd edition, CRC Press/AK Peters, Wellesley, MA (2012).
Project Origami: Activities for Exploring Mathematics 2nd edition, CRC Press/AK Peters, Wellesley, MA (2012).
Publications
鈥 A rigid origami elliptic-hyperbolic vertex duality, Results in Mathematics, Vol. 80, No. 5, Article 138, 2025 (13 pages).
鈥 The stamp folding problem from a mountain-malley perspective, with Adham Ibrahim, Jacob Paltrowitz, Natalya Ter-Saakov, and Grace Wang, Discrete Mathematics & Theoretical Computer Science, October 8, 2025, Vol. 27, No. 3 (15 pages).
鈥 Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean, with R. Foschi and J. Ku, Physical Review E, Vol. 106, No. 5 (2022), 055001鈥055011.
鈥 Maximal origami flip graphs of flat-foldable vertices: properties and algorithms, with Manuel Morales, Sarah Nash, and Natalya Ter-Saakov, Journal of Graph Algorithms and Applications, Vol. 26, No. 4 (2022), 503鈥517.
鈥&苍产蝉辫; Rigid folding equations of degree-6 origami vertices, with Johnna Farnham and Aubrey Rumbolt, Proceedings of the Royal Society A, Vol. 478, No. 2260 (2022), 20220051 (19 pages).
鈥&苍产蝉辫; Folding points to a point and lines to a line, with H. Akitaya, B. Ballinger, E. Demaine, and C. Schmidt, in He, Sheehy eds., Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021), 271鈥278.
鈥 Counting locally flat-foldable origami configurations via 3-coloring graphs, with A. Chiu, W. Hoganson, and S. Wu, Graphs and Combinatorics, Vol. 37, No. 1 (2021), 241鈥261.
鈥 Face flips in origami tessellations, with H. Akitaya, V. Dujmovi膰, D. Eppstein, K. Jain, and A. Lubiw, Journal of Computational Geometry, Vol. 11, No. 1 (2020), 397鈥417.
鈥 Rigid foldability is NP-hard, with H. Akitaya, E. Demaine, T. Horiyama, J. Ku, and T. Tachi, Journal of Computational Geometry, Vol. 11, No. 1 (2020), 93鈥124.
鈥 The flat vertex fold sequences, with E. Chang, Origami5: Fifth International Meeting of Origami in Science, Mathematics, and Education, Lang et al. ed., A K Peters (2011), 599鈥607.
鈥 Solving cubics with creases: the work of Beloch and Lill, American Mathematical Monthly, Vol. 118, No. 4 (2011), 307鈥315.
鈥 Configuration spaces for flat vertex folds, in Origami4: Fourth International Meeting of Origami Science, Mathematics, and Education, Lang ed., AK Peters, (2009), 361鈥370.
鈥 Folding regular heptagons, in Homage to a Pied Puzzler, Pegg et al., ed., AK Peters (2009), 181鈥191.
鈥 The stamp folding problem from a mountain-malley perspective, with Adham Ibrahim, Jacob Paltrowitz, Natalya Ter-Saakov, and Grace Wang, Discrete Mathematics & Theoretical Computer Science, October 8, 2025, Vol. 27, No. 3 (15 pages).
鈥 Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean, with R. Foschi and J. Ku, Physical Review E, Vol. 106, No. 5 (2022), 055001鈥055011.
鈥 Maximal origami flip graphs of flat-foldable vertices: properties and algorithms, with Manuel Morales, Sarah Nash, and Natalya Ter-Saakov, Journal of Graph Algorithms and Applications, Vol. 26, No. 4 (2022), 503鈥517.
鈥&苍产蝉辫; Rigid folding equations of degree-6 origami vertices, with Johnna Farnham and Aubrey Rumbolt, Proceedings of the Royal Society A, Vol. 478, No. 2260 (2022), 20220051 (19 pages).
