Very cool. It would be neat to invert the solver by sending balls FROM the hole in every possible speed and direction. This would give you a 4D map of all possible starting locations + speeds + directions that go in. Not directly visualizable, but you could slice it like an MRI to look up the solution for any given shot location…

I love when analytics like this confirm what are traditionally held ideas. Typically a golfer is taught to line up their put such that the ball will come to rest about 18 inches beyond the hole. This is fast enough for the ball the sink, but not so fast that there is a chance that the ball skips out of the cup. Should you miss you will have an easier second shot at the hole vs a missed bullet shot. And since golf is a sequence of shots with the goal of completing the course in the fewest shots possible rather than a series of independent shots than clearly it is advantageous to set up an easier subsequent shot.

Minigolf designers: "WRITE THAT DOWN! WRITE THAT DOWN!

Excellent video! Simulations like these are super fun! By the way, I'm pretty sure the more accurate term would be "parameter space". Phase space does imply dynamics, where here you are plotting initial conditions.

I always thought it would be neat to draw out the phase space of a roulette wheel. Since roulette wheels are physical devices the outcome is completely deterministic (under classical mechanics) and depending on how we choose to simulate our roulette wheel, could depend on just a few starting variables, like the speed of the ball and it's starting angle with the wheel. Because roulette wheels are designed to be "random" I suspect the phase space to be quite chaotic, with fractal like boundaries, but it would be interesting to see if that were actually the case.

It's interesting seeing this, the engineer's solution of just simulating it a million times where so many math youtube channels would try to describe it with equations. Keep up the good work!

Innacurate, you didn't simulate me not hitting the ball on every single shot.

This video has CRIMINALLY low views! Very well made! Also, is your simulation code available anywhere?

you could make a phase space with cooking, the combination of the temperature and time is all you can controle, but there are multiple correct outcomes that cook it correctly I mean for this example there are formula's of course if you ignore that you can burn the food or cook it unevenly but yea xD still an interesting idea

I feel like this channel is the second generation of what KZfaq could be.

watching people do such beautiful engineering analysis for an oddly specific question no one has ever asked is my jam. i'm so glad i found this channel.

Easy course, the flags in a dip. Pro tip, don't take the shot with a frictionless ball.

I never knew Golf could be this interesting

If you look up "magnet pendulum fractal" you get videos of another situation like this. You can also plot firing a ball in space with planets, and plot what planet the ball lands on at each angle in 360 degrees.

In theory, you can control the initial "spin" of the ball

I remember watching my grandad hit a ball like a bullet into the flagpole, making the pole bend, then the ball falling straight down and directly into the hole. It was quite a ways away too.

I’m glad KZfaq is recommending me your older videos that I haven’t seen yet, every vid you’ve put out is SO good I don’t have the firmest grasp on machine learning but I’m pretty sure it’s somewhat similar to this. you have to find a minimum of the loss function (the minimum being like the hole in golf) and the parameters are the weights/biases for each node. When you’re learning about machine learning you see examples in 3 dimensions because they only show 2 parameters to make it simple, but there can be thousands of nodes, and the problem is to find the “hole” in the 10000-dimensional green I think.. it’s been a while since I learned that stuff

In Golf, you have control over the speed, the initial direction, and the spin of the ball. 0:33

"I decided to let my CPU do all the work" sums up just about anything I do that is mildly difficult. also, this is an incredibly well made video. it's crazy that it doesn't have more views. thanks for continuing to put out great content and putting effort into each video. hope you get the recognition you deserve

This would be of great interest to a miniature golf course designer.

Really interesting video, thanks for this. I’m a curler, and in that sport we spend a lot of time thinking about phase spaces actually. We don’t use that term, but in assessing possible shots given the current rocks in place and, crucially, how the ice is playing we think through a similar two dimensional space. We have a small wedge of possible angles and a range of ‘weights’ or speeds we can throw with. Part of what makes curling so interesting is the fact that the faster you throw the straighter the rock will travel, and slower shots will curl more. Once you have other stones in play you can find interesting places in that phase space, like maybe there’s a spot that I can get to throwing light where my opponent won’t be able to hit me out because if you throw hard enough for a hit you can’t get to that same spot. Phase space seems like a great tool for describing curling, I guess I’m off to go draw some diagrams!

I love how you built up the simulation resolution, it's honestly interesting to see the lower resolution before the higher ones. The crazy phase spaces at the end were so cool! As for other applications, the thing that comes to mind immediately would be for coffee brewing, temperature vs. technique/grind/amount/flavoring. It's probably not super exciting, but I'm sure it's been done by some coffee nerds out there!

