Playing Sports in Hyperbolic Space - Numberphile
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- เผยแพร่เมื่อ 17 พ.ค. 2015
- Dick Canary on baseball and golf in hyperbolic space.
A second part, looking at soccer and more baseball: • More Hyperbolic Sports...
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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.
Numberphile did a video about hyperbolic space some time ago.
xisumavoid Will it help you if I tell you that kale is essentially a hyperbolic area curled up into euclidean space?
Wow. Did not expect Xisuma commenting on a numberphile video...
xisumavoid Well, i just went from watching Xisuma's latest episode to here, only to find out I can't get away.
xisumavoid The most relatable part to me was the drawing around 4:25. If I interpret it correctly, the field is still a constant angle wide. Applying that to the golf scenario, you see that a constant 1 degree angle error becomes an exponentially growing error in horizontal distance. Whereas in Euclidean geometry (e.g. the regular baseball field), the horizontal distance grows linearly for a constant angle.
If you want something to relate to, consider the elliptical geometry of spheres. For example, great circles (the fastest route for air travel can look like an unnecessarily long curve on a normal map). Or speaking of golf, consider that you are at the 'north pole' of a sphere and putt the ball such that it will roll along the ground until it reaches the latitude of the hole. If your putt is off by 1 degree, the ball will be 1 degree of longitude off in the end. The corresponding distance along the surface that your ball is away from the hole varies non-linearly with latitude (and distance putted). To the point that, if your hole is on the opposite hemisphere, the distance error will start to reduce. And if your hole is on the south pole (opposite point of the sphere from you), you will obviously make the shot no matter which direction you hit. More to the point, a hole very close to the south pole would result in a very small distance error even if you have a huge error in shot angle. Again, this is elliptical geometry, not hyperbolic, but it's non-euclidean and actually relatable!
A good way to imagine hyperbolic space is this: in the middle you have a special height. And the further you go to the "boundary" the smaller you get. Then your steps would be smaller and you would never reach the limit.
Also, if you want to make a shortcut, like brady suggested, you wpuld get further away from the middle, therefore you would get smaller, and the way would take longer to go.
superliro100 I think this analogy actually helped me understand this a lot better. So it's like, the farther from the center you are, the harder travel is, such that going the shortest "distance" might take longer than making a detour towards the center so that travel is easier?
Or, like looking down from on top of a mountain? Two paths that look like they are the same distance from your perspective (i.e., they both take up the same amount of your view) could actually be very different distances in real space, because things that are farther away look smaller from your perspective, so things that look like they are the same size from your perspective must be different sizes if they are different distances.
superliro100 Thankyou. I have watched all the number phile videos on this topic with no understanding. This explanation is so simple, I don't know why they never took the time to say this.
superliro100 So it's actually impossible for the golf ball to reach the boundary? (Meaning there is no boundary?)
superliro100 Your explanation made way more sense than the video. :)
Nathan Richan The boundary of the world doesn't exist, it's just a limit, but the boundary of the circular golf course certainly does.
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?
Dryued So for all we know we do exist in hyperbolic space.... ahem gravitational fields that warp time and space around them.
Dryued Spot on. This video is overly obsessed with the weirdness of Poincaré disk model used to visualise hyperbolic space -- if you were actually in hyperbolic space, everything would look normal (as long as you don't look too far) and you'd be able to play golf just fine.
TheHue's SciTech It's still true that if you fire a shot far enough away from the hole (relative to the spatial curvature, which they totally never really addressed here, they just say "300". Okay, 300 whats? And how curved is the space in those units?) that being off by only one degree would put you only a few thousandths of a percent closer to the hole than shooting 180 degrees off would.
You would experience this as trying to walk towards the hole and the imaginary line representing your optimal shot would veer far away, and eventually behind you compared to the imaginary line you are really following to get to your ball.
Happ MacDonald Absolutely, the translation from "300 feet" to "300" is a massive problem. Like I said in a separate comment here, that's equivalent to discussing elliptical geometry (think: the earth's surface) and translating "300 feet" to "300 radians", and concluding that if we lived on a sphere (imagine that) every golf shot would orbit the earth dozens of times.
TheHue's SciTech
That brings up an interesting point. Radian is absolutely a unit tied to the curvature of Riemannian/elliptical space. Do you know of any complement for hyperbolic spaces?
The best idea I can come up with are along the lines of "the distance at which you can fit X non-overlapping equilateral triangles all sharing a vertex". I know that for X = 4 on a sphere that distance is π/2 radians, and for all distances in flat space X = 6.
Is it a special unit like that which has to be plugged into those hyperbolic area and circumference equations? :o
This video wasn't that... "straight"forward.
Hahaha, see what I did there?
...
I'll see myself out
Robert Yang If I bend you over, will that straighten you out?
