Feb. 17, 2015 | By Kira

The recent advancements in 3D technology, from in-your-face movie screenings to the 3D printing stories that we report on everyday, certainly endow the third-dimension with a sense of novelty and the promise of the future. However, the truth is that we have always lived in a three-dimensional environment; we are hard-wired to see and imagine things in 3D. The concept of the fourth-dimension (4D), on the other hand, still eludes even the most qualified mathematicians, even after two centuries of research.

Although we cannot physically create 4D objects in our given universe, artist and mathematician Henry Segerman of the Oklahoma State University in Stillwater has devised a creative way to show us what the shadows of a 4D cube—also known as a hypercube or Tesseract—would look like.

Segerman unveiled his method last weekend at the annual meeting of AAAS (the American Association for the Advancement of Science) in San Jose, California. During the presentation, available in an hour-long clip in YouTube, he compares us trying to visualize a 4D object to a person living in a 2D ‘flatland’ trying to imagine a cube—it would be difficult, but not impossible. One way to explain the concept of a cube would be to shine a light above it in order to project its shadow. Segerman explains: “just as the shadow of a three-dimensional object squishes it into a two-dimensional plane, we can squish a four-dimensional shape into three-dimensional space.”

The process of ‘squishing’ these shadows utilizes a concept similar to stereographic projection, whereby light hits a three-dimensional object (such as a sphere) and projects its image onto a flat surface (such as a wall). Stereographic projection has been used for centuries to make maps of the Earth and sky.

With 3D objects, stereographic projection is pretty straightforward. What Segerman did was to take it a step further by 3D printing sculptures that represent a 4D hypercube in order to project their shadows. Although the shapes of the sculpture are necessarily distorted (they appear to grow in size as they move outward from the middle), it’s the closest we can get to actually visualizing 4D objects.

Segerman’s sculpture, titled More Fun than a Hypercube of Monkeys,” shows the projections of monkeys hooked up in a hypercube shape and may take several minutes to wrap your mind around (if you manage to at all), however in the mathematical world, radial projection moves the monkeys onto the 3-sphere (the unit sphere in 4D space), then stereographic projection moves the monkeys to 3D space. According to Segerman, this may be the first sculpture ever with the quaternion group as its symmetry group.

If you’ve got an hour and truly want to grasp the math and science behind Segerman’s project, be sure to watch the video of his presentation, or read this detailed explanation on the Scientific American’s Roots of Unity blog. Or, if you’d rather get your hands dirty and experiment with some mathematical 3D prints of your own, check out Segerman’s Shapeways and Thingiverse pages, which offer several models that beautifully combine complex mathematics with eye-grabbing art.



Posted in 3D Printing Applications


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