cool facts about hyperbolic geometry: almost every path is close to straight equidistants (paths that stay the same distance from lines) converge to a horocycle (circle of infinite radius) as the guiding line goes away

Ironic how mathematicians discovered hyperbolic geometry before discovering the geometry they were already doing every time they drew or looked at a map, which violates the parallel postulate as well. (It also violates the first postulate, and the second and third postulates are dicey, but it's still its own geometry.)

@mostly_mental

Жыл бұрын

History is funny like that. It's fairly common for mathematicians to find something extremely complicated before realizing that it also shows up in a simple context they've been looking at for years. My favorite example is that complex numbers were discovered in the context of solving cubic equations, even though the mathematicians had been solving quadratics for centuries and just ignoring the roots of negative numbers.

@angeldude101

Жыл бұрын

@@mostly_mental Complex numbers are usually just seen as the square roots of negatives, but really they're just rotations. They're painfully geometric despite being said to detach algebra from geometry.

@mostly_mental

Жыл бұрын

@@angeldude101 Very true, although I've more often heard them described as the natural connection between algebra and geometry. But they were discovered as an algebraic curiosity a few decades before anyone considered algebra and geometry might be connected at all.

@angeldude101

Жыл бұрын

@@mostly_mental At least according to Veritasium's video, at the time they were invented, algebra _was_ geometry, and Complex numbers were the trigger to sever the connection, since there is no square that can have an area of -1, even if its side lengths were negative. Treating i geometrically does require a completely different approach though, with multiplication as composing geometric transformations rather than as forming rectangles from side lengths.

@mostly_mental

Жыл бұрын

@@angeldude101 I pulled out my closest math history book (A History of Mathematics by Victor Katz) and a bunch of online sources to check, and I definitely misremembered a few points. Here's the summary: In the 800s, Al-Khwarizmi (whose name got corrupted into "algorithm") invented what would become modern algebra. He used geometric methods in his proofs, and he didn't even deal with negative numbers, let alone complex. In the 1200s, Leonardo of Pisa (better known as Fibonacci) brought the ideas from the Islamic world to Europe. In the 1400s, European mathematicians started using symbols and equations instead of words to express algebra (and by the late 1500s, it looked a lot like what we have today). And they started to use negative numbers in their calculations, even when they dismissed them in the final answers. In the 1400s and 1500s, it became common practice for mathematicians to issue each other challenges (with the winners earning professor positions, rather than a tenure system). Many of these problems were algebraic, and by this point algebra was pretty solidly its own field (even if proofs were largely geometric). In 1545, Cardano published a method of solving cubics, and he noticed that sometimes it required taking square roots of negative numbers. This upset him, but got the right answers, so he put up with it. Over the next few decades, Bombelli figured out the rules for working with imaginary numbers (published around his death in 1572). In the 1600s, Fermat and Descartes independently constructed analytic geometry, which reunified geometry and algebra. Descartes' method became the Cartesian plane. Descartes was also the one to name them "imaginary" numbers. Later in the 1600s, Wallis and de Moivre put negative and complex numbers on the same footing ass positive numbers as part of the same framework. In the 1700s, Euler put the complex numbers into the Cartesian plane, where all their nice geometric properties really shine. His work was largely based on intuition, without the full formalism. He also called the imaginary unit "i". Finally, in the 1800s, Gauss nailed everything down rigorously. He was the one who named them "complex numbers".

Amazing video!

@mostly_mental

2 ай бұрын

Glad you like it. Thanks for watching!

Great topic! Still really like that game! I wonder how difficult it would be to make a version of HyperRogue that corrects the holonomy to make navigation slightly more intuitive. (Idea: Always rotate the disk such that a fixed point on the boundary stays "north".)

@mostly_mental

2 жыл бұрын

That would definitely be possible, and the game's open source if you want to try github.com/zenorogue/hyperrogue/. But the devs went in a different direction and used the holonomy instead of correcting for it. There's actually a land (the Burial Grounds) where that's the primary gimmick. You can acquire an Orb of Sword, which always points the same way, and you need to rotate yourself with the holonomy to get it to point the way you want.

@cannot-handle-handles

2 жыл бұрын

@@mostly_mental True! The Yendor Quest would also be less challenging. :-D

♾

Hyperbolic geometry is very different from euclidean geometry. The fact is that the 2 parallel lines in question are not straight lines. So, parallel lines are not parallel straight lines.

@mostly_mental

Жыл бұрын

They don't look like straight lines if you try to embed them in Euclidean space. But I'd argue they're still straight lines. If you look at the crochet example at 7:13, you could flatten out the surface along each of those lines and make the line look straight. Yes, it would cause the rest of the surface to bunch up, but that's only because we're trying to squish a hyperbolic thing into Euclidean space. In a hyperbolic space, all those lines would simultaneously be straight. If you try out the game, you'll pretty naturally find yourself walking in straight lines.

is there a link to that game ? what great visual add. thanks for the vid

@mostly_mental

16 күн бұрын

I'm glad you liked it. HyperRogue can be downloaded at zenorogue.itch.io/hyperrogue (or it's available on Steam).