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Hello everyone, welcome to the Magnificence of Mathematics. I'm your host, Eddie Kingston.

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Last time we touched on a simple, yet beautiful geometry problem from Paul Lockhart's mathematician's

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lement. You may remember geometry from your high school days as finding areas and

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volumes and working a lot with triangles and circles and whatnot, but what if I told you

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the rules of geometry as you know them behave differently on different surfaces?

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Today we'll be diving deep into the history of the geometry you know and love or hate,

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talking about Pythagoras and Euclid and finishing off with what's known as non-Euclidean geometry.

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Now most of you listeners have probably at least heard of the Pythagorean theorem, a

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squared plus b squared equals c squared. Here's what this means so that you can try to visualize

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it. Take the length of one side of a right triangle, either the horizontal or the vertical

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side, and multiply it by itself. Take the length of the other side, whichever of the

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horizontal and vertical side wasn't used before, and add them together. That is equal

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to the length of the hypotenuse, the diagonal side, multiplied by itself. But where did

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that come from? Well, it might surprise you to know that it wasn't even Pythagoras who

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discovered this theorem. It was first discovered by the Babylonians in Mesopotamia, which is

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modern day Iraq, Syria, Turkey, and Kuwait, between roughly 2000 and 1800 BCE, together

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with other known math at the time for certain crafts in astronomy. Nowadays it's used

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for things like architecture, painting, engineering, navigation, forestry, etc. Fun fact, did you

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know that there's over 370 proofs of the Pythagorean theorem? That's over 370 different

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ways to demonstrate why the theorem is true. Some of these proofs were written by 12-year-old

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Albert Einstein, Leonardo da Vinci, former US President James Garfield, and of course

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Pythagoras and his students. The most recent of these proofs was discovered in April 2023

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by two high school students from New Orleans, Louisiana, Kelsey Johnson and Nakaiya Jackson.

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That's something that I really like about math. There's so many ways to solve a problem,

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and there's new proofs still coming out from problems that were posed millennia ago. A

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couple centuries after Pythagoras' time, Euclid came along and revolutionized geometry,

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inspiring mathematicians over the next couple thousand years to study his work. In about

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300 BCE, he wrote his magnum opus, The Elements, a collection of 13 books spanning 2D and 3D

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geometry, as well as elementary number theory. In the first book, Euclid described five postulates

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from which the rest of his work follows, although some scholars disagree that everything follows

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from just these five axioms. There's probably more that are needed. These axioms, at least

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the first four, might seem pretty straightforward. First, given any two points on a 2D plane,

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you can draw a line connecting those two points. Second, you can extend that line to go on

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forever. Third, you can describe any circle given its center and radius. Fourth, all right

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angles are equal to each other. 90 degrees is 90 degrees. The fifth postulate, called

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the parallel postulate, is rather infamous. Take a straight line and a point not on that

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line. Then there is exactly one line going through that point, such that the two lines

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never touch each other, even when those lines are extended infinitely. People tried proving

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the parallel postulate for quite a while, but then it was discovered that by tweaking

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it a bit, or abandoning it altogether, one could describe entirely different, yet still

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valid systems of geometry that behave differently from Euclidean geometry. You know the angles

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on the triangles that you're used to add up to 180 degrees, but did you know that there's

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surfaces on which the angles of a triangle add up to more than 180 degrees, and some

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on which the angles add up to less than 180 degrees? This is where various Riemannian

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geometries such as hyperbolic and elliptic geometry come into play. The underlying feature

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here is the idea of curvature. An intuitive way to think of curvature of a space is that

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it's a measure of how much space opens up or closes in on you as you move across it.

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In Euclidean geometry, space is completely flat, that is, there's no curvature involved.

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If you walk across a flat surface, it'll continue to appear flat for you. For hyperbolic geometry,

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pretend you're standing on a giant horse saddle or a giant Pringles chip. If you walk across

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this surface, you'll notice the surface appearing to open up away from you. This is what's known

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as negative curvature. For elliptic geometry, pretend you're standing on a ball or sphere.

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If you walk across this surface, it'll appear as though the space is bending towards you

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or closing in on you. This is an example of positive curvature. Curvature in this sense

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was first quantified by Bernhard Riemann in the mid-1800s via what's known as a curvature

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tensor which ends up being quite complicated when you get into the nitty gritty, which

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I won't do here. Riemannian geometries are kind of like Euclidean geometry but without

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the parallel postulate. Let's say we have a line and a point not on the line. In hyperbolic

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geometry, there are an infinite number of lines going through the point that don't

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intersect the original line, whereas in elliptic geometry, any line through the point intersects

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the original line, so there's no parallel lines in elliptic geometry. So in hyperbolic

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geometry, you end up with triangles that are strictly less than 180 degrees. Meanwhile,

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in elliptic geometry, you end up with triangles that are strictly greater than 180 degrees.

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For example, say you're on a sphere in one location and start walking before taking a

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turn 90 degrees to the right. Let's say you walk the same distance as before and take

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another 90 degree turn to the right and walk the same distance that you have before. You

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actually end up back where you started, but the angles in the triangle you just walked

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in add up to 270 degrees, 90 degrees for each side, instead of the 180 degrees with 60 degrees

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on each side that you'd walk if you were in flat space. If you were to try walking

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on a hyperbolic surface, you would need to turn somewhat less than 60 degrees each time

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to end up back where you started. If you'd like a visualization of walking in hyperbolic

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space, I highly recommend checking out the game Hyperboloca by Code Parade on Steam,

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or at least watching gameplay videos of it on YouTube. I personally haven't played

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it yet, but I think this is a great complement to what I've been describing here.

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Thank you for tuning in to another episode of the Magnificence of Mathematics. Next time

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I'll talk about my personal favorite math subfield, probability. See you then!

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Algid Productions LLC Outro. Thank you for listening!