SAN JOSE, CALIFORNIA—How can you visualize a 4D object in our 3D world? The answer involves some tricky projections and a 3D printer. Mathematician and artist Henry Segerman of Oklahoma State University, Stillwater, described his method for representing a hypercube—a 4D cube—today at the annual meeting of AAAS (which publishes Science). Visualizing a 4D object is a mind-bending task, so to understand how Segerman does it, it helps to imagine a person who lives in a 2D “flatland,” or a plane with no thickness (think Nintendo character Mario in the side-scrolling games of the 1980s, unable to escape his TV set). How might one explain to a flatlander what a cube is? One method you could use is to shine a light above the cube, projecting a shadow onto a 2D surface. This is essentially what Segerman does, although he goes a bit further. He uses a stereographic projection, first projecting the cube onto the surface of a sphere, and then casting its shadows onto a plane. Like a Mercator projection, commonly used to make maps of Earth, a stereographic projection is a method for representing a sphere on a plane. The 3D-printed sculpture pictured above shows how a stereographic projection translates from a sphere to a plane—the curves on the sphere become straight lines on the plane. Using the same projection with a cube would illuminate the concept of a cube for a flatlander. Now imagine doing this with a hypercube instead of a cube, projecting it into three dimensions, and 3D printing the result. That’s how you make a hypercube.
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