The artist M. C. Escher was well known for his drawings that showed perspectives defying the logic of geometry as it is usually experienced. His woodcut work called “Circle Limit IV”, is one of a series showing hyperbolic planes.
A new virtual reality (VR) headset transports people to a reality twisted by hyperbolic geometry. This is a typical example where Math and technology meet. This combination is used extensively in animation and there are animation schools that teach their students how to master this art.
Sabetta Matsumoto, an applied mathematician and physicist at the Georgia Institute of Technology, co-created the VR headset program as a visual tool to help researchers explore geometries that differ from the norm seen every day.
The virtual space’s graphics fills your world with color and can tempt people with the most math-oriented minds to crawl, roam, or slither about. Matsumoto collaborated with mathematician Henry Segerman from Oklahoma State University. When either of them do weird looking physical activities, they’re actually investigating specific geometric niches.
People normally experience the world through Euclidean geometry and seldom consciously see hyperbolic geometry, which is what the VR headset depicts.
If you would like to see more of the warped rainbow weirdness yourself, go to Hypernom. Although you can navigate it on a computer in 2D using the arrow keys, it’s much more fun to experience it with your smart phone or VR headset via a webVR interface. Be warned though – when walking around the 3D version, turning corners is very different from what we do in everyday life. The hyperbolic space does not provide visual balance orientation, as it doesn’t have a floor.
That simulation will give any non-mathematician an idea of how the minds of physicists and mathematicians are strained when they mentally picture non-Euclidean geometries.
Matsumoto and Segerman collaborated with a collective of mathematician-artists, called eleVR, on the hyperbolic virtual reality experience. This made the work of the geometry experts more productive and easier.
Matsumoto, an assistant professor in Georgia Tech’s School of Physics noted that visualizations generally prove invaluable in proving theorems that are purely abstract, as it helps physicists get a feel for what’s going on. The virtual reality makes something you can interact with from what would normally only live in a set of equations.
EUCLID VS. WARP DRIVE ENGINES
Sci-fi fans will know that the Starship Enterprise was able to travel at multiples of the speed of light when its “warp drive” engines curved space-time. When it did so, everything inside the ship was shaped and moved “normally.”
Although what we’re describing is fiction, it makes for a convenient bridge to hyperbolic geometry, as it explains how this new VR program takes viewers from the Euclidean geometry experience of everyday life, to the hyperbolic plane.
The geometry taught in high school today was developed by the mathematician Euclid of Alexandria about 2,300 years ago. The basics consists of flat points, straight lines, angles, and planes that extend infinitely. Within this are rectangles, triangles, spheres, circles, cubes, etc. This geometry is recognizable when we look around us, be it at buildings, chairs or water glasses.
If you think of curving (or warping) a Euclidean plane like a Pringle’s potato chip to give it hyperbolic curves, hyperbolic geometry starts making sense. As we live in a Euclidean reality, the warped plane would look like a three-dimensional object to us, although it is still a plane, so it stays two-dimensional.
Principles change when we warp things. A rectangle as we know it does not exist, triangles have warped lines and parallel lines bend away from each other. Furthermore, when the plane is warped, all of space is warped at the same time.
The new VR program senses head motions in 3D Euclidean space and warps them into virtual movement in 3D hyperbolic space. It does however give the wearer a visual output that is “normal”, i.e. Euclidean.
AIRPLANES AND RELATIVITY
Hyperbolic geometry describes some actual physics and is a prime example that there is often more to reality than meets the eye.
Gravity from massive heavenly bodies bends light rays. As a result, hyperbolic geometry helped in formulizing the Theory of Relativity, which describes realities of space-time outside the sphere of human perception, but not completely out of perceptual reach.
Thanks to air travel, people already often experience a geometry called spherical geometry. When one stands on the ground, the surface is experienced as level. That is why our ancestors believed that the Earth was flat. When one looks out the window of an airplane however, it may feel as if one is moving over a Euclidean plane, while one is in fact over a spherical plane. That has geometric consequences.
Matsumoto explained that if a journey starts at the North Pole and takes any route south, then turns 90 degrees left and goes a quarter of the way around the world, then turns 90 degrees left again and goes the same distance, you’ll arrive back at your starting point. The same would not happen if the journey were done on a Euclidean plane.
THE UNIVERSE’S SHAPE
The three dimensions we perceive can be broken up into eight different geometries. This helps to unlock physical realities even more mind-blowing than the Earth being round, or Relativity.
Matsumoto noted that the topology of the universe is unknown. We don’t know if it has holes in it, is hyperbolic, or an expanding sphere.
In the late 1800s, the math of hyperbolic geometry was firmed up. Decades later, artist M. C. Escher demonstrated how warped geometry could reveal twisted reality. His drawings are renowned for conflicting perspectives flawlessly married in a single motif, like odd aqueducts or stairways.
Escher’s circles with animals that interlace perfectly is a representation of a Poincaré disk. This disk depicts a hyperbolic plane in 2D, with same-size repeating shapes appearing large in the middle and increasingly smaller toward the circle’s edges. This represents the hyperbolic warp stretching to infinity. Such shapes are called “tiling.”
Tessellation is the repetitive tiling of an area to express its characteristics. Tessellation has become a widely used tool in geometric illustration. In Segerman and Matsumoto’s VR, animated 3D tessellation conveys hyperbolic space.
Other notable mathematical artists use 3D or animated tessellation today. Jeff Weeks inspired Matsumoto and Segerman by developing three-dimensional hyperbolic space visualization software, while Jos Leys uses attractive computer graphics.
Matsumoto also has a passion for art inherent in geometry going back to childhood. That was when she first developed the instincts she wanted to convey with the VR program.
She explained that her mom always knitted and sewed while she loved math. This resulted in her knowing how to piece together parts of a dress from a very early age. She only later realized that these things are actually complex geometry.
Matsumoto ultimately wants to make the leap into a weird “actual reality” experience of hyperbolic geometry. She foresees that she will end up making a museum installation where everything is hyperbolic and people can walk through it doing things like playing pool or basketball.