Playing Sports in Hyperbolic Space - Numberphile

So I guess in eucledian space, you can use the function f(x)=x to measure distance. Every unit is one unit. In hyperbolic space, you can use g(x)=x^2, explaining how distances is going up rapidly the further out you're on the circle.

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

Nice

It feels so weird when a scientist uses imperial units

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.

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.

You could potentially still play baseball if you cut off some of the excess sides, and in golf light bends as well so it would look straight.

Aren't all the calculated values dependent on the curvature of the hyperbolic space? It might be fun to examine what sports would be like in a hyperbolic space with much larger radius of curvature, so warping is noticeable but not so extreme.

Keep in mind that professor here assumes that the hyperbolic unit of length is equal to 1 ft. If you put say hyperbolic unit = 100 ft, your local geometry will be way closer to Eucledian (less curvature).
Btw it is an open question if we actually live in hyperbolic space, just the curvature is so negligible that we can't notice it.

You should do this in elliptic space

Im sorry but I laughed so hard when i checked out his name in description!😂👌

This hyperbolic 2d space is way too curved for the sports examples! Please reduce the curvature significantly or otherwise reduce the scale of all distances (which is equivalent)! It's no fun with this extreme negative curvature, just like it's no fun with travelling large distances (e.g. several kilometers) on a small ball with extreme positive curvature (e.g. radius 1 centimeter). But a wee little bit of negative curvature is fun to get slightly unfamiliar results from both flat Euclidean 2d space, and from slightly positively curved 2d space on say a large ball (or projective plane). And it would be fun with such a computer game, where light travelled in hyperbolic straight lines (shortest paths), the landscape shifting subtly from what we are used to, when moving around.

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.

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 ?

It's mind boggling

Once my ug tutor in iitb asked :
If you go from A-> B
would you opt for longer path or shorter
Me: longer
Tutor : yes but I am talking about the normal people.
Eventually I dropped out

Imagine hitting the ball in golf almost to the hole and now there’s superclusters of galaxies between your ball and hole...

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.

😁

Do you really expect average viewers to visualize this? Your explanations do not help much.

But how do you put 300 feet in direction of radius, if it takes infinite time to get there?

I think a thing that people keep forgeting about hyperbolic space is that any movement through space would create a streching tidal force on an object, an object moving fast enough trough space would rip apart whereas in eucledian geometry, the object would just keep moving through space with no issues.

Thanks!

Doesn't it depend on scale?

Comparing two differently sized circles on a ball surface, their area and circumference compare almost in the Euclidian way, if they are very small compared to the curvature of the ball surface in which they are embedded. Wouldn't the same happen with hyperbolic space? I did not catch the video discussing the amount of the space curvature.

One of my favourite videos to this day.

what someone explain this
Someone should make a Wii sports but in hyperbolic geometry so that I can understand it

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.

vim pelo felps, super interessante

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

so it's like a fractal's edge.

There is a cool video that helps you intuitively visualize why the shortest path in hyperbolic space are those 90 degree intersecting circles. It's called "iluminating hyperbolic geometry" and a guy projects shadows of wireframe type objects to show it, and u see around the edges of the circle, the space has very dense lines in the shadow, vs less near the center, so u kinda wanna walk away from the dense edge part first, while keep moving closer to your new location, then return to the dense part bc the destination is there and ur forced to. But if u took the straight line path, it mostly travels through the dense area covering more space.

Is outer space therefore hyperbolic

Hory sheeto

How high was the guy that created hyperbolic space?

Thanks!

The nerdiest sports talk in the world

But that's just with unit curvature. With less curvature it would be more humane

never knew Pedro Pascal was on numberphile

Doesn't a hyperbolic space have a curvature term? As in, the surface of a sphere can have different curvatures related to the diameter, so wouldn't there be a similar variable for a hyperbolic surface?

Very interesting, but I have one issue.

The professor uses formulae that equate the curvature constant to the unit of measurement, which causes the plane to be significantly more curved for smaller units of measurement.

The best way to think of this is to compare it to Spherical geometry. If the curvature constant is equal to 1ft, then the plane would be very small. But if it were far smaller than 1ft, on the order of 10^-80 when compared to 1ft, then to a human it would be effectively indistinguishable from Euclidean geometry, like living on a ball so big that you can't see the curvature.

So for hyperbolic geometry, while the professors calculations are correct, they are showing the reality of an extremely curved plane.

In the game hyperrogue, the curvature is determined by the tiling used. For simple tilings, like the default truncated {7,3} (I think that's the correct terminology), the curvature is very slight, allowing for a gameplay experience that doesn't immediately break your brain. However, of you modify the tiling, say to {8,8} or something like that, the curvature is far larger, becoming far more difficult to project and comprehend. (The engine is also capable of things like {3,∞}, which has some interesting gameplay consequences).

Essentially, I'm just a little miffed that the professor neglected to mention the degree of curvature he was working with. I'd left comments years ago to try and articulate this, but I hadn't the knowledge or words to do so.

Didn’t know Pedro Pascal was a math professor

Hey if you guys want to do more maths like this yourselves and find the areas of hyperbolic circles that are bigger than a standard calculator can handle....

Y=log(pi*e^X) is very close to
Y=13/30*X + 0.4975

Since my calculator can't handle powers above 10^99

But with this approximation I can say you would need around 6*10^126 outfielders for baseball ⚾

Oh come on, in a universe with this kind of extreme curvature your golf ball would get ripped apart before it reaches the hole.

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

"He's going to aim a little to the left of the cup and... oh no. Oh no, that is MILES away."

The baseball rip video

This space is huge!

All of this assumes a unit of curvature of 1 foot, which is very extreme. It's like living on a sphere with a radius of 1 foot, but with the opposite curvature