![]() ![]() So Earth is not flat(because it's an example of a sphere, thus, non-Euclidean), but the 3d space sure seems flat Having a curvature means it's non-Euclidean, and being flat means it's Euclidean. ![]() So we can discuss curvature of our 3d space as well. In hyperbolic geometry, it's almost as if there are more directions packed in, more space to get lost in.įor physics perspective, you notice that this geometry doesn't actually really require 2d plane, we can make triangles just the same in 3d space. My favorite example is that if you played golf on a space that had hyperbolic geometry, unless you manage perfect hole in one, you'll notice that it's almost as if your ball went almost exactly the opposite direction of the hole. In it, angles of a triangle add up to less than 180 degrees, and basically things get weird. Angles of a triangle add up to more than 180 degrees, and there are no parallel lines. One of these geometries you get is basically the same as doing geometry on a surface of a sphere. And to differentiate between them, the geometry that follows Euclids original 5 postulate logic is called "Euclidean", and these new twists are thus "non-Euclidean". So mathematicians tried what would happen if you tweaked it to say "there are no parallel lines", and "there are infinitely many parallel lines" instead of "exactly one". ![]() So a quick recap: parallel postulate says that there's exactly one way to make a line that's parallel to another line, when you've been given a point this new parallel line has to pass through. So what did mathematicians do? Well, since 5th postulate was fundamental to geometry as we knew it, after all, they obviously wanted to try out what happens if you change it! Thousands of years passed, and the quest continued, until one day a bright young mathematician managed to prove 5th postulate was necessary! So began the hunt to remove parallel postulate from the required postulates. That would mean you only needed 4 postulates to do geometry! Euclid managed to prove a shitton of things using only the 4 first postulates, never touching the 5th one, and it seemed like with just some more effort, he could've proved parallel postulate, the fifth postulate, from the other postulates. The final, fifth, postulate seemed almost unnecessary. Anything true in geometry would be in accordance with those postulates, and anything in accordance with those postulates would be true in geometry. He had 5 postulates(axioms in todays terms) that captured plane geometry and all the fancy stuff within it. So to simplify, geometry was made rigorous by a guy name Euclid. There was a website I found really useful when I was learning about it for a research project, I’ll try and find it again, and if I’ve got anything wrong or something doesn’t make sense, please reply and I’ll try fixing it The angles must be the same, and since two are 90 degrees, they’re all 90 degrees. You have three lines with right angles between them, which in Euclidean Geometry makes sense, it’s a square with a side missing.īut we defined a case where parallel lines meet, so it ends up being a triangle, and if the third line is the same length as the first two, the triangle is equilateral. ![]() In elliptical geometry, the original line and the line though the point intersect, and it makes sense that you could draw a third line that cuts across both these points, and is perpendicular to both of them (think of two points, and draw parallel lines through them, then connect the points up) There are lots of different things you can do and tonnes of different words associated with what things means, and to best understand it all you just have to read up about it.įor example, it’s best not to think in straight and curved lines, rather think of them as “geodesics” - the shortest distance between two points (in Euclidean Geometry, this is a straight line)Ī popular idea is a triangle with three 90 degree angles: When no lines through the point intersect the original line, you get hyperbolic geometry. With having more than one line through the point that intersects the original line, you get Elliptical Geometry. This assumption is a weird one with no reason to assume it, and mathematicians decided to see what would happen if you changed it - Non-Euclidean Geometry. Euclidean Geometry assumes that if there’s a line with a point next to it, there will be exactly one line through the point that won’t intersect the original line - a parallel line. ![]()
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