Playing Sports in Hyperbolic Space - Numberphile
video description
Also folded variants of hyperbolic space, like the 2d klein's quartic space (tileable by 24 regular heptagons, three in each corner, topologically a 3-torus, is a fun limited area version to play with, just like the flat curvature torus is a fun limited area version of Euclidean 2d space. The oriented sphere of (constant) positive curvature (elliptic space) may be folded into a projective space (plane) of constant positive curvature, likewise i think the flat torus may be folded into a Klein bottle of constant zero curvature. It is possible that some multitorus manifolds might be folded into non-oriented manifolds of constant negative curvature, thus providing yet other examples of hyperbolic spaces (planes.
Also we can take a Moebius band (with its edge at infinity) and give it an indefinite metric that makes it a dual space to the Euclidean plane, in the sense that all paths of locally shortest distance close into straight line loops when continued, but angles around corners never close, but go on and on forever, not repeating at a full circle, but going from one infinity to another infinity along a hyperbola, this geometry also giving us a sense of opposite edges instead of opposite angles, and parallel corners/angles instead of parallel lines/edges. Variants of this geometry similar to elliptic and hyperbolic curved space ought to exist as well, on manifolds with Moebius strip topologies (instead of disc topologies as with usual 2d geometry) or something similar.
We could of course go on into higher dimensional spaces like 3d and 4d, where various mixes of indefinite metrics are possible, plus there are more exotic ways to fold the (possibly infinite) geometry into finite volume/hypervolume geometries with varying topological -holes-.
There is also a way to reconcile hyperbolic geometry on a disc with some hyperbolic-like indefinite geometry on a Moebius strip into a single unified geometry of two regions of a projective plane, separated by a circle at infinity (according to the metric, letting us get imaginary distances between points at either side of the border, and certain other amusing properties, this is explored by the Universal Hyperbolic Geometry theory of Wildberger, which imho is extremely promising, yet has not been explored so much, probably because mr Wildberger is alienating himself from his mathematical peers due to his completely unreasonable stances about actual infinities and real numbers (he seems to be using real algebraic numbers instead, claiming them to be -non-existent-, and his unconventional uses of quadrance and spread instead of distance and angle (although these are related by simple mathematical formulas, and personally i find the two new concepts strange but useful.
There are very many possibilities more i suppose.
Date: 2022-04-08
Related videos
Comments and reviews: 9
---
Notes:
The opposite of hyperbolic geometry is elliptic geometry, which can be easily visualized by a sphere. On a sphere, an angle has smaller distance, w. r. t. the distance you go, further you go, and eventually it becomes zero, which simply means you reach the other pole. If you put sphere on the plane, obviously you can't with euclidean distance, the -distance- of two point is closer than it looks.
Back to hyperbolic geometry. With the picture of elliptic geomery, now it's -easier- to imagine what happens in hyperbolic geometry. Further you go, further the distance become between angles, again w. r. t. the distance you go. And eventually it will be infinity. If you put it on the plane you get the strange thing in the video.
To understand better why the -straight lines- (called geodesics in math) are not straight. Imagine you are a lifeguard at the beach, and usually swimming speed is much slower than running speed. So when someone (say Tom) is drawning, the best you can do, in order to reach Tom faster, after some easy calculation, you will find out that, it is better to first run toward the shore line closest to Tom, then you swim, which is not a straight line, right? This happens because you cannot move at the same speed.
More obvious example will be, if you want to travel 100 meters as fast as you can, either through running or swimming. Of course everyone will know running is faster. Then if you must start in water and finish in water, you will again end up with path that is not straight line.
This difference in speed when you travel gives you these paths with shortest time that are not straight lines. And you can actually find similar geometry, in the trajectory of sound wave, or light in cosmology.
reply
Notes:
The opposite of hyperbolic geometry is elliptic geometry, which can be easily visualized by a sphere. On a sphere, an angle has smaller distance, w. r. t. the distance you go, further you go, and eventually it becomes zero, which simply means you reach the other pole. If you put sphere on the plane, obviously you can't with euclidean distance, the -distance- of two point is closer than it looks.