鈥&苍产蝉辫; Folding points to a point and lines to a line, with H. Akitaya, B. Ballinger, E. Demaine, and C. Schmidt, in He, Sheehy eds., Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021), 271鈥278.
鈥 Counting locally flat-foldable origami configurations via 3-coloring graphs, with A. Chiu, W. Hoganson, and S. Wu, Graphs and Combinatorics, Vol. 37, No. 1 (2021), 241鈥261.
鈥 Face flips in origami tessellations, with H. Akitaya, V. Dujmovi膰, D. Eppstein, K. Jain, and A. Lubiw, Journal of Computational Geometry, Vol. 11, No. 1 (2020), 397鈥417.
鈥 Rigid foldability is NP-hard, with H. Akitaya, E. Demaine, T. Horiyama, J. Ku, and T. Tachi, Journal of Computational Geometry, Vol. 11, No. 1 (2020), 93鈥124.
鈥 Topological kinematics of origami metamaterials, B. Liu, J. L. Silverberg, A. A.
Evans, J., C. D. Santangelo, R. J. Lang, T. C. Hull, and I. Cohen, Nature Physics, Vol. 14, 2018, 811鈥815.
鈥 Self-foldability of monohedral quadrilateral origami tessellations, with T. Tachi, Origami7: Proc. of the 7th International Meeting on Origami Science, Mathematics, and Education, Tarquin (2018), 521鈥532.
鈥 Rigid foldability of the augmented square twist, with M. Urbanski, Origami7: Proc. of the 7th International Meeting on Origami Science, Mathematics, and Education, Tarquin (2018), 533鈥543.
鈥 Self-foldability of rigid origami, with T. Tachi, ASME Journal of Mechanisms & Robotics, Vol. 9, No. 2 (2017), 021008-021017.
鈥 Box Pleating is Hard, with H. Akitaya, K. Cheung, E. Demaine, T. Horiyama, J. Ku, T. Tachi, R. Uehara, in Akiyama et al. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science, Vol. 9943, Springer (2016), 167-179.
鈥 Rigid origami vertices: conditions and forcing sets, with Z. Abel, J. Cantarella, E. Demaine, D. Eppstein, J. Ku, R. Lang, and T. Tachi, Journal of Computational Geometry, Vol. 7, No. 1 (2016), 171鈥184.
鈥 Self-foldability of monohedral quadrilateral origami tessellations, with T. Tachi, Origami7: Proc. of the 7th International Meeting on Origami Science, Mathematics, and Education, Tarquin (2018), 521鈥532.
鈥 Rigid foldability of the augmented square twist, with M. Urbanski, Origami7: Proc. of the 7th International Meeting on Origami Science, Mathematics, and Education, Tarquin (2018), 533鈥543.
鈥 Self-foldability of rigid origami, with T. Tachi, ASME Journal of Mechanisms & Robotics, Vol. 9, No. 2 (2017), 021008-021017.
鈥 Box Pleating is Hard, with H. Akitaya, K. Cheung, E. Demaine, T. Horiyama, J. Ku, T. Tachi, R. Uehara, in Akiyama et al. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science, Vol. 9943, Springer (2016), 167-179.
鈥 Rigid origami vertices: conditions and forcing sets, with Z. Abel, J. Cantarella, E. Demaine, D. Eppstein, J. Ku, R. Lang, and T. Tachi, Journal of Computational Geometry, Vol. 7, No. 1 (2016), 171鈥184.
鈥 Origami structures with a critical transition to bistability arising from hidden degrees
of freedom, 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鈥393.
鈥 Programming reversibly self-folding origami with micro-patterned photo-crosslinkable
polymer trilayers, J. Na, A. A. Evans, J. Bae, M. Chiappelli, C. D. Santangelo, R.
J. Lang, T. Hull, and R. C. Hayward, Advanced Materials, Vol. 27, No. 1 (2015), 79鈥85.
鈥 Locked rigid origami with multiple degrees of freedom, with Z. Abel and T. Tachi, in Miura et al. (eds.), Origami6: Proc. of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 131鈥138.