While not quite everyday (unless you're an astrophysicist, aerospace engineer, or Kerbal Space Program player), DeltaV graphs are a perfect example of phase space, and I never knew it. Are prospective space program wanting to send something to another planet can be viewed as only being able to control two things about the launch: the date of departure and the date of arrival. The result is a phase diagram where each value is the DeltaV (change in velocity, can be thought of as fuel, however fuel and DeltaV don't have a constant ratio) required by the minimum energy trajectory for a given departure and arrival date. They're often times called 'porkchop plots', because the graph is bimodal, and has two adjacent regions of low DeltaV requirements, which somebody thought looked like porkchops.

My god this video and channel is a hidden gem in youtube's space.. I think we don't have to wait for much longer until this channel explodes with millions of subs.

When testing and debugging silicon products, I regularly express the properties of the test results in phase spaces for frequencies, voltages and signal delays providing insights into testing margins and others. Very fun stuff 👌🏻

One of my favorite videos I’ve seen in a while! I’ve thought about this exact idea on the course before and it was awesome to see it done so well

There's a local paddle tennis league in my hometown and a friend of mine is in it. Every time I go see him play, I get these visualizations that I've been trying to verbalize and this fuckin video has finally given me the words to do it. I basically imagine how every time a player hits the ball, there's an imaginary shape made up of all the vectors that result in a valid bullet shot (meaning it hits the floor first and then the wall), and because the edge of the court is a straight line, there's a very defined angle at which these shots cross into invalidity. This shape gets weirder than what I can imagine when we start considering volley shots at lower speeds, which would allow for higher angles to end in the court, because they would curve back down. Thank you so fucking much for this video.

I learned this intrinsically over playing for a while, and this perfectly describes the mental process I came up with to understand how to get better at putting. Good video.

This video was surprisingly AMAZING. So glad I watched it. Thanks!!

A year or two I went down a massive rabbit hole of making bread. I started creating phase space diagrams for baking temp and time. I then realized that temperature could be varied throughout the baking process, which would turn it into a 4D vector field plot. I never went this far, but it was fun. Phase space diagrams are also very common for motors and engines. RPM and %throttle (or, for electric motors, current and phase) on the X and Y axes, and efficiency (or just about anything else) on the Z axis. These are extremely useful to visualize the most efficient (or best in some other way) operating regions.

Fascinating. I will definitely try applying the phase space idea in my work. Thanks for sharing!

Phase space/parameter space has to be one of my favorite concepts in physics. Another great example of a change in mathematical perspective making a process a lot less complex-looking and a lot more intuitive.

This crosses over with a great video I saw explaining the interaction of chaotic systems and fractals. The inverted pyramid in this video might be a great example of that, where similar putts continually have different outcomes, even with miniscule changes, when viewed from a phase graph perspective

I think this approach could be used in F1 or other racing, as the combination of speed, gear and RPM (and param of steering wheel angle) showing perfect line to drive (as simulation f.e.)

Thanks for visiting Pinehurst! Love your videos man!

You gotta love people in academia that have passion. I met way to many people in my ME undergrad that just couldn't care less. Keep up the content!

Wow! This is so interesting and so well done. You're simulations are phenomenal. An excellent introduction on phase space. Also, so many aspects of the sport I never thought of. I'm subscribed.

Found your channel recently, really amazing content! Love your approach to these complex concepts

This is one of the best KZfaq videos I’ve watched in a long time

this is the same thinking that led to machine learning algorithms as well. its basically building a phase space, but instead of colors as you used, you assign a 'cost' to each node in phase space based on how far it is from the solution you want, then you minimize the cost function by following the gradient of cost through phase space to arrive at a solution. amazing way of looking at the world.

Thanks for this. It was interesting and fun. As a golfer I would mention that putts wobble a good deal as they lose speed/energy, which makes a big difference in line and direction.

this is gonna get millions of views now the algorithm has chosen it

Without knowing it until now...I 100% think of each putt in phase space. I take the angle that my mind assumes is the highest chance of a make with the lowest rollout on a miss. I see the angle in my head and then once I'm aimed...I have a little bit of leeway on the speed. The most important aspect is that I give enough speed to create a made shot, without going too far passed on a miss. That's cool and I fully plan on nerding out hard tomorrow when I'm playing

I really like Numberphile, but your approach to the different topics suits me better. I like watching simulations, and I love prototyping. I have never played or been interested in golf at all, but you just made me watch a 14 minute video about golf, and I found it very interesting. This channel is massively underrated when it comes to subscribers. Greetings from Norway!

God I love the KZfaq algorithm, Matlab enthusiast+golf psychopath=phase space putting sim Thank you for this, really fascinating simulation

hope the algorithm blesses this video, this was awesome.