Jeremy Joachim You can bend me over anytime ;) Nah, I'm already straight.
Robert Yang Hyperbolic space is nothing but an exaggeration.
Legend has it that Robert Yang is still on his way out
not funny
didn't laugh
sorry :(
So in other words, hyperbolic golf is pretty much like normal golf is for me.
Your profile picture fits that just right.
They should make a game of golf computer game in hyperbolic space.
There's a neat roguelike set in hyperbolic space called HyperRogue!
CodeParade is making Hyperbolica. You can try that out once it's out
Instead of suggesting an unrelated hyperbolic game, I'm going to agree with you! There could be two versions - Euclidean golf translated into hyperbolic space (just a gag game basically, maybe it always tells you your distance from the hole and you just watch it get bigger with each stroke lol. Then you get powerups which increase accuracy but it still doesn't help 😂 ) and an actual playable version where the the distances are all super small. It's basically just tiny puts, but it's still super hard. Maybe there's a third version so that the player is scaled down to a size where the physics make more sense and you can actually drive.
On the apple App Store there’s pool and other things in 2d hyperbolic space called hyperbolic it’s pretty similar to golf
Sports are *impossible* in hyperbolic space.
Brady you troll, actually used his drawing of the golfer :D
Thanks you made my day with that :)
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?
Penny Lane In what way do they suck? You can make the rules for a good game on virtually any regular grid.
Sprite Guard Alpha Squares either don't allow diagonal movement or you have that nasty factor of ~1.4142 that you either have to pretend doesn't exist or somehow incorporate in your game.
Hexagons don't allow straight movement in one direction, although they do in the other. That's absurd if you think about it.
Don't get me wrong, I like me my hex grids but _if_ octagons tessellated the Euclidean plane, suggesting a game on a hex grid would sound like the worst idea ever.
Penny Lane Ok but a theoretical octagon 'grid' allows straight movement in 8 directions but you still can't move at, for example, 30 degrees from the 'horizontal'. You're always going to be forced to zig-zag and take a longer path when traveling towards the corners of any polygon grid.
However, it is true that as you increase the number of sides of the polygon, these 'zig-zag' paths become closer and closer to linear. The logical course of action is, therefore, to use a spherical geometry.
So basically just bring a ruler and forget the grid.
Jason Slade You can play your board games without me, sorry. I'm not going to calculate my ass off and get into fights about measurement inaccuracies with your brilliant continuous approach. I like me my discrete turn-based games thank you very much.
Penny Lane You can design a game to any grid with good results. Go only works on a 4-connected grid, Chess only on an 8-grid, there are lots of games that only work on a hex grid.
In what game is the difference in scale of checkerboard axes actually relevant to anything?
i don't understand baseball nor hyperspace but i still watch this video
Minh Nhân Just want to correct a bit of your semantics if you don't mind. Hyperspace is a space beyond the dimensions we work in. So currently for us hyperspace is 4+ dimensions. Hyperbolic space is based on space which is bounded within the curvature of a circle. So I think you were referring to hyperbolic space based on the content of the video.
tehlolzfactor thanks i get it now
tehlolzfactor I know you two are trolling, but just to be clear -- you can travel in a straight line forever in hyperbolic space, and you never reach an "end" or "limit". The dotted circle on the page is just the end of the projection (which corresponds to infinity in the space).
If you see my other comment on this page you will see that I do have a proper understanding of the topic and that I am indeed not trolling. I may have been a bit unclear when I wrote that hyperbolic space is "bounded" by a circle. My intended meaning is that an infinite plane in euclidean space becomes something of an "infinite circle plane" in hyperbolic space where the coordinates are shifted from cartesian to hyperbolic. I hope I have made myself clear here. Cheers.
tehlolzfactor OK sorry, I completely missed the point you were trying to make in your first post, I stand corrected, sorry. But hyperbolic space is definitely infinite; the only thing that's bounded here is the map/chart we might use to view it (in the case of this video, the Poincare disk model). Hyperbolic space: infinite. Poincare disk model: bounded by disk. I know you know all this already, but I think it's a very important distinction to keep very clear.
I've concluded that hyperbolic....anything... is really fucking weird.
***** Just think that if you stand at the north pole (or anywhere), and you walk in one direction, and your friend walks in exactly the opposite direction... starvation and drowning aside, you would meet right back up at the south pole (or the antipodes of wherever you started). That's because Earth is a sphere(-ish) / it has positive curvature; hyperbolic space is just the opposite where you have negative curvature.
So wouldn't you just be on the inside of a sphere, and why is a straight line infinitely long?
Nabre Labre No, if you're inside a sphere, you'll still find that the angles of a triangle add up to more than 180 degrees. Being on the inside is the same as being on the outside; you're still constrained to the *surface* of the sphere regardless.