鈥 Rigid flattening of polyhedra with slits, with Z. Abel, R. Connelly, E. D. Demaine, M. L. Demaine, A. Lubiw, and T. Tachi, in Miura et al. (eds.), Origami6: Proc. 6th Int. Meeting on Origami Science, Mathematics, and Education, AMS (2015), 109鈥117.
鈥 Symmetric colorings of polypolyhedra, with sarah-marie belcastro, Origami6: Proc. of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 21鈥31.
鈥 Coloring connections with counting mountain-valley assignments, Origami6: Proc. of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 3鈥10.
鈥 Minimum forcing sets for Miura folding patterns, with B. Ballinger, M. Damian, D. Eppstein, R. Flatland, and J. Ginepro, ACM-SIAM Symposium on Discrete Algorithms (SODA15), (2015), 136鈥147.
鈥 Locked rigid origami with multiple degrees of freedom, with Z. Abel and T. Tachi, in Miura et al. (eds.), Origami6: Proc. of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 131鈥138.
鈥 Rigid flattening of polyhedra with slits, with Z. Abel, R. Connelly, E. D. Demaine, M. L. Demaine, A. Lubiw, and T. Tachi, in Miura et al. (eds.), Origami6: Proc. 6th Int. Meeting on Origami Science, Mathematics, and Education, AMS (2015), 109鈥117.
鈥 Symmetric colorings of polypolyhedra, with sarah-marie belcastro, Origami6: Proc. of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 21鈥31.
鈥 Coloring connections with counting mountain-valley assignments, Origami6: Proc. of the 6th International Meeting on Origami Science, Mathematics, and Education, AMS (2015), 3鈥10.
鈥 Minimum forcing sets for Miura folding patterns, with B. Ballinger, M. Damian, D. Eppstein, R. Flatland, and J. Ginepro, ACM-SIAM Symposium on Discrete Algorithms (SODA15), (2015), 136鈥147.
鈥 Using origami design principles to fold reprogrammable mechanical metamaterials, J.
L. Silverberg, A. A. Evans, L. McLeod, R. C. Hayward, T. Hull, C. D. Santangelo, and
I. Cohen, Science, Vol. 345, No. 6197 (2014), 647鈥650.
鈥 Counting Miura-ori foldings, with J. Ginepro, Journal of Integer Sequences, Vol. 17 (2014), Article 14.10.8.
鈥 Counting Miura-ori foldings, with J. Ginepro, Journal of Integer Sequences, Vol. 17 (2014), Article 14.10.8.
鈥 The flat vertex fold sequences, with E. Chang, Origami5: Fifth International Meeting of Origami in Science, Mathematics, and Education, Lang et al. ed., A K Peters (2011), 599鈥607.
鈥 Solving cubics with creases: the work of Beloch and Lill, American Mathematical Monthly, Vol. 118, No. 4 (2011), 307鈥315.
鈥 Configuration spaces for flat vertex folds, in Origami4: Fourth International Meeting of Origami Science, Mathematics, and Education, Lang ed., AK Peters, (2009), 361鈥370.
鈥 Folding regular heptagons, in Homage to a Pied Puzzler, Pegg et al., ed., AK Peters (2009), 181鈥191.
鈥 H.P. Lovecraft: a horror in higher dimensions, Math Horizons, February (2006), 10鈥12.
鈥 Counting mountain-valley assignments for flat folds, Ars Combinatorica, Vol. 67 (2003), 175鈥188.
鈥 The combinatorics of flat folds: a survey, in Origami3: Third International Meeting of Origami Science, Mathematics, and Education, Hull ed., A K Peters (2002), 29鈥38.
鈥 Counting mountain-valley assignments for flat folds, Ars Combinatorica, Vol. 67 (2003), 175鈥188.