This is a great analysis to get an objective measure of the difficulty of a course. The third dimension is angle up since you can pop the ball up and not just roll it flat across the ground. The next step is to build a machine that systematically shoots golf balls to see where they end up on a real course.

I'm with other people watching this. This video is indeed criminally underrated. As a physics undergrad that has been searching for the more in depth videos in science, I'm subbing.

Man, this video was AW-SOME! You should definitely do more simulation stuff on the channel. Also, really cool and intuitive explanation of a commonly encountered Diffy Qs concept.

This video isn't exactly new, but if you're still reading comments on this, a phase space I'd love to see would be feeds and speeds of a mill, you have many potential variables but you can get away with keeping some constant, here's a list in order of importance: - Feed (Velocity of cutter) - Speed (RPM of cutter) - Flutes (# of cutting surfaces per rotation) - Diameter (Diameter of cutter) - Material engagement (How deep the cutter is in a material) The reason you want to proportion these variables is for the finish on the material, feed per tooth (amount of metal cut by each flute every rotation), and forces applied to the cutter. Though, after writing this it does seem a bit complicated, lol.

Omg this is so interesting please do more phase maps and golf maps please!(finding a phase map that looked like a fractal , or working backwards from having a fractal as a phase map and building a golf turf around it would be fun)

5:00 is where this relates to real golf. I like seeing that where we are told to aim statistically makes sense. Because even if we aim the ball perfectly we still have to hit it perfectly.

Very cool. Once again an awesome video that breaks down a complex problem into something accessible to us non-physics types. When the iPhone gets a depth camera with decent range in a year or two it’s conceivable that an AR app will let you walk around a hole and build a 3D model of it and essentially create the Rodney Dangerfield putter from Caddy Shack. The app basically runs your AlphaPhoenix simulation, computes the high probability angle and tells you when you are addressing the ball correctly. It’ll give you clues on how hard to hit it too. Maybe it also incorporates environmental data to know if the greens are wet and slow. Of course this would be kinda cheating but a fun problem to contemplate.

This is such a rad channel. Thanks!

i've never golfed in my life. but i'm an engineer and i like simulations so i watched. glad i did. wish i knew how to model stuff like this

really enjoyed the visualizations, thanks!

As a computer programmer, I frequently think of how inputs lead to outputs. It's nice to have a word for it, so now I can look it up.

Design of Experiments - Taguchi method comes to mind. This, as I recall, allows for multidimensional variables and a reduction of "number of tests/experiments" to get to the answer needed. Great video, great simulation - Python is cool.

This is awesome! I’ve recently been obsessed with disc golf, and an hoping to (eventually) produce similar analysis/simulations of possible disc flights!

I’m a golfer but I definitely learned something new from this! I always wondered if it was possible for there to be no way to make a putt because I sure have had some tough ones 😂

very well done and quite insightful. Thanks!

Dude! That's an awesome insight! As a physicist I see myself thinking vagely in these terms, but not usually so explict. You mentioned you're not sure if that's called a phase space, I use the term "parameter space" to describe these things in more abstract problems (like your crystal growth example). In the golf case, it's actually a map of initial conditions, so I'm not sure whether "phase" or "parameter" space are adequate terms, but they are descriptive enough given your explanation. Cheers!

Really enjoyed this video! Mustve been recommended to me as a golfer - loved it

Reminds me of a problem in computer graphics. (Explanation of rendering equation and path tracing, skip if you already know what these things are) There is an equation called the rendering equation, in which one of the parameters is the total energy of all light hitting a given point. To calculate this variable, one would have to add up the energy of all light, from every angel, and every possible combination of bounces. Since this is basically impossible to compute, rendering programs must make approximations. This is done by selecting random paths between a point and a light source, and then averaging their intensity. This is where the part related to the video comes in: sampling random paths is a rather inefficient method to get an accurate result, so most algorithms will take samples that are believed to be representative of the total samples instead. There are algorithms that attempt to estimate a "Phase Space" plot of light intensity vs every possible path in order to identify the most preventive samples. Some use AI, while others use more traditional optimizer. An interesting side note: Caustics (those weird reflections that waves make on the bottom of the pool) are a lot like the white portions of the plot in this video. There are only a narrow range of possible inputs that lead to them, so they can be hard to sample accurately (they are usually removed to prevent visual artifacts).

Interesting to hear it's scientifically beneficial to shoot slightly past the hole. Good golfers also do this so that if they miss, they can at least get a "read" of the putt coming back the other way.

Love the mix of maths, physics, engineering, and coding ideas. Thanks for such an interesting video! Subbed

Wow!! Bravo again! You’re so brilliant 👏👏👏👏

i met your channel today and I really love it!

