Playing Sports in Hyperbolic Space - Numberphile 

Its interesting, but without an explanation of what hyperbolic space is i found most of this to be a bit trivial since i couldn't relate the math to anything.

This video wasn't that... "straight"forward. Hahaha, see what I did there? ... I'll see myself out

Wait a second. Should there even be any confusion about the shortest path on the golf course, if light would travel in hyperbolic patterns, too, so you'd *see* that path as straight?

So in other words, hyperbolic golf is pretty much like normal golf is for me.

It seems like almost every euclidean sport would be fundamentally unplayable, as minor deviations compound exponentially. The question then is, can we invent a sport for hyperbolic geometry?

I've concluded that hyperbolic....anything... is really fucking weird.

My heads hurts! What is the best way of actually visualising a hyperbolic plane in my head?

"Imagine you are a golf player here...with a golf club coming out of your freaking chest!"

Brady, you should get in touch with the people who crochet hyperbolic plane representations. Great video as usual!

You forgot to mention, the unit the calculations are done refers to the curveture of the hyperbolic plane. Depending on the curveture you choose, the numbers you get become more Euclidian or more hyperbolic.

7:49 a great thing to mention is that "your euclidian eyes" would see the light also travel the shortest path, you would see it being 590ish feet away, not a misleading 5 or so, but a daunting 590.

Someone should make video game where you play pool, billiards, in hyperbolic geometry. People are going to play it regardless of how difficult it is, like solving a 4-D rubik's cube games

I have no idea what he's talking about but I love putting on videos like these when someone is in my room and nod my head pretending that I'm understanding it perfectly, makes me look really smart.

What I get from this video is basically "The further away you are from the center, the bigger the distances become." That's why the shortest distance between two points is curved towards the center, because this way you are passing through areas where distances are shorter.

now that one can experience this in game "hyperbolica", it is useful

I did not get a visceral understanding. If some one did a FPS game where you were in hyperbolic space, maybe that would be visceral. But there would be the bigger problem of how physics would work, or if it would work. Would the rule for the combination of force vectors make any sense, because it depends on the Pythagorean theorem, which depends on space being Euclidean. Would gravity behave anything at all like what we have in Euclidean space, because it depends on the way forces combine? Would light reflect the same way in hyperbolic space?

I think it is very logical. The only thing I did not understand is why the heck the ball is supposed to travel as a straight line in this curved space? I think it should perfectly follow the shape of the space which for the inside observer will not be distinguishable from the straight line since the light will also travel at curves.

I think you should mention that these paths make sense when you take into account that it's how it works also for paths on the surface of spheres or saddle-shape planes. When you draw "straight" lines on the Unit Disc it's as if you did the same on the a flat map of the world. For example: go to google maps and make a straight line between London and Miami. But because the World is actually rounded, this line is actually a longer distance IRL than if you made a curved path approaching Greenland and curving back to Miami. The actual short paths on a sphere are called arcs of Great Circle. They are found by having an euclidian plane intersecting both London, Miami and the center of the sphere. This way you get the only largest circle that connects these points, thus the name. This abstract maths surrounding hyperbolic space is still very useful because there are more situation in the Universe where we need to calculate weirdly shaped curved surfaces than just spheres. The best example is the geometry of Space-Time which is a 4-dimentional entity which surface we stand on, thus all the unusual paths we perceive when travelling at great speed or distance (or within great gravity fields).

If we consider that the border of the circle is actually an infinitely far away horizon, I think the distance to the hole itself (that you placed near the border) is already pretty huge so it’s okay to miss a lot

There actually is a way to win hyperbolic golf and that's to keep taking shots of such a small distance that the ball is more likely to get closer to the hole than further away aiming for the hole in each of those shots. The more skilled you are at controlling the direction you shoot the ball, the further you should shoot it in each shot.

two great ways to imagine hyperbolic space visually: 1. youtube the game "hyper rogue" and look at the gameplay 2. look at a video of the home screen on the Apple watch. very similar

I clicked because it sounded like this might explain the ending of that space movie, where there is baseball at the end and weird floating cities and stuff.

My head really, really hurts. (Math is awesome!)

This could make baseball watchable. Not any easier to understand, but watchable.

Interesting to note, in both hyperbolic space and euclidean space, the circumference of a ball is the derivative with respect to R of the area.

