“We don’t know what topology the universe has, whether it’s an expanding sphere, if it’s hyperbolic, if it’s got holes in it.”įor example, gravity from massive celestial bodies bends rays of light. It describes some actual physics, and is yet another example that there’s more to reality than meets the eye. Hyperbolic geometry isn’t just hypothetical. And it gives the VR wearer a visual output that is Euclidean, i.e. Matsumoto and Segerman’s new virtual reality program detects head motions in 3D Euclidean space and warps them into virtual movement in 3D hyperbolic space. And when you’ve warped the plane, it warps all of space at the same time. That warp changes principles: Parallel lines curve away from each other triangles have warped lines, and there’s no such thing as a rectangle as we know it. Since we usually live in a Euclidean reality, the warped plane would now look to us like a three-dimensional object, but it’s still a plane, so it’s really two-dimensional. If you warp a Euclidean plane like a Pringle’s potato chip, giving it hyperbolic curves, you get an idea of hyperbolic geometry. Why close-up photos make us seem ‘warped’ We recognize this geometry when we look at buildings, desks, or coffee cups. There are triangles, rectangles, circles, spheres, cubes, etc. The basics are: Points, straight lines, angles, and planes that are flat and extend infinitely. Some 2,300 years ago, mathematician Euclid of Alexandria developed the geometry commonly taught today in high school. That’s fiction, but it makes for a nice bridge to hyperbolic geometry and how this new VR program takes viewers hyperbolic from the much more customary Euclidean geometry experience of everyday life. Meanwhile, inside the ship, everything is shaped and moves “normally.” Sci-fi fans may remember hyperspace, created when “warp drive” engines curve spacetime so that the Starship Enterprise can travel at multiples of the speed of light. Most people have never consciously seen hyperbolic geometry, as opposed to Euclidean geometry, which is how we usually experience the world. The researchers have posted papers ( first, second) on the math and perceptual considerations behind their work on arXiv. “The virtual reality takes something that would normally live in a set of equations, and makes something you can interact with.” “Visualizations can help to prove theorems that are purely abstract, and physicists want to get an intuition for what’s going on,” says Matsumoto, an assistant professor in Georgia Tech’s School of Physics. ![]() Segerman and Matsumoto collaborated on the hyperbolic virtual reality experience with a collective of mathematician-artists called eleVR to make the work of the geometry experts easier and more productive. That weirdness can give the non-mathematician an idea of how picturing non-Euclidean geometries mentally can strain even the minds of mathematicians and physicists. ![]() “It never stops, just keeps going, and you never get to the back side of it.” ![]() He slides around a diamond-like shape in VR hyperbolic space, describing it. “If you walk around in this space, things that started out horizontal and vertical become twisted and weird,” Segerman says, as he dons a VR headset. Or you can peruse it on a computer in 2D using the arrow keys.īut be a little careful walking around the 3D version, as the hyperbolic space doesn’t have a floor to provide visual balance orientation, and turning corners is very different from in everyday life. You can navigate it with your VR headset or smart phone via a webVR interface. If you’d like a peek at the warped rainbow weirdness yourself, go to h3. When physicist and applied mathematician Sabetta Matsumoto of Georgia Institute of Technology or her collaborator, mathematician Henry Segerman from Oklahoma State University, do that, they’re actually exploring particular geometric nooks. The virtual space’s colorful graphics can entice even the most math-phobic mind to roam, crawl, or slither around. The program is a visual aid for researchers exploring geometries that deviate from the everyday norm. ![]() Math meets “warp drive” in a virtual reality headset that transports anyone who wears the visor into a reality twisted by hyperbolic geometry. University Georgia Institute of Technology
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