You just managed to make golf interesting. Well played, sir!

Wow! That one dimensional phase space pyramid putt made me so happy for some reason! I think there must be some interesting dimensional math puzzles with theoretical putting green shapes, could be a numberphile collaboration!

awesome video! I play a video game called Peggle with my parents and had been thinking a lot about this concept but just now learned the term "phase space"

If you compute this phase space diagram for every green friction value, you can tween through the successive diagrams like an MRI slicer. That'd be a cool segment to see in media coverage of a golf tournament. A weather man green screen style presentation of the phase space of each drive or putt at the introduction to a hole or course.

This is awesome! Keep this up, so impressive.

Amazingly enough, I was thinking about this EXACT problem for the past couple of months. Specifically, I was interested in whether there was a theoretical green that had spots on it that you couldn't possibly hit no matter how hard you tried. I'm somewhat surprised I managed to find this video completely accidentally (KZfaq algorithm actually deserves some praise, for once.) but thank you for answering that question for me.

Oh... Bryson's going to be all over this scientific approach!

This was surprisingly interesting. Thank you!

Thanks to you, golf is now even MORE complicated.

Honestly this would be a really cool graphic to have on broadcasts or something to understand the complexity of a putt. Shows the amount of break, how difficult it is to stop near the hole, how much margin there is to avoid it running off down a big slope, etc.

We all know the "fast straight" shots, and a lot of the ones close to them would just go over or flick out of the hole.

Sir you have been blessed by the algorithm. Congrats.

You could pair a drone to do an aerial topography scan with a putting robot and implement this with it.

Can't wait to see the sign on the wall say 2^20! What a fascinating channel!

Good start. 8:40 What's your logic as far as speed of ball which goes in? The faster the speed of the ball as it hits the cup, the less chance the ball goes in. The hole size effectively becomes smaller when the ball is going faster (e.g. either skips over the hole, or catches an edge and spins out rather than falls in). That's why there's an "optimal speed" of a ball going into the hole. In addition, while a specific phase space might increase your chances of a single putt, a player must also recognize the odds of making the next shot if he misses. See "strokes gained" for odds on making putts from certain distances.

beautiful visuals, excellent explanation, very interesting topic. nice.

As a golfer and math nerd, I loved this video! Subscribed! Oh, and I know fast greens. Very well.

I always forget about phase space and then am mind blown by a novel application of it. So cool!

This was a really cool topic in thermodynamics/stat mech., but I didn’t think it could be broadened to modeling “optimal putting shots”! Awesome video! Makes me wonder on the similarities and differences with a phase diagram would look in a game of bowling (assuming no bumpers). We could claim there are 3 degrees of freedom: angle & speed of throw, and the thrower’s position on the bowling lane entrance. “Changing the terrain of the putting” could map to a change in pin arrangement of the bowling lane (e.g maybe you have a split, or half the pins are standing in many different ways, etc…) which would yield many cool simulations. Food for thought and something cool to try perhaps.

Really cool video! Being a golfer and a maths / Matlab nerd, it would be really interesting to somehow model an actual ball falling into a hole. I'm sure there must be some physics equation that will allow you to calculate things like lip-outs, horesehoes, and the speed limit of "ram-in" puts. Really really cool video, thanks!

When you really think about it, professional golfers (or any person who has become highly skilled at something requiring precise motor coordination) have essentially mapped out that phase space in their brains. They just develop the intuition to know that they can make a particular shot at one of two angles, one of which is more forgiving in terms of the power of the shot. Additionally, their brains have far more variables to consider. They aren't directly controlling the angle and power of the shot-their brains have to simultaneously control hundreds of muscles in just the right way in order to swing the club as intended. It would be cool to see phase space diagrams for bowling. You would probably have to find a way to add another dimension to the diagram to include the spin of the ball, but it would be interesting to see how the diagrams would look for splits, strikes, etc..

Ball spin, the bane of my golf game, adds a new dimension to a simulation like this.

This is the third video I have been suggested by you. Some pretty good content you have going here.

This was a great approach to demonstrate the concept.

Absolutely freaking beautiful.

actually an incredibly interesting video

This is the first video of yours I have watched. It was excellent. Thanks. Trying to think of everyday phase spaces now. 😆

Great job, I loved this video!

An explanation of what most golfers learn without any understanding of the maths/physics ☺️

I used a similar method to find orbital trajectories.I have 2 parameters, start time and deltaV and I want to check where the space craft will end up for each combination. Made it easy to find rendezvous encounters

This is super cool. I would love to see a similar video about how phase space diagrams might look on a pool table!

You, sir have just earned a subscriber.