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What does origami have in common with electronics? Here, math once again proves to be a universal language, spanning not just cultures but disciplines.
The discovery of the mathematical underpinnings of folded paper art helped Robert Lang leave a 20-year engineering career, including over four years at NASA’s Jet Propulsion Laboratory in Southern California, to pursue his lifelong passion for turning paper into impossibly intricate three-dimensional forms.
“Over the years of solving mathematical problems to describe lasers and optoelectronics, I built up a toolkit to use as I worked on a hobby basis on this problem of computational origami design,” said Lang. The Altadena, California-based artist holds dozens of patents for optoelectronics — technology that combines light and electricity — but after years of innovating in both fields, the tools he designed for origami are the ones he chose to move ahead with.
In the Microdevices Laboratory at JPL in the late 1980s and early ’90s, Lang worked on integrating components like semiconductor lasers and spatial light modulators onto chips, with the ultimate goal of building an optical computer — one that uses light, rather than electricity, to transmit information and carry out calculations.
Steady advances in electronic computing have since removed some of the incentives to develop optical computers.
“One of the theoretical fields I learned about at JPL turned out to be the key to being able to plug in a description of a shape you wanted and then find the best possible design in great detail — every single crease you needed to make that shape,” said Lang. “And that turned out to be nonlinear constrained optimization.”
It’s All About the Numbers
A simple nonlinear constrained optimization problem would be the challenge of packing several different-sized balls into the smallest possible box, Lang explained. The constraint is that the balls can’t overlap each other, and the solutions are nonlinear because the balls can be any distance from each other. The optimization is in making the box as small as possible.
Designing lasers and other components required a similar calculation to minimize energy consumption, the amount of semiconductor material, and other costs, said Lang. In origami, he said, optimization means creating the largest form possible out of a given sheet of paper. Design begins with mapping the points on that sheet that will become features like a head and limbs. “I found there was an equation that said the distance between any of those two points had to be greater than or equal to a mathematical function that related to where they were in the shape I was after,” Lang said. “And that was really the breakthrough, was figuring out how to mathematically describe that constraint for every possible pair of points in the crease patterns.”
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