This means that a triangle drawn on a surface of a sphere will have sum of its angles greater than 180 degrees. A sphere is an example of a surface with POSITIVE curvature. We all know one example of a non-Euclidean geometry the spherical geometry that governs distances and angles on the surface of Earth, at least if the distances involved are large enough. ![]() HyperRogue is a game played in one of these geometries - a hyperbolic geometry - and before you can hope to understand it, you must first understand some basic facts about hyperbolic geometry and the way it's represented in-game. But in the process of finding out that the Fifth Postulate is, indeed, essential for Euclidean geometry, a whole new world of non-Euclidean geometries was discovered. "If you have a straight line and a point outside that line, you can construct exactly one parallel line through that point."Ĭompared to other Euclid's postulates (like "All right angles are equal"), this seemed unneccesarily complex. "Sum of the angles in a triangle is 180 degrees." This postulate has been stated in many equivalent forms, with two of the best-known are: ![]() ![]() ![]() For millenia, best mathematical minds were puzzled by one problem: so-called "Euclid's Fifth Postulate", one of the cornerstones of geometry.
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