Beyond Euclidean Limits

Venture with Eddie Kingston in this riveting episode of "The Magnificence of Mathematics" as he contrasts the familiar realm of Euclidean geometry with the intriguing world of non-Euclidean geometries. Delve deep into the foundational postulates and experience the mathematical breakthroughs that expanded our geometric horizons. Whether you're a math enthusiast or just curious, Eddie's insights promise to shed light on dimensions of thought previously unexplored.

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Episode Transcript

(00:00):
Hello everyone, welcome to the Magnificence of Mathematics. I'm your host, Eddie Kingston.

(00:18):
Last time we touched on a simple, yet beautiful geometry problem from Paul Lockhart's mathematician's
lement. You may remember geometry from your high school days as finding areas and
volumes and working a lot with triangles and circles and whatnot, but what if I told you
the rules of geometry as you know them behave differently on different surfaces?

(00:42):
Today we'll be diving deep into the history of the geometry you know and love or hate,
talking about Pythagoras and Euclid and finishing off with what's known as non-Euclidean geometry.
Now most of you listeners have probably at least heard of the Pythagorean theorem, a
squared plus b squared equals c squared. Here's what this means so that you can try to visualize

(01:03):
it. Take the length of one side of a right triangle, either the horizontal or the vertical
side, and multiply it by itself. Take the length of the other side, whichever of the
horizontal and vertical side wasn't used before, and add them together. That is equal
to the length of the hypotenuse, the diagonal side, multiplied by itself. But where did

(01:25):
that come from? Well, it might surprise you to know that it wasn't even Pythagoras who
discovered this theorem. It was first discovered by the Babylonians in Mesopotamia, which is
modern day Iraq, Syria, Turkey, and Kuwait, between roughly 2000 and 1800 BCE, together
with other known math at the time for certain crafts in astronomy. Nowadays it's used

(01:48):
for things like architecture, painting, engineering, navigation, forestry, etc. Fun fact, did you
know that there's over 370 proofs of the Pythagorean theorem? That's over 370 different
ways to demonstrate why the theorem is true. Some of these proofs were written by 12-year-old

(02:08):
Albert Einstein, Leonardo da Vinci, former US President James Garfield, and of course
Pythagoras and his students. The most recent of these proofs was discovered in April 2023
by two high school students from New Orleans, Louisiana, Kelsey Johnson and Nakaiya Jackson.
That's something that I really like about math. There's so many ways to solve a problem,

(02:30):
and there's new proofs still coming out from problems that were posed millennia ago. A
couple centuries after Pythagoras' time, Euclid came along and revolutionized geometry,
inspiring mathematicians over the next couple thousand years to study his work. In about
300 BCE, he wrote his magnum opus, The Elements, a collection of 13 books spanning 2D and 3D

(02:54):
geometry, as well as elementary number theory. In the first book, Euclid described five postulates
from which the rest of his work follows, although some scholars disagree that everything follows
from just these five axioms. There's probably more that are needed. These axioms, at least
the first four, might seem pretty straightforward. First, given any two points on a 2D plane,

(03:16):
you can draw a line connecting those two points. Second, you can extend that line to go on
forever. Third, you can describe any circle given its center and radius. Fourth, all right
angles are equal to each other. 90 degrees is 90 degrees. The fifth postulate, called
the parallel postulate, is rather infamous. Take a straight line and a point not on that

(03:41):
line. Then there is exactly one line going through that point, such that the two lines
never touch each other, even when those lines are extended infinitely. People tried proving
the parallel postulate for quite a while, but then it was discovered that by tweaking
it a bit, or abandoning it altogether, one could describe entirely different, yet still

(04:03):
valid systems of geometry that behave differently from Euclidean geometry. You know the angles
on the triangles that you're used to add up to 180 degrees, but did you know that there's
surfaces on which the angles of a triangle add up to more than 180 degrees, and some
on which the angles add up to less than 180 degrees? This is where various Riemannian

(04:24):
geometries such as hyperbolic and elliptic geometry come into play. The underlying feature
here is the idea of curvature. An intuitive way to think of curvature of a space is that
it's a measure of how much space opens up or closes in on you as you move across it.
In Euclidean geometry, space is completely flat, that is, there's no curvature involved.

(04:47):
If you walk across a flat surface, it'll continue to appear flat for you. For hyperbolic geometry,
pretend you're standing on a giant horse saddle or a giant Pringles chip. If you walk across
this surface, you'll notice the surface appearing to open up away from you. This is what's known
as negative curvature. For elliptic geometry, pretend you're standing on a ball or sphere.

(05:10):
If you walk across this surface, it'll appear as though the space is bending towards you
or closing in on you. This is an example of positive curvature. Curvature in this sense
was first quantified by Bernhard Riemann in the mid-1800s via what's known as a curvature
tensor which ends up being quite complicated when you get into the nitty gritty, which

(05:33):
I won't do here. Riemannian geometries are kind of like Euclidean geometry but without
the parallel postulate. Let's say we have a line and a point not on the line. In hyperbolic
geometry, there are an infinite number of lines going through the point that don't
intersect the original line, whereas in elliptic geometry, any line through the point intersects

(05:54):
the original line, so there's no parallel lines in elliptic geometry. So in hyperbolic
geometry, you end up with triangles that are strictly less than 180 degrees. Meanwhile,
in elliptic geometry, you end up with triangles that are strictly greater than 180 degrees.
For example, say you're on a sphere in one location and start walking before taking a

(06:17):
turn 90 degrees to the right. Let's say you walk the same distance as before and take
another 90 degree turn to the right and walk the same distance that you have before. You
actually end up back where you started, but the angles in the triangle you just walked
in add up to 270 degrees, 90 degrees for each side, instead of the 180 degrees with 60 degrees

(06:39):
on each side that you'd walk if you were in flat space. If you were to try walking
on a hyperbolic surface, you would need to turn somewhat less than 60 degrees each time
to end up back where you started. If you'd like a visualization of walking in hyperbolic
space, I highly recommend checking out the game Hyperboloca by Code Parade on Steam,

(07:00):
or at least watching gameplay videos of it on YouTube. I personally haven't played
it yet, but I think this is a great complement to what I've been describing here.
Thank you for tuning in to another episode of the Magnificence of Mathematics. Next time
I'll talk about my personal favorite math subfield, probability. See you then! 77 00:07:43,920 -->00:08:41,920 Algid Productions LLC Outro. Thank you for listening!