The mathematics of how diffraction works to separate white light into component colors is a well-studied, predictable phenomenon. But how some iridescent butterfly scales keep showing brilliant iridescent colors when viewed from different angles has been a long-standing mystery. The answer to that question is revealed in a paper by Saranathan et al. (2010), who found that butterflies, through millions of years of selection, evolved a geometric nanostructure within their scale cells, previously known only to mathematicians. The first three-dimensional picture of the arrangement was computer generated by NASA materials scientist A. H. Schoen in 1970. He named the almost impossible-to-describe geometric shape a gyroid. Probably the best way to introduce students to the complexities of a gyroid is to let them climb through one – at least the educators at San Francisco’s Exploratorium think so. They built a 10-foot-high gyroid in the institute’s outdoor Geometry Playground,...

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