Why is a straight line infinitely long in hyperbolic space? Because it has no particular boundary to stop it. The fact that lines on a sphere run right back to their own starting point is the special case here.
+TheHue's SciTech so is hyperbolic space a way of viewing the cartesian coordinates vanishing into the horizon.
My heads hurts! What is the best way of actually visualising a hyperbolic plane in my head?
TheDiggster13 Like a hamster wheel.
TheDiggster13 The inside of a hollow sphere.
Thanks guys! Makes more sense now :)
livarot1 So it's like a trumpet? With the origin infinitely far from the part where the sound comes out?
James Oldfield That's not true, google "Pseudosphere"
wow this is my real analysis professor at university of michigan, interesting stuff
I never liked golf anyway
Dev Mehta Golf sucks
economíamatemática screw you it's a fun sport, and a good bonding experience since I play it with friends and family
PraetorianCuber But rallycross is cheaper and more fun.
get out of here Jezza, this is BBC property!
"Imagine you are a golf player here...with a golf club coming out of your freaking chest!"
Really interesting, but in Hyberbolic space would you still be able to hear the WOOOOSH sound that this made as it went right over my head...
[add] I guess it would have gone around my head...
brady i still think that is still needed I full program about noon Euclidean geometries with a historic background included. I wouldn't mind at all if that video lasted 1 hour or even more.
migfed That would be perfect!
migfed Agreed. There should be a video where a professor makes his best effort to break down non-Euclidean geometry for non-mathematicians.
migfed Not much to say. Spheric and Euclidean geometry have been studied for many many hundreds of years. The hyperbolic space was not studied for equally long because many considered it to be "bad". Therefore the study of hyperbolic space started at 1800-1900's. It started with Bolyai and Lobachevsky(which is also why hyperbolic geometry sometimes is referred to as "Bolyai-Lobachevskian geometry") which stated a successful new parallell postulate.
Is hyperbolic space hard to study? Well, yes. For me it was just a bunch of formulas thrown in my face and we had to happily accept it and learn it. I found the hyperbolic geometry the most boring simply because it is so intense in formulas. Everything you want to do needs a frickin' formula.
@numberphile pleaseee
2:44 "hyperbolic sin times R"??
Brady, you should get in touch with the people who crochet hyperbolic plane representations. Great video as usual!
Hyperbolic Geometry is like the absolute coolest thing in the frickin' universe!!!
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.
you're absolutely fantastic brady thank you dearly for making these videos
Words cannot express how much I love this man's name.
Jokes don't even need to be written.
... Coal mine.
My sincerest apologies, Mr. Canary.
I dig it
Ohhh bboooyyyy!!!😂😂👌👌
This is one of the most interesting videos on numberphile!
I cannot even. I have lost my ability to can.
blackkittyfreak Use a flask or a jar instead.
But can you odd?
This video was one of your best in a while(at least in my opinion). I haven't looked into hyperbolic geometry before so this was really cool.
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.
Loved this vid! A little heaviness now and then is great. Thanks Brady.
Thank you, Numberphile, once again for another eye-opening video!
Man, hyperspace always got me wiggling.
I keep forgetting that you just have to imagine it as 'stretched' space to the end and 'compressed' space to the centre. The reason lines curve inward is because there every 1 length becomes 'worth more' length further outward.
Space being worth more space depending on it's place in space.
Neat.
Yay! More from this guy please, this is much more interesting than the usual "weird number series" videos.
One of my favourite videos to this day.
do a video on every millenium prize problem
Incomprehensible for me as most of these videos but something keeps me coming back for more!
So this is why we don't learn about this in school...
I've learnt about in further maths for my A-Levels! It's more simple than this though haha
***** I learned about this 2 weeks ago, but I'm in university so that is probably why.
Alex Shelley I was confused for a second. In Australia (in Victoria, to be precise) further maths is actually the easiest maths in year 12.
***** in England, further maths is the hardest maths you can do
Before a degree that is
I stopped understanding at 0:10
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.
Dick Canary is amazing. I laughed so hard at this video, best of Numberphile!
most interesting video in a while! keep it up :)
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'm not sure it's techically playable in the first place though
Sergio Garza You should check out hyperrouge. A rougelike game in hyperbolic geometry.
SpySappingMyKeyboard
That's awesome! Thank you, I'm definitely gonna play this game!
Sergio Garza Try making an FPS in H^3 and let me know how you set gravity up....
Cooper Gates
That would depend on what kind of surface you want to be on top of and how that would be shaped. If gravity is the result of the bending of spacetime, then we would have to discuss both what kind of surface we're standing on and how intense the global hyperbolic geometry.
Im sorry but I laughed so hard when i checked out his name in description!😂👌
now that one can experience this in game "hyperbolica", it is useful
7:56 "this path isn't open to me"
Hyperbolic dude: yea but it's 10^100
Nice
Please make more of these videos
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?
