12:31 Actually two such people will diverge linearly from each other. After all, if they're both 100m from the starting point, then by the triangle inequality they're at most 200m away from each other. That is, person A could reach person B by first walking 100m to the starting point, and then 100m to person B. Since the distance between two points is the length of the shortest path between them, that means person A and person B were at most 200m apart. However, the circular arc between them will grow exponentially. And the precision needed to find each other again will increase exponentially as well.
I guess the geodesic between them in this projection will be a curve that swoops way in to somewhere near the origin and bends back out again. I recall someone speculating that the profusion of little-contacted neighborhoods within the land of Oz in L. Frank Baum's books, even though distances were short enough to be walked in a couple of days, suggested that Oz was a negatively curved manifold.
@@MattMcIrvin Exactly. In fact, no matter how far the two people walk, the line between them will pass fairly close by the origin. Even if all three ends of a triangle are at infinity, in hyperbolic space, the edges stay fairly close together, and the area stays finite. Look up the infinite-order triangular tiling on Wikipedia for an example of such triangles. A game inspired by the land of Oz, set in a hyperbolic space, could certainly be interesting.
It is not gone. Just your chances of finding it are exponentially decreasing with you getting away from it, because of just how much larger the hyperbolic space is in area.
What you call pseudorthographic projection is called the Gans model ( en.wikipedia.org/wiki/Hyperbolic_geometry#The_Gans_Model ). Moving the center along a straight line while using the Gans model gives a very strange optical illusion of looking into a crater, although unfortunately the effect is quite weak in your video because the zoom factor is too big. (We don't know why this illusion appears.)
In this projection, distance is preserved along circles centered around the projection center, just as in the orthographic projection of a sphere. Thus, for H^2, distances toward or away from the center appear greatly magnified.
Just looked at a video of it... seems to me that it's because the distortion of shapes is much like the parallax effect of a cone being tilted. (Which makes sense, since this model is literally an orthographic projection of a hyperboloid, in Minkowski spacetime, with the shape of a blunted cone.) To me, I can make it subjectively flip between looking at a crater or a mountain.
I was showing a bit of this to my 10th grade geometry students the other day. (Substitiles on with no sound worked much better for them.) They REALLY liked it. Is the transcript available? I'm very interested in all this. Currently reading Geometry of Surfaces by John Stillwell with whatever time I have. Thanks!!
A small note regarding what you say at the end about the equidistant projection: can one really say that the heptagons one can reach in a few steps are spaced far apart ? By the triangle inequality the line segment between any of them is quite short even if the circular arc between them is long.
You're right, my statement about the heptagons being vastly spaced apart is misleading (it's not sufficiently precise to be clearly "wrong") because of the triangle inequality. There used to be an overlaid video annotation correcting this mistake, but a TH-cam upgrade must have removed it.
I wonder how it would look to put the hemisphere model on a sphere and make it so that when the sphere is rotated, the hemisphere model moves accordingly.
I think I'm slowly starting to understand what hyperbolic space is and how it works. The important words here are "slowly" because progress in comprehension of this topic is, well, slow; "starting", because I'm only at the very beginning of getting an understanding; and "think", because I'm really not sure.
I believe the Earth surface is hyperbolic. It is pretty clear, when you look closely at plane routes. Also I tried to walk along some people in the street at a slight starting angle difference, constantly measuring distance btw us and guess what... the distance seemed to increase exponentially!!!
![Non-Euclidean Geometry [Topics in the History of Mathematics]](/img/n.gif)
Best explanation of non Euclidean geometry, spent hours trying to understand this
helps to keep the name of the projection on the screen while describing it. better for those without good written-memory
straining... other things to do in life... never took calculus... must... learn..
12:31 Actually two such people will diverge linearly from each other. After all, if they're both 100m from the starting point, then by the triangle inequality they're at most 200m away from each other. That is, person A could reach person B by first walking 100m to the starting point, and then 100m to person B. Since the distance between two points is the length of the shortest path between them, that means person A and person B were at most 200m apart.
However, the circular arc between them will grow exponentially. And the precision needed to find each other again will increase exponentially as well.
I guess the geodesic between them in this projection will be a curve that swoops way in to somewhere near the origin and bends back out again.
I recall someone speculating that the profusion of little-contacted neighborhoods within the land of Oz in L. Frank Baum's books, even though distances were short enough to be walked in a couple of days, suggested that Oz was a negatively curved manifold.
