Artist M. C. Escher was famous for drawings with perspectives that defied the logic of geometry as we usually experience it. He dedicated a series of woodcuts to depictions of hyperbolic planes. This one is titled “Circle Limit IV.” . Credit: The M.C. Escher Company – the Netherlands. All rights reserved. Used by permission. www.mcescher.com
Math just met “warp drive” in a virtual reality headset to transport anyone who dons the visor to a reality twisted by hyperbolic geometry. The program was co-created by Sabetta Matsumoto, a physicist and applied mathematician at the Georgia Institute of Technology as a visual aid to researchers exploring geometries that deviate from the everyday norm.
Splashed in color, the virtual space’s graphics can seduce even the most math-phobic mind to roam, crawl or slither about. When Matsumoto or her collaborator, mathematician Henry Segerman from Oklahoma State University, do that, they’re actually exploring particular geometric nooks.
“If you walk around in this space, things that started out horizontal and vertical become twisted and weird,” Segerman said, as he donned a VR headset. He slid around a diamond-like shape in VR hyperbolic space, describing it. “It never stops, just keeps going, and you never get to the back side of it.”
Most people have never consciously seen hyperbolic geometry, as opposed to Euclidean geometry, which is how we usually experience the world. We’ll go into the difference between them in the next section.
In the meantime, if you’d like a peek at the warped rainbow weirdness yourself, go here: h3.hypernom.com. You can navigate it with your VR headset or smart phone via a webVR interface. Or you can peruse it on a computer in 2D using the arrow keys.
But 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.
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. 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.
“Visualizations can help to prove theorems that are purely abstract, and physicists want to get an intuition for what’s going on,” said Matsumoto, an assistant professor in Georgia Tech’s School of Physics. “The virtual reality takes something that would normally live in a set of equations, and makes something you can interact with.”
EUCLID VS. STAR TREK
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. Meanwhile, inside the ship, everything is shaped and moves “normally.”
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.
Some 2,300 years ago, mathematician Euclid of Alexandria developed the geometry commonly taught today in high school. The basics are: Points, straight lines, angles, and planes that are flat and extend infinitely. There are triangles, rectangles, circles, spheres, cubes, etc.
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Credit: Georgia Institute of Technology
Mathematicians can break up the three dimensions we perceive into eight different geometries, which could help unlock more physical realities even more mind-bending than Relativity or as simple as the Earth being round.
“We don’t know what topology the universe has,” Matsumoto said, “whether it’s an expanding sphere, if it’s hyperbolic, if it’s got holes in it.”
A few decades after the math of hyperbolic geometry was firmed up in the late 1800s, pioneering artist M. C. Escher illustrated how warped geometry can reveal twisted reality. His drawings are famous for contradictory perspectives seamlessly married in a single motif, like odd stairways or aqueducts.
His circles with animals that interlace perfectly were a take on a Poincaré disk, a depiction of a hyperbolic plane (2D), in which repeating shapes, all the same size, appear large in the middle and increasingly smaller toward the circle’s edges to represent the hyperbolic warp stretching to infinity.
Such shapes are called “tiling.” Repetitive tiling of an area to convey its characteristics is called “tessellation,” which has become a common tool in geometric illustration. Animated 3D tessellation conveys hyperbolic space in Matsumoto and Segerman’s virtual reality.
Other notable mathematical artists employing animated or 3D tessellation today are Jos Leys, who uses attractive computer graphics, and Jeff Weeks, who developed a three-dimensional hyperbolic space visualization software, which was an inspiration to Matsumoto and Segerman.
IN ESCHER’S FOOTSTEPS
Matsumoto’s own passion for art inherent in geometry goes back to childhood, when she first developed tangible instincts like those she wants to convey with the VR program.
“I’ve always loved math, and my mom always knitted and sewed. So, from a very early age, I knew how to piece together parts of a dress. It turns out these things are really, actually complex geometry.”
In the future, Matsumoto, like Alice in Wonderland, a metaphor Matsumoto employs herself, wants to make the leap into a weird “actual reality” experience of hyperbolic geometry.
“The end game of this is going to be to make a museum installation where people can walk through, like a house where everything is hyperbolic, and you can do things like play basketball or play pool.”
The research team has also created a virtual reality experience combining hyperbolic and Euclidean geometry (H2 X E), which can be found at: h2xe.hypernom.com. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of any sponsoring agencies.
Source: Phys