Back to hyperbolic geometry. With the picture of elliptic geomery, now it's -easier- to imagine what happens in hyperbolic geometry. Further you go, further the distance become between angles, again w. r. t. the distance you go. And eventually it will be infinity. If you put it on the plane you get the strange thing in the video.
To understand better why the -straight lines- (called geodesics in math) are not straight. Imagine you are a lifeguard at the beach, and usually swimming speed is much slower than running speed. So when someone (say Tom) is drawning, the best you can do, in order to reach Tom faster, after some easy calculation, you will find out that, it is better to first run toward the shore line closest to Tom, then you swim, which is not a straight line, right? This happens because you cannot move at the same speed.
More obvious example will be, if you want to travel 100 meters as fast as you can, either through running or swimming. Of course everyone will know running is faster. Then if you must start in water and finish in water, you will again end up with path that is not straight line.
This difference in speed when you travel gives you these paths with shortest time that are not straight lines. And you can actually find similar geometry, in the trajectory of sound wave, or light in cosmology.
reply
Robert
Every time I see one of these hyperbolic space things, they always try to assert the direction of motion different than direction of sight. If you lived in a hyperbolic space, then the path of light would travel from a point in all directions. and the one that reaches your eyes first will be the brightest. which means it took the shortest distance, whether it was on a curve or not. This means that if you want to shoot for a hole, you will shoot directly at the brightest return to the hole. and while the ball is moving, you will see it move in a straight line to it because the wave of light will still bend to meet your eyes in the same way. There is no way to differentiate between flat space and hyperbolic space in entity motion while being trapped within the confines of that space. except in 1 way. since light travels in all directions at the same time and only reaches your eyes based on distance traveled according to the angle you view, you will see an overlay of every object in space everywhere at the same time (like an overexposed rotating camera) if we lived in hyperbolic space (kinda like if you were drunk with blurred vision.
Here's something to think about though. what if we did live in a slightly low hyperbolic space and the stars we see in the night are simply our own solar system bending itself. Since it takes a long time, we see it in various stages of its life. The accuracy of angles would be within the confines of this universe which means 90 is still 90 to us and there's no way we can verify otherwise.
reply
Every time I see one of these hyperbolic space things, they always try to assert the direction of motion different than direction of sight. If you lived in a hyperbolic space, then the path of light would travel from a point in all directions. and the one that reaches your eyes first will be the brightest. which means it took the shortest distance, whether it was on a curve or not. This means that if you want to shoot for a hole, you will shoot directly at the brightest return to the hole. and while the ball is moving, you will see it move in a straight line to it because the wave of light will still bend to meet your eyes in the same way. There is no way to differentiate between flat space and hyperbolic space in entity motion while being trapped within the confines of that space. except in 1 way. since light travels in all directions at the same time and only reaches your eyes based on distance traveled according to the angle you view, you will see an overlay of every object in space everywhere at the same time (like an overexposed rotating camera) if we lived in hyperbolic space (kinda like if you were drunk with blurred vision.
Here's something to think about though. what if we did live in a slightly low hyperbolic space and the stars we see in the night are simply our own solar system bending itself. Since it takes a long time, we see it in various stages of its life. The accuracy of angles would be within the confines of this universe which means 90 is still 90 to us and there's no way we can verify otherwise.
reply
christosvoskresye
I hate to say this, but the video is WRONG. In Euclidean spaces, we have no inherent length scale, but in hyperbolic space, there is a length scale. If the radius of curvature of the space is one light year, a baseball diamond would look exactly like it does to us. If the radius of curvature were 1 kilometer, it would begin to look a little weird. Only if the diamond were much larger than the radius of curvature -- say, the radius were 1 cm -- would it look like he portrays.