Hi Numberphile,can you do a episode on the caesar cipher? great episode once again btw

Please make more of these videos

Also this has made me think it's weird how hyperbolic functions like sinh and cosh diverge rapidly, whereas Euclidean equivalents like cos and sin just cycle round. And Euclidean trig can be imagined using the horizontal and vertical components of a line connecting the origin to the circumference of a unit circle. Is there a similar way you can visualise how hyperbolic trig functions work?

For those of you better versed in hyperbolic geometry, I have a few questions. 1. If being even 1 degree off puts my ball further (almost twice as far) from the hole than where it started, then how accurate do I have to be for my ball to end up closer than before I hit it from 300ft away? 2. How does this change if the target is closer? If I only need to putt the ball 6ft, how accurate do I have to be to put the ball closer to the hole than before I hit it? 3. It seems to me that the best strategy in hyperbolic golf is to only hit the ball a short distance that will scale with my accuracy, to put the ball closer to the hole than before I hit it. If my accuracy is within 6 degrees, how far can I hit it from 300ft to ensure it ends up closer? 4. What if my accuracy is within 3 degrees? 5. How does this change as I get closer? Once I get it half way to the hole, 150ft, can I aim further, even if my accuracy is the same?

What confused me the most (until further research) was the statement at 0:45. In hyperbolic space, those aren't semicircles, they are just fractions of a circle. Also, from 7:20 onwards, they refer to a diagram that shows a curve that is not circular, which further compounded my confusion. This was a nice video, but as many people have already said, the description of hyperbolic space in this video was very lacking.

I really loved the video, but I would appreciate if they showed how you actually have to play golf in hyperbolic space. The current style showed is based on Euclidean assumptions. How would a golfer hit the same ball in the hyperbolic space to get the result they get in the Euclidean space?

Next topic, golf in hypergolic space with Professor Poliakoff :D (I have a vague memory of something called hyperbolic space from some math class decades ago.)

Loved this vid! A little heaviness now and then is great. Thanks Brady.

Ok, i thiiiiiiiiiiiink i understand it? What I got from the video was that space was essentially more dense the farther away from the center you got, so what looks like 1 foot really far the away from the center is actually the same as a lot more feet really close to the center, which is why its easier to go into the center, move to your destination and then move back towards the edge then continue along the edge. This is all pretty much a guess tho, someone please correct me.

At 5:07 you can see a second copy of the professor's "golf guy"

The best strategy for hyperbolic golf world be hit it what looks like half way to you, then move to the ball and hit it half way again until you are right next to it

A hyperbolic outfielder could probably cover a much larger area. A euclidean outfielder has 1256 square feet within 20 feet of them, whereas a hyperbolic outfielder has about 3 million square feet within 20 feet of them.

I’m pretty sure if we existed in hyperbolic space distances would still look straight it just feels further away or space feels more “voluminous”. This is because of the way light travels in hyperbolic space.

Is all hyperbolic space the "same" in the sense that the golf ball mistake would result in the ball being the same distance away every time? Are there hyperbolic spaces that curve the ball more or less away, so that the number is different? Also, this sort of curving phenomenon happens everywhere in hyperbolic space, so there isn't a "center" to the curvature, right?

so in hyperbolic golf, if you miss by 1 degree of a circle, you are about googol meters away from the hole if the ball goes exactly 100 yards?

Some similarity to the universe, is the space we live inside a hyperbolic space in universe scale? We might be on the relative flat part of it and near the edge the infinite long make it kind enclosed. In a sense it sounds similar to the universe expansion theory.

Never knew the h in cosh, sinh, stood for hyperbolic haha. Just goes to show I wasn't listening in class.

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.

What I don't understand is, wouldn't light travel the same route as the golf ball, meaning that no matter how where you hit it it'll still appear to travel in a straight line and travel the same distance as in Euclidian space?

Maybe in hyperbolic space the area covered by each outfielder also grows?

is the equation for the circumference of the hyperbolic ball not the derivative of the equation of the ball's area?

It's mind boggling

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.

Just last week I thought it would a good idea to build an elliptic pool table as an educational tool.

I think another problem you skimmed over is the 3rd dimension. No one hits the ball directly towards the hole except when putting. Most people hit in into the air, at a significant angle away from the hole. Does that mean you will be screwed regardless?

A man walks into a hyperboloic bar, he walks straight towards the bar but never gets there.

Futurama should make an episode based around this.

