![]() ![]() To have a smooth sphere you must lose the excess space, resulting in parallel lines eventually meeting, like the lines of longitude that start parallel at the equator meeting at the two poles. And if you try to make a flat map into a globe, you end up with excess paper wrinkling up as you curve it into a sphere. You can get a feel for the basics of curved spaces by imagining-or actually playing around with-pieces of paper or maps.įor instance, the surface of a globe (or any ball) is an example of a two-dimensional positively curved space. The mathematical definitions of flat and curved are similar to the day to day meaning when applied to two dimensions. In a curved space, these basics of geometry no longer hold true. In flat space, the shortest distance between two points is a straight line, and parallel lines will never intersect-no matter how long they are. ![]() “At this point, it's unforeseeable what all can be done, but I expect that it will have a lot of rich applications and a lot of cool physics.” “There are really many applications of these experiments,” says JQI postdoctoral researcher Igor Boettcher, who is the first author of the new paper. ![]() The situation isn’t precisely like Alice falling down the rabbit hole, but these experiments are an opportunity to explore a new world where surprising discoveries might be hiding behind any corner and the very meaning of turning a corner must be reconsidered. With these tools, researchers can better explore the topsy-turvy world of hyperbolic space. Their new toolbox includes what they call a “dictionary between discrete and continuous geometry” to help researchers translate experimental results into a more useful form. The research builds on Kollár’s previous experiments to simulate orderly grids in hyperbolic space by using microwave light contained on chips. In a recent paper in Physical Review A, a collaboration between the groups of Kollár and JQI Fellow Alexey Gorshkov, who is also a physicist at the National Institute of Standards and Technology and a Fellow of the Joint Center for Quantum Information and Computer Science, presented new mathematical tools to better understand simulations of hyperbolic spaces. But scientists can still mimic hyperbolic environments to explore how certain physics plays out in negatively curved space. Even a two-dimensional, physical version of a hyperbolic space is impossible to make in our normal, “flat” environment. One type of non-Euclidean geometry that is of interest is hyperbolic space-also called negatively-curved space. Physicists are interested in new physics that curved space can reveal, and non-Euclidean geometries might even help improve designs of certain technologies. Non-Euclidean geometries are so alien that they have been used in videogames and horror stories as unnatural landscapes that challenge or unsettle the audience.īut these unfamiliar geometries are much more than just distant, otherworldly abstractions. These environments overturn core assumptions of normal navigation and can be impossible to accurately visualize. In such a world, four equal-length roads that are all connected by right turns at right angles might fail to form a square block that returns you to your initial intersection. Or it could expand so that they forever grow further apart. Space might contract so that straight, parallel lines draw together instead of rigidly maintaining a fixed spacing. If you could explore non-Euclidean environments, you would find perplexing landscapes. Spaces that have different geometric rules than those we usually take for granted are called non-Euclidean. “But, any place where there's actually a laboratory is very weakly curved because if you were to go to one of these places where gravity is strong, it would just tear the lab apart.” “We know from general relativity that the universe itself is curved in various places,” says JQI Fellow Alicia Kollár, who is also a professor of physics at the University of Maryland (UMD). But studying how physics plays out in a curved space is challenging: Just like in real estate, location is everything. And in curved space, normal ideas of geometry and straight lines break down, creating a chance to explore an unfamiliar landscape governed by new rules. Thanks to Einstein, we know that our three-dimensional space is warped and curved.
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