Illuminating hyperbolic geometry

You would understand my work! Its hard to explain to people that incoming light on the retina maps onto half a geodesic dome of roughly hexagonal (6 triangles) tiling, which is then rescaled by the thalamus onto the V1 Gyri, like an image. As far as the brain is concerned the gyrii bump is geometrically perfect. Same for all the other sulcii and gyrii, as long as each cortical column has wired correctly with its neighbors, then tasks can be performed such as read/write operations, graphical construction, pixel encoding, moire pattern animations.

Please do this with colors.

Like fr 😑

is there a model where lengths aren't distorted (regardless of angles)

In serial geometry we can walk in parallel!

astounding demonstration

Amazing video!!Could you share how you made some of these beautiful models? I would love to print some of these for my high school class!

The city of R'lyeh is said to have non-Euclidian geometry. Theoretically, how would a model of it look?

my favorite giga nerds

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.

fun fact: normal non-euclidean spaces without hyperbolic or something are easy to imagine a visualization with a brain video(imaginary video of a cube that changes whats inside depending on the angle or another thing), but 4d visualizations are very hard to visualize(at least to me)

Omgosh I did a project like this for my 3d class, with a light bulb but I did was that I made two Mandelbrot geometric spheres with alternative concentric angles with their geometry one would rotate inside the other sphere and when both spheres rotate in opposite polarities the shadows would start interchanging like crazyyy,
It was the wildest experience manipulating the shadows

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!

Thank you gentlemen. Realizing this particular video is 7 years old, and I’m just learning the subject, I have to ask, how does this, translate to real world work. Aka, in geometry speak, where are these models applied and used? Thanks again.

illuminating!

one question:
how can i join your cult?

all I see are cool shapes

._. what

if there would be an index like 1/amount of videos with similiar topics like the indexed video (each one of this channel), im sure this channel would have a lot of videos in the top 100. (sry no math expert here, but i think you know what i mean :)) and the channel itselft would be in the top 3. respect.

Interesting

I always find your presentations interesting and informative. And they are delivered in a concise and professional manner.

What I found odd in this particular episode was when Henry was holding the hemispheric model above the huge white board which was being supported at one end by an assistant. I thought for sure that you would then move the model away and see how the pattern changed on the white board?!

But that didn't happen so my question is why not show us how the image changed with the model higher above the board? Otherwise why use such a big board at all?

Are those conformal mappings?

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

So there is no model that can accurately depict all aspects?

The other guys voice sounds familiar...
Did he by any chance voice "Chaos" by Jos Leys?

dmt hallucinations have hyperbolic geometry

Outro: when you have one minute left to do your homework:

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

Is there any way to get the gans model using the hemisphere model and a light source?

I didn't understand the essence of this. Is a hyperbolic plane actually a half sphere made of triangles? Or is that just a model that represents some characteristics of the hyperbolic plane? Can a hyperbolic plane be visualized in 3-space at all? Is the hyperbolic plane more realistic for cause/effect physics than the flat Euclidean plane? Where do I get such questions answered?

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

i neeeeeed moreeeee

Cool!

So wait a minute...

If 3d space isn't curved its Euclidian. But when we add time into the mix, to create spacetime with 4 dimensions, then that spacetime is a 4D hyperbolic space?

Is that correct?

What would Maxwell's Equations look like in Lobachevsky (Hyperbolic) space? I really like your videos!

These projections are so satisfying.

Far out

perfect for people designing lamps

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

Him: talking about math and science stuff
Me: “Oooo pretty patterns”

...Thank you so much for posting this......

It reminds me of fractal geometry...

perfectly odd and informative video!

dash

could you make some outdoor lamps that cast a tennis court (through the shadows)?

This is all well and good but what does hyperbolic mean

That last frame made it seemed like they are part of some kind of math cult.....but then I'm making the assumption math isn't one massive cult with subsections.

This is what appears in your recommendations after watching way too many geometry dash videos

this video was very illuminating