Excellent examples. But as a physicist, I must point out that he abused the units. And I think I know where he did that. He never mentioned the degree of curvature (a quantity measured in length). That would have to go into his equations somewhere and would have likely cancelled the R so that the units worked correctly.

Thanks!

So there"s like more 'space in the space' (visualised on the disc) when you go out from the centre? So if you want to cross space out near the boundary then it's shorter/straighter to go via the inner section. Can you have hyperbolic time?

In the golf example wouldn't the ball travel in a curve as opposed to a straight line because of the least action principle?

He explicitly said the radius is infinitely long, why then is the hyperbolic area, defined in terms of an exponential of radius, not also infinite?

Numberphile What kind of spacial geometry do we live in, and how do we know?

Are there any games which simulate hyperbolic space with a decent degree of accuracy? A minecraft mod that did that would be amazing.

Shouldn't the curvature of the hyperbolic space effect equations for area and circumference?

I'm a tad confused. Is Hyperbolic space a single defined set of rules or is it a broad category for any theoretical system that alters Euclidean rules?

vim pelo felps, super interessante

Great Video. Also by the way the annotations to the next video haven't been updated yet.

So does this mean hyperbolic space has an objective centre, unlike Euclidian space where coordinates are relative and not definitive. Please correct me if I got something wrong.

Maybe its me that has the settings wrong or something, but i am never able click on the links at the end of all the Numberphile videos... Whats wrong? :-)

Can someone explain to me how we know that "hyperbolic space" and "higher dimensions" are even real? Do they even exist? I can follow the math and the abstract/conceptual thought process that leads us to our understanding of hyperbolic space and higher dimensions, but I just don't see the purpose of studying them, where they might lead us, or how we can even prove that they're real.

Can you do a video about the 4th dimension? It would be really fun.

what happens when zeno walks half way to the golf hole, then half way of the remainder half... etc... hyperbolic zeno's paradox?

Is it natural to sit here and think "What the hell is this supposed to be good for..?!"?

3:12 "This grow quadratically" or "This grows quite radically"? :))

Do they use these kind of things for navigating airplanes and such because the earth is shaped like a sphere and when you want to represent that '3D' navigated route on a '2D' plane, it should be curved right? (Maybe it's a dumb question, sorry. xD)

It is so hard to get your head around it, if you cant attach a shape or an object to the hyperbolic space. I mean how the f. is that little distance 10^100 ?

How about ree kicks in football/soccer in hyperbolistan?

Which odd number is the most even?

I wonder if there's a creature in hyperbolic space right now making a video pondering the unusualness of Euclidean space.

Is there anything such as Parabolic space though?

Is this similar to polar coordinates? also, hi xisumavoid :D

Links in the video and not in the description :-( sad face. I can't get to them on my iPhone. They don't seem to be listed anywhere either. :-( ?!?

Umm... so how does hyperbolic space work? Can you give a better definition than 'a parabolic curve is the shortest distance from *a to *b'?

Can you do a video on Squeeze Theorem?

Это геометрия Лобачевского?

One question, where in the universe can we find hyperbolic space?

Is there a way to think about hyperspace to make it more intuitive or imagine it more easily, or do you just have to tell your brain to stop asking questions and accept that this just works mathematically?

That's weird. I like it. :)

I'm from the city of Euclid in Ohio, and I think our teams would have better chances playing by the rules of hyperbolic geometry...

7:46

I'm REALLY disappointed there was no talk of a hyperbolic time chamber...

The nerdiest sports talk in the world

Dont circles and balls look differently in hyperbolic space?

This is really interesting but what I was hoping for was for you to ask him what the point of hyperbolic space is. What do we use it for in maths and science etc? Is there anything that's best described by hyperbolic space in nature? the inside of a black hole or the path of a superstring though the universe or something?

He's like a mathematical Zach Galifianakis. By that I mean his appearance and voice.

6th! 301 club!

My head hurts

Before watching this video,I had troubles with geometry.Thanks for confusing me more.

so a map of the earth is sort of like hyperbolic because the shortest path is actually a curve on the map?

so it's like a fractal's edge.

So... is 300 feet too long in hyperbolic space?

Please, subtitle the video in english? I can read, but I can't understand only listening them.

Nice

Guys, I LOVE your videos, but I hate hate hate hate HATE the sound that marker pens make on rough paper :( Any chance of you switching to pencils?