You can make this kind of mistake with hyperbolic geometry because the hyperbolic plane allows for infinite lengths and infinite areas. If you tried this in spherical geometry, the problem would be immediately apparent. -Imagine you're playing golf, and you're 100 meters from the pin. You make a terrible shot and hit the ball at an angle of 90 degrees from the straight line distance to the pin. How far are you from the pin now? By the way, the radius of the planet you're on is 1 cm. - No matter how bad your shot is, you're not going to be more than a few radii away! On the other hand, real golf is obviously played on an approximately spherical surface, and that causes no problem, because the radius of the sphere is much larger than the length of the course.
Distance is not a pure number, and it should not be treated as though it were. In hyperbolic geometry it must be specified how any length compares with the plane's radius of curvature.
reply
I hate to say this, but the video is WRONG. In Euclidean spaces, we have no inherent length scale, but in hyperbolic space, there is a length scale. If the radius of curvature of the space is one light year, a baseball diamond would look exactly like it does to us. If the radius of curvature were 1 kilometer, it would begin to look a little weird. Only if the diamond were much larger than the radius of curvature -- say, the radius were 1 cm -- would it look like he portrays.
You can make this kind of mistake with hyperbolic geometry because the hyperbolic plane allows for infinite lengths and infinite areas. If you tried this in spherical geometry, the problem would be immediately apparent. -Imagine you're playing golf, and you're 100 meters from the pin. You make a terrible shot and hit the ball at an angle of 90 degrees from the straight line distance to the pin. How far are you from the pin now? By the way, the radius of the planet you're on is 1 cm. - No matter how bad your shot is, you're not going to be more than a few radii away! On the other hand, real golf is obviously played on an approximately spherical surface, and that causes no problem, because the radius of the sphere is much larger than the length of the course.
Distance is not a pure number, and it should not be treated as though it were. In hyperbolic geometry it must be specified how any length compares with the plane's radius of curvature.
reply
KingGrio
It's a nice video but what really bugs me in this presentation is that Euclidian premises are setup (you're still hitting the golf ball in Euclidian straight lines) and then you say -oh but that's really far and will take forever-.
I'd say if we're playing hyperbolic golf, let's go full hyperbolic. Build a hyperbolic topology for the field, gimme a hyperbolic driver to hit the ball with and make hyperbolic paths. Probably hyperbolic physics and gravity would be nice to help me. And Euclidian definition of a ball and then calculate the area of it? Gimme a hyperbolic definition of a ball that preserves pi-R-2.
And doesn't all of these definitions I'm choosing basically making hyperbolic geometry back into Euclidian geometry? By choosing definitions that preserve Euclidian properties and formulas?
reply
It's a nice video but what really bugs me in this presentation is that Euclidian premises are setup (you're still hitting the golf ball in Euclidian straight lines) and then you say -oh but that's really far and will take forever-.
I'd say if we're playing hyperbolic golf, let's go full hyperbolic. Build a hyperbolic topology for the field, gimme a hyperbolic driver to hit the ball with and make hyperbolic paths. Probably hyperbolic physics and gravity would be nice to help me. And Euclidian definition of a ball and then calculate the area of it? Gimme a hyperbolic definition of a ball that preserves pi-R-2.
And doesn't all of these definitions I'm choosing basically making hyperbolic geometry back into Euclidian geometry? By choosing definitions that preserve Euclidian properties and formulas?
reply
Aaron
I'm not quite sure how I got from this video to the question I'm going to ask that was inspired by it, but I'm curious anyways:
If you zoom in to a hyperbolic shortest distance line (which is a curve, right, it will look like a euclidean shortest distance line (ie straight, instead of a curve. are there geometries where when you zoom in, the curvature of a line is preserved?
What I'm asking sounds weird. a line with an invariant curvature as you scale in and out would have to be attached to different points on the plane I would think, but is it possible to construct a space that follows that rule and is internally consistent?
reply
I'm not quite sure how I got from this video to the question I'm going to ask that was inspired by it, but I'm curious anyways:
If you zoom in to a hyperbolic shortest distance line (which is a curve, right, it will look like a euclidean shortest distance line (ie straight, instead of a curve. are there geometries where when you zoom in, the curvature of a line is preserved?