鈥 The combinatorics of flat folds: a survey, in Origami3: Third International Meeting of Origami Science, Mathematics, and Education, Hull ed., A K Peters (2002), 29鈥38.
鈥 In search of a practical map fold, Math Horizons, February (2002), 22鈥24.
鈥 Modelling the folding of paper into three dimensions using affine transformations, with s-m. belcastro, Linear Algebra and its Applications, Vol. 348 (2002), 273鈥282.
鈥 Classifying frieze patterns without using groups, with s-m. belcastro, The College Mathematics Journal, Vol. 33, No. 2 (2002), 93鈥98.
鈥 Defective list colorings of planar graphs, with N. Eaton, Bulletin of the Institute of Combinatorics and its Applications, Vol. 25 (1997), 79鈥87.
鈥 A note on 鈥渋mpossible鈥 paperfolding, American Mathematical Monthly, Vol. 103, No. 3 (1996), 242鈥243.
鈥 On the mathematics of flat origamis, Congressus Numerantium, Vol. 100 (1994), 215鈥224.
鈥 Modelling the folding of paper into three dimensions using affine transformations, with s-m. belcastro, Linear Algebra and its Applications, Vol. 348 (2002), 273鈥282.
鈥 Classifying frieze patterns without using groups, with s-m. belcastro, The College Mathematics Journal, Vol. 33, No. 2 (2002), 93鈥98.
鈥 Defective list colorings of planar graphs, with N. Eaton, Bulletin of the Institute of Combinatorics and its Applications, Vol. 25 (1997), 79鈥87.
鈥 A note on 鈥渋mpossible鈥 paperfolding, American Mathematical Monthly, Vol. 103, No. 3 (1996), 242鈥243.
鈥 On the mathematics of flat origamis, Congressus Numerantium, Vol. 100 (1994), 215鈥224.
Grants & Awards
I am interested in, have studied, and published in almost all areas of origami-math.
My research on rigid origami is currently being supported by the National Science
Foundation,
and has been in the past,
and has been in the past,
Sampling of Courses Taught
MAT 109 Calculus I - Introduction to the basic concepts of calculus and their applications. Functions,
derivatives, and limits; exponential, logarithmic, and trigonometric functions; the
definite integral and the Fundamental Theorem of Calculus.
MAT 110 Calculus II - Techniques of integration, applications of integration, separable first-order differential equations, convergence tests for infinite series, Taylor polynomials and Taylor series.
MAT 211 Introduction to Higher Mathematics - A course designed as a transition from calculus to advanced mathematics courses. Emphasis on developing conjectures, experimentation, writing proofs, and generalization. Topics will be chosen from number theory, combinatorics and graph theory, polynomials, sequences and series, and dynamical systems, among others.
MAT 229 Linear Algebra and Differential Equations - Systems of linear equations and matrices, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, nth order linear differential equations, systems of first-order differential equations.
MAT 439 Nonlinear Dynamics - Analysis of nonlinear systems of differential equations with a focus on trajectories of solutions in the phase plane. Topics include bifurcations, limit cycles, and transition to chaos. Applications to physics, biology, economics, and other fields will be explored.
MAT 110 Calculus II - Techniques of integration, applications of integration, separable first-order differential equations, convergence tests for infinite series, Taylor polynomials and Taylor series.
MAT 211 Introduction to Higher Mathematics - A course designed as a transition from calculus to advanced mathematics courses. Emphasis on developing conjectures, experimentation, writing proofs, and generalization. Topics will be chosen from number theory, combinatorics and graph theory, polynomials, sequences and series, and dynamical systems, among others.
MAT 229 Linear Algebra and Differential Equations - Systems of linear equations and matrices, vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, nth order linear differential equations, systems of first-order differential equations.
MAT 439 Nonlinear Dynamics - Analysis of nonlinear systems of differential equations with a focus on trajectories of solutions in the phase plane. Topics include bifurcations, limit cycles, and transition to chaos. Applications to physics, biology, economics, and other fields will be explored.