From Quantamagazine
By Rachel Crowell
Machine Heart Compilation
Machine Heart Editorial Department
This result may help researchers answer a more important question about how to flatten objects from the fourth dimension to the third dimension.
Computer scientist Erik Demaine and his artist and computer scientist father Martin Demaine have been pushing the limits of origami for years. Their intricate origami sculptures are in the permanent collection of the Museum of Modern Art in New York. Ten years ago, PBS also aired an art documentary featuring them.
The duo began working together when Erik was 6 years old, and today Erik is a professor at MIT. "We have a company called Erik and Dad Puzzle Company that makes and sells puzzles to toy stores in Canada," he said.
Erik learned basic math and visual arts from his father, but Martin also learned advanced math and computer science from his son. "Now we're all artists and mathematicians/computer scientists," Erik says, "and we've collaborated on a lot of projects, especially those that span a lot of disciplines."
Their latest work is a mathematical proof published in the journal Computational Geometry last October.
Thesis link: https://www.sciencedirect.com/science/article/abs/pii/S0925772121000298
In the paper, titled "Continuous Flattening of All Polyhedra Manifolds Using Innumerable Creases," Erik et al. say they demonstrated that if the standard folding model is extended to allow for the appearance of innumerable creases, any finite polyhedra manifold in 3D can be continuously flattened to 2D while preserving the inherent distance and avoiding crossover.
This result answers a question posed by the Demaine father and son and Erik phD supervisor Anna Lubiw in 2001. They wondered if it was possible to take any finite polyhedra (or flat-sided) shapes (such as cubes, rather than spheres or infinitely large planes) and flatten them with creases.
Of course, you can't cut or tear shapes. In addition, the inherent distance of the shape must be maintained, "that is, 'You can't stretch or shrink this material,'" Erik says. And he notes that this type of folding also has to avoid crossovering, which means "we don't want paper to pass through itself" because that wouldn't happen in the real world. "When everything moves continuously in 3D, meeting these limits can be very challenging." Taken together, these constraints mean that simply squeezing the shape won't work.
Research by Erik & Sons et al. shows that you can accomplish this kind of folding, but only if you use an infinite folding strategy. But before that, several authors proposed another practical technique in a paper published in 2015.
Thesis link: https://erikdemaine.org/papers/FlatteningOrthogonal_JCDCGG2015full/paper.pdf
In this paper, they studied a simpler class of folding problems for shapes: orthogonal polyhedra whose faces intersect at right angles and are perpendicular to at least one of the x-, y-, and z-axes. Meeting these conditions forces the face of the shape to be rectangular, which makes folding much simpler, just like folding a refrigerator box.
"This situation is easier to calculate because every corner looks the same. It's just two faces intersecting vertically." Erik said.
After its success in 2015, researchers began using this flattening technique to process all the finite polyhedra. However, the faces of a non-orthogonal polyhedron may be triangular or trapezoidal, and the crease strategy that applies to refrigerator boxes does not apply to pyramids. And for non-orthogonal polyhedra, any finite number of creases always produces some creases that intersect at the same vertex.
So Erik et al. considered using other methods to circumvent this problem. After some exploration, they found a way to solve the problem of flattening non-convex objects — the cube lattice, which is a three-dimensional infinite mesh. At each vertex of the cube lattice, many faces intersect and share an edge, which makes it very difficult to achieve flattening at any one vertex.
But the researchers eventually found a solution. First, they find a point that is "far away from the vertex" and can be flattened, and then they find another point that can be flattened, repeating the process over and over again, getting closer to the problematic vertex, and flattening more positions as they move.
This process needs to be continuous, and once it is interrupted, there will be more problems to solve. Jason Ku, one of the authors of the paper, said: "Near the apex in question, using the method of making the slices smaller and smaller will be able to flatten each slice."
"In this case, the slice is not an actual cut, but a conceptual slice used to imagine breaking the shape into smaller pieces and flattening it. Then we conceptually 'glue' these little slices together to get the original surface." Erik Demaine said.
The researchers applied the same method to all non-orthogonal polyhedra. By migrating from finite "concept" slices to infinite "concept" slices, they created a program based on the idea of mathematical limits, got an unfolded plane, and solved the initial problem.
Joseph O'Rourke, a computer scientist and mathematician at Smith College in the United States, praised: "I never thought of using infinite creases, they changed the standards that make up the solution in a very clever way."
Erik Demaine attempts to apply this infinite folding method to more abstract shapes. O'Rourke recently suggested using this method to flatten four-dimensional objects into three dimensions. Meanwhile, Erik Demaine says they still want to explore whether polyhedra can be flattened with a limited number of creases, optimistically believing it's possible.
A prodigy playing with origami on a computer
It is not an exaggeration to say that Erik Demiane is a child prodigy. He went to Canada at the age of 12 and graduated early with a bachelor's degree at the age of 14. He taught at MIT at the age of 20, became a professor at the age of 21, published his doctoral dissertation at the University of Waterloo at the age of 23, and received the Canadian "Governor-General Gold Medal" and the NSERC Doctoral Fellowship, and received the MacArthur Award in the same year.
Before the age of 12, Erik was at home taught cultural knowledge by his father, Martin Demaine. Although Martin only had a high school degree, he was a well-known artist and mathematician.
Erik's main research interests are origami algorithms and computational theory, and he now teaches at MIT with his father, Martin. They do a lot of algorithmic simulations in computers, simulate the process of origami, and design real-world origami artwork based on this. At the same time, by creating origami artworks, Erik Demiane is able to invert the improved algorithm, which in turn further stimulates origami art creation, thus forming a real-virtual, algorithmic-art cycle.
Reference links: https://www.quantamagazine.org/father-son-team-solves-geometry-problem-with-infinite-folds-20220404/
https://www.x-mol.com/paper/1386871662666866688/t
http://www.archcollege.com/mobile.php?m=index&a=appDetails&id=28655