What I'm asking sounds weird. a line with an invariant curvature as you scale in and out would have to be attached to different points on the plane I would think, but is it possible to construct a space that follows that rule and is internally consistent?
reply
Sandy
This makes me think of the math that would be used to plot the flight path between objects through accelerating space, like for super long space voyages. In our space that is expanding, you'll never reach the end, so that matches. Also, going from one single point to another far point, across very vast differences or vast speeds, you would have to make adjustments for the acceleration of the universe. Your destination would have moved and your 'draw' to the destination would follow a curve - not a straight line - as you came into alignment with its
directional acceleration.
reply
This makes me think of the math that would be used to plot the flight path between objects through accelerating space, like for super long space voyages. In our space that is expanding, you'll never reach the end, so that matches. Also, going from one single point to another far point, across very vast differences or vast speeds, you would have to make adjustments for the acceleration of the universe. Your destination would have moved and your 'draw' to the destination would follow a curve - not a straight line - as you came into alignment with its
directional acceleration.
reply
Noah
This is showing hyperbolic geometry of a very large curvature (1 is a whole lot bigger than the 0 or slightly larger curvature of the universe, which is quite a jump from Euclidean geometry. Most forms of hyperbolic geometry that are used to make it more understandable, they would use a very slightly curved space, such as the hexagonal-heptagonal tiling used in hyperrogue.
We need more documentation on how to relate a curvature number to distance relationships (like some way of determining the edge length of a certain tiling based on the curvature of the space it is within)
reply
This is showing hyperbolic geometry of a very large curvature (1 is a whole lot bigger than the 0 or slightly larger curvature of the universe, which is quite a jump from Euclidean geometry. Most forms of hyperbolic geometry that are used to make it more understandable, they would use a very slightly curved space, such as the hexagonal-heptagonal tiling used in hyperrogue.
We need more documentation on how to relate a curvature number to distance relationships (like some way of determining the edge length of a certain tiling based on the curvature of the space it is within)
reply
AlphaSquadZero
For the baseball equation(assuming it took place on a non-euclidean sphere with the radius of earth): (pi/2(cosh. 09144km/6371km) - cosh. 03048km/6371km)-6371000m = 0. 000915225 m-2
Conversely, it will be the in-field that will need more players (it is over 10km-2.
For the game of golf (under the same assumptions as before):
pi/180-sinh. 09144km/6371km) - 6371000m = 1. 59593m
Which is exactly the same distance in non-euclidean space. You only really see a lot of deviation in the arc length once you travel around the entire world nearly half a time.
reply
For the baseball equation(assuming it took place on a non-euclidean sphere with the radius of earth): (pi/2(cosh. 09144km/6371km) - cosh. 03048km/6371km)-6371000m = 0. 000915225 m-2
Conversely, it will be the in-field that will need more players (it is over 10km-2.
For the game of golf (under the same assumptions as before):
pi/180-sinh. 09144km/6371km) - 6371000m = 1. 59593m
Which is exactly the same distance in non-euclidean space. You only really see a lot of deviation in the arc length once you travel around the entire world nearly half a time.
reply
Joel
OK here's my takeaway from this: In Euclidean space all straight lines are geodesics (shortest routes) no matter where you draw them. But in non-Euclidean space a straight line is only a geodesic if drawn from the origin. Any visual representation of non-Euclidean space is meaningless unless you specify the origin.
Now that I think about it, it's like the unwritten rule of non-Euclidean space: The Origin Is Everything. I feel like it's one of those things that those in the know forget to point out to those of us who are still scratching their heads
reply
OK here's my takeaway from this: In Euclidean space all straight lines are geodesics (shortest routes) no matter where you draw them. But in non-Euclidean space a straight line is only a geodesic if drawn from the origin. Any visual representation of non-Euclidean space is meaningless unless you specify the origin.
Now that I think about it, it's like the unwritten rule of non-Euclidean space: The Origin Is Everything. I feel like it's one of those things that those in the know forget to point out to those of us who are still scratching their heads
reply
Add a review, comment
Other channel videos