@@MattMcIrvin Exactly. In fact, no matter how far the two people walk, the line between them will pass fairly close by the origin. Even if all three ends of a triangle are at infinity, in hyperbolic space, the edges stay fairly close together, and the area stays finite. Look up the infinite-order triangular tiling on Wikipedia for an example of such triangles.
A game inspired by the land of Oz, set in a hyperbolic space, could certainly be interesting.
The last phrase pretty much explained the entire video to me. :D
Really wish I would've viewed this while in my intro to Maps and GIS courses lol Great video!
Great job, David! Thx for sharing. Keep up the good work...
12:16 So if you leave a heptagon, the heptagon is heptagone?
It is not gone. Just your chances of finding it are exponentially decreasing with you getting away from it, because of just how much larger the hyperbolic space is in area.
What you call pseudorthographic projection is called the Gans model ( en.wikipedia.org/wiki/Hyperbolic_geometry#The_Gans_Model ).
Moving the center along a straight line while using the Gans model gives a very strange optical illusion of looking into a crater, although unfortunately the effect is quite weak in your video because the zoom factor is too big. (We don't know why this illusion appears.)
Thanks!
In this projection, distance is preserved along circles centered around
the projection center, just as in the orthographic projection of a sphere.
Thus, for H^2, distances toward or away from the center appear greatly magnified.
Just looked at a video of it... seems to me that it's because the distortion of shapes is much like the parallax effect of a cone being tilted. (Which makes sense, since this model is literally an orthographic projection of a hyperboloid, in Minkowski spacetime, with the shape of a blunted cone.) To me, I can make it subjectively flip between looking at a crater or a mountain.
I was showing a bit of this to my 10th grade geometry students the other day. (Substitiles on with no sound worked much better for them.) They REALLY liked it. Is the transcript available? I'm very interested in all this. Currently reading Geometry of Surfaces by John Stillwell with whatever time I have. Thanks!!
click on the 3 dots that should be above the subscribe button, then there will be an option for transcript.
A small note regarding what you say at the end about the equidistant projection: can one really say that the heptagons one can reach in a few steps are spaced far apart ? By the triangle inequality the line segment between any of them is quite short even if the circular arc between them is long.
You're right, my statement about the heptagons being vastly spaced apart is misleading (it's not sufficiently precise to be clearly "wrong") because of the triangle inequality. There used to be an overlaid video annotation correcting this mistake, but a TH-cam upgrade must have removed it.
I wonder how it would look to put the hemisphere model on a sphere and make it so that when the sphere is rotated, the hemisphere model moves accordingly.
I believe the "pseudo-orthographic" projection of the hyperbolic plane is called the Gans model.
Underrated video.
In the last projection it's not possible to choose a heptagon on the right which goes near the center.
You can use not ex.: 1m13s, you can use ex.: 1:13 or 0:13
How did U make this video? what type of software/s U used?
U did not make this video. David did.
One single flatlander didn't like the video.
I think I'm slowly starting to understand what hyperbolic space is and how it works.
The important words here are "slowly" because progress in comprehension of this topic is, well, slow; "starting", because I'm only at the very beginning of getting an understanding; and "think", because I'm really not sure.
Clickable timestamps:
0:17 • 1:16 • 2:33 • 3:59 • 5:23
6:27 • 8:54 • 10:19 • 11:20 • 11:55
Anyone else end up here because of the game "HyperRogue"?
Yes
Awesome 🙂
This what i see when i close my eyes fr, just closeup in like a neon rainbow color that is really dim
this actually kinda helped wow
gracias
David Madore lmao
ok
Anyone else get here from a scientific paper into the geometry people see when taking DMT/Ayahuasca?
Good shit, the like/dislike bar is quite Yoda
Now gimme nil, sol, and the other Thurston geometries :0
#8 oh yah because all that about lorensian 3 space made perfict sense
wew trippy
Even the most dim witted individual can understand this! As long as they have a highest degree in hyperbolic topology :P
I believe the Earth surface is hyperbolic. It is pretty clear, when you look closely at plane routes. Also I tried to walk along some people in the street at a slight starting angle difference, constantly measuring distance btw us and guess what... the distance seemed to increase exponentially!!!
The coriolis effect can do that.
Pretty sure I've seen this video projected on the wall inside a night club
🤭💖🙄
im the 777th like and there is 7 dislikes
ok Kermit x
Fuck it, I'll buy a globe.
I like hyperbolic space but WHAT?!
Ow my tooth
Not super clear explanations.