Interesting and well explained !!!
Hi Numberphile,can you do a episode on the caesar cipher? great episode once again btw
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.
Wouldn't the baseball players bend together with the space so they would cover the field just like they did on euclidean space?
There's a youtuber called CodeParade who's releasing a game on Steam that takes place in hyperbolic space. He's got a great TH-cam channel as well.
And he mentionned hyperrogue in his latest video.
And I can confirm everything said in that numberphile video, the slightlest deviation quickly lead to an enormous change of path....one goal of the game is to find an orb of yendor, first you find an orb, and it'll create 100 tiles away a corresponding key. You got that key? Great , now go back to the orb.....wait from where I came from?
Learning this kind of stuff in my astro course at uni. It's so hard to get my head around, conceptually, as to what is actually happening in hyperbolic space.
***** I imagine it's a lot easier to imagine with 3d modeling, looking at it on a 2-D surface really can't do it justice.
Please make more videos about hyperbolic and spherical geometry!
On a side note, it is refreshing to see a mathematician on this channel who isn't a natural artist with perfect penmanship.
I like the sound of this one nice title mate
In the last part, the golf stuff, I think if we could see the ball at the hyper space, before hitting it, we'll see a straigth path between the ball and the hole, this is because the light travel at the shortest path, well i guess.
great video, thanks brady, I like very much this channel
I found this program called "magic tile" which has some hyperbolic puzzles, this helps people understand why these crazy distances happen.
This was badass
very interesting, and exposed in a simple and fun way!
are you going to do something similar with elliptic space?
Giancarlo d'Alessandro admitting it could have sense, that I really don't know, I admit!
Never really studied hyperbolic spaces in a serious way so that was a bit of a punch in the face for me. Very interesting though relating it to real life situations. Thanks!
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.
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.
That Stick figure pasted in the right side of the euclidean golf, killed me, ROFL.
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.
Imagine hitting the ball in golf *almost* to the hole and now there’s superclusters of galaxies between your ball and hole...
but a "curve" of the ball can make it go into the hole like a normal golf
Thanks!
It's so weird to me that the boundary of that hyperbolic space is right there, and yet you have to think about it as being infinitely far away. It's RIGHT THERE!
I can't wait to play Riemann-Surface Tennis...
Yes! Finally!
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.
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.
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?
My head really, really hurts. (Math is awesome!)
Shouldn't all these numbers depend on the size of the hyperbolic "disc" it's played on? In my imagination, the area around the center is very euclidean-like and it becomes weirder towards the edge. But that may be wrong.
You're talking about the curvature of the hyperbolic space. Right. This whole video assumes a space with bretty extreme curvature.
It's not exactly like that, because if you stand in point A and look right in to point B the light will go in the fastest possible way, so that will be our "curve" direction. It means that nothing unexpected won't happen - the shortest way to point B is in direction of point B.
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
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.
Great Video. Also by the way the annotations to the next video haven't been updated yet.
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 ⚾
That's the most badass golfer I have ever seen in my life!
"He's going to aim a little to the left of the cup and... oh no. Oh no, that is MILES away."
Do more videos about hyperbolic space..
I think I died when they put his abstract golfer in at 5:14
This could make baseball watchable. Not any easier to understand, but watchable.
Very interesting :D
Far more interesting then the stuff we do in 8th grade right now . ( Germany)
Can you do a video about the 4th dimension? It would be really fun.
I like watching these even though I don't understand
nice vid
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?
This would give a completely different result if he worked in metric :) which needs the question: what if we defined the unit length as the longest length a available to us in each model, then it would be infinitely shorter distance calculated (I think)
The area of a circle can also be found by using the formula A=C squared/(4pi)
Interesting to hear about it, keen to know what its place in maths is?
It's mind boggling
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).
The hyperbolic baseball game would be possible if the measures of the radii of both quadrants were not 100 and 300 feet. They only needed to be quite smaller. It would only be a little weirder though, because it would be a field that is not that long, but will be somewhat wide. If the radii were 4 and 12 (respecting its 3:1 proportion), the field would have an area of 127784.
It sounds like playing sports in some sort of weirdly frustrating dream o.0
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.
This dude must really love numbers.
"Numberphile"
I can't wait for them to explain how hyperbolic space relates to compounded interest.
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.)
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.
If anyone is as confused as I, after reading a tone of comments and googling some stuff the way I came to understand it is if you have a binary tree, IE.
6
3,9
1,2,4,5,7,8,10,11
Where each level grows exponentially.
The fastest way to transverse from 1 - 11 is not to go along the bottom, 1-2-4-5-6-8-10-11 but to step up and back down the tree, 1-3-6-9-11.
And the tree is wrapped into a circle, where higher tiers are toward the centre, and the lowest tier is the circumference.
The numbers are arbitrary.