Illuminating hyperbolic geometry

Among your videos this is my favourite so far.

I honestly want to use one of those as a lamp

using light to show off those various projections is really cool

hey this is very interesting and all but why is the outro so creepy

This is well above my knowledge level, but i'd love to understand properly one day

Great video! I would love to see more schools use your models to teach these concepts.

i kinda have understood it but actually i haven't

Here from CodeParade. Great video!

one of my favorite projections is taking a euclidean plane, pulling back a gnomonic projection to the half-sphere, and parallel projecting to a disk in the plane. it maps lines in the plane to half-ellipses tangent to the boundary of the disk at two opposing points, which makes it very well suited to conceptualizing projective geometry. of course, you still have to implicitly equate opposing points on the boundary. (edit: i suppose you could stereographically project the half-sphere to the disk instead; that would map lines to circular arcs which intersect the boundary at opposing points) you can even model this projection in a graphing calculator like desmos, which means you can graph proper functions and see how they behave. i suppose it shouldn't be surprising that the point at which they intersect the circle at infinity is closely related to the limit of the slope as x goes to infinity (if it exists), so most common functions (polynomials, exponentials) intersect at the top/bottom of the circle (i.e. straight vertical from the origin). as a final example, sin and cos do not have limiting slopes, but they are bounded between two horizontal lines, and so must intersect the point at infinity where those lines do: the horizontal point, corresponding to 0 slope lines.

This is so good. It's helpful to explain because it is the real deal. No 'imagine a ray that blabla', here you just show it and say. And now we show you how to make it on paper. Thanks!

I watched this to understand my absurdly confusing dreams. it helped a bit and the switching voice and outro are fittingly eerie

Love how eccentric you guys are, and excellent visual portrayal of the material, new sub & fan, much love, thank you! :)

The idea that different projections are the result of light sources in different positions and directions is rather striking

Great video. I found it a little too fast for someone uninitiated. I loved the 3D printed models and the demonstrations - pretty cool.

Hey cool! I love hyperbolic geometry. While I have already been introduced to these three projection types before (at least the ones on the flat plane, the hemisphere model is new to me even though it's such a great exchange medium between the other three!), this is the first time I've seen them compared to one another and their most important shape conservation properties discussed in full. Thanks guys! :D

I like your funny words, magic man~ this is way above my grade 12-level knowledge of euclidian/non-euclidian planes, but I can tell this is cool stuff!

These projections are so satisfying.

These are really some thought inducing videos! Great work gentlemen!

This is so cool, thank you for taking the time to explain all of this!

when he says "we're gonna use the sun" 😂