Erik and Martin Demaine on the Mathematics of Curved-Crease Sculpture

Erik Demaine is an MIT computer science professor; his father Martin Demaine is an MIT researcher and artist in residence. Erik brings a background in mathematics and computing, Martin one in art and making. Together they describe themselves as both origami theorists and artists: they study how sheets fold and make intriguing sculptures from paper.

When the Demaines began studying origami mathematics, they were advised that it was a niche recreational subject and not a promising research area. They ignored the advice because the questions were too compelling. The mathematics of their curved-crease forms remains an active subject, with foundational existence questions still unproved.

Art and mathematics inspire each other in their work. Trying to build a sculpture can raise a mathematical question about whether a shape exists or how to generalize it. A mathematical result, in turn, can become a physical sculpture that people can appreciate beyond a screen.

Their process is often improvisational. One might add a few pieces and then the other adds more; the sculpture grows as they respond to each other and to the material. They may have several projects in progress at once, moving to another when stuck. Some remain unfinished, but still produce artistic or mathematical insights.

The Demaines credit their collaboration to communication that is honest and humorous, as well as deep trust. They have written hundreds of papers and made hundreds of sculptures together. Martin describes Erik as his best friend and their work as a shared undertaking.

They spend much of their time brainstorming what a piece should be. The process starts with a design on a sheet of paper, which is scored and folded; they repeat that to make multiple pieces and then assemble them into a three-dimensional sculpture. The assembly is improvised, and its outcome is not fully known in advance.

They use technology for much of the design, but insist on assembling the sculptures by hand. As with building blocks, they make adjustments while working and sometimes take a piece apart and start over.

For a new piece, they discuss overlapping sheets and making dissection a theme, with mathematical problems incorporated into the design. They call the sculpture Solve Me, a reference to the 'eat me' and 'drink me' prompts in Alice in Wonderland. The finished work is displayed in Jane Street's Hong Kong office.

In many of their sculptures, a viewer can count the pieces. Solve Me is designed to make that difficult: the twisting and weaving patterns lead the eye through an intricate whole. The artists want viewers to wonder how it was made and whether it really is paper.

The printed text hides mathematical problems the artists think about, including a couple that remain unsolved. Such problems lie on the frontier between what mathematicians know how to solve and what they do not yet know.

The sculpture incorporates two mathematical fonts the Demaines designed. In the Tetris font, each letter is made from all seven Tetris pieces. In the two-piece dissection font, each letter can be divided into two sets and rearranged into a six-by-six square.

The layout of the problems also follows Tetris: bold outlines form the game pieces. One surprise for the artists was that the printing on the paper helped the piece shape itself, blending components into a complex form that makes them hard to distinguish.

The Demaines say they could not have imagined these geometries when they first began folding paper. Building Solve Me generated new puzzles and mathematical problems that may guide future research. For them, the attraction of mathematics is how much remains unknown and open to discovery.