Art-inspired Mathematics and Technology

Art and artists have inspired mathematics and science for aeons, usually by prefiguring many ideas and artifacts at an intuitive level long before we realized that there was even a scientific question to which art had already provided an answer. This can be seen in many instances in perspective painting, chiarascuro,  trompe l’oeil, origami and its cousins, and a range of crafts such as glass-blowing, weaving, knitting, crocheting etc.   We have thought about these questions from a perceptual and predictive perspective, with a particular interest in the paper arts of origami and kirigami.

Origami is an art form that probably originated with the invention of paper in China, but was refined in Japan. Kirigami is a related cousin of origami that replaces folds by cuts in a thin sheet and enables even more remarkable shapes to be generated. Artists have an intuitive way to work with these forms. A natural challenge is to ask if mathematics help quantify aspects of the design principles behind these art forms? And can engineering help realize and (mass) produce the shapes?

The ability to create complex origami and kirigami structures depends on folding (cutting) thin sheets along creases, a natural consequence of the large-scale separation between the thickness and the size of the sheet. This allows origami (kirigami) patterns to be scaled; the same pattern can be used to design human architectures and graphene gadgets. But how can we design the number, size and location of folds (cuts) on a sheet of paper that will enable us to fold (cut) it into a given shape?

Inspired by the mathematical analogies between the two art forms that manipulate geometrical and topological degrees of freedom, we have posed the inverse problem of origami and kirigami as  constrained optimization problems or evolutionary algorithms and solved them in specific instances. Some questions of continuing interest in these areas include the quest for impossibility theorems  (or their converse),  the physical behavior of these unusual auxetic meta-materials, and the mathematical relation between these art forms, the Nash-Kuiper-Gromov convex integration program and its generalizations in terms of the smoothness requirements for the class of feasible solutions, and the exploration of floppy origami and floppy kirigami as mechanical memories, conductors, diodes, amplifiers, and perhaps even computers.

Other work includes trying to rationalize the paintings of Jackson Pollock from a fluid-dynamical perspective,  quantifying the blooming of a flower (a perennial inspiration for poets and artists), the dynamics of writing with ink etc.

Related Publications

Self-organized origami, L. Mahadevan and S. Rica,  Science , 307, 1740, 2005. [View PDF] [Download PDF]
Geometric mechanics of curved crease origami M.A. Dias, L.H. Dudte, L. Mahadevan, C.D. Santangelo,  Physical Review Letters , 109, 114301, 2012. [View PDF] [Download PDF]
Geometric mechanics of periodic pleated origami Z.Y. Wei, Z.V. Guo, L. Dudte, H.Y. Liang, and L. Mahadevan, Physical Review Letters, 110, 215501, 2013 . [View PDF] [Download PDF]
Programming shape using kirigami tessellations G. P. T. Choi, L. H. Dudte and L. MahadevanNature Materials 18, 999–1004. 2019. [DOI] [View PDF] [Download PDF]
Programming curvature using origami tessellations L.H. Dudte, E. Vouga, T. Tachi and L. Mahadevan,  Nature Materials , 15, 583-88, 2016. [View PDF] [Download PDF]
Deterministic and stochastic control of kirigami topology Siheng Chen, Gary P. T. Choi, L. Mahadevan, PNAS (2020) 117 (9) 4511-4517. [DOI] [View PDF] [Download PDF]
Topological and statistical control of kirigami S. Chen, G.P. T. Choi, L. MahadevanarXiv:1910.07090 [ONLINE ARTICLE] [View PDF] [Download PDF]
Rigidity percolation and geometric information in floppy origami S. Chen, L. Mahadevan, PNAS 116 (17) 8119-8124, 2019. [ONLINE ARTICLE] [View PDF] [Download PDF]
Hydrodynamics of writing with ink
J. Kim, M-W. Moon, K-R. Lee, L. Mahadevan, and H-Y. Kim, Physical Review Letters,  107, 264502, 2011.  [View PDF] [Download PDF]
Painting with drops, jets, and sheets
A. Herczynski, C. Cernuschi, and L. Mahadevan,  Physics Today  June 2011, 31-36.  [View PDF] [Download PDF]
Growth, geometry and mechanics of the blooming lily H-Y. Liang and L. Mahadevan,  Proceedings of the National Academy of Sciences , 108, 5516-21, 2011. [View PDF] [Download PDF]