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…
Nice to see you here! Also - cool idea! Somebody else on this thread suggested that you would use this style of algorithm to make an automated golf machine but you’d need like a lidar 3D model of the green… I got nerd sniped real hard by your second lock video and according to the hastily scribbled sketches from about 4am the next day, I totally think it could be done with 4 concentric rings with a few connecting sheet metal parts in the back. Maybe or maybe not cheaper to machine… Haven’t had time to CAD yet cause ive been moving… to Raleigh…
that would be very interesting as the ball would have to accelerate more the further from the hole it gets and given that shots break at a higher rate the slower the ball is moving that would create a very interesting visual effect.
@onetoone It just has to simulate the physics backwards. Like, instead of simulating the ball rolling up the hill, simulate it _unrolling down_ the hill. Which looks like rolling up, but gives numbers for rolling down.
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.
@@Arrica101 Funny, but also true for all kinds of races: the runner who spends the least time running wins the race; the driver who drives for the shortest time wins…
@@hutchmusician This rule could also be generalized to account for some proffesions that exist to delete the object of their work (for example, the object of having a police force is to have nobody break the law).
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 think you're right that these should be called parameter spaces. Phase space officially I think I've only seen in multi-pendulum examples with something like angular velocity plotted over time? It still maps out forbidden regions and stable regions. Phase diagrams (as in states of matter or different crystals formed by the same alloy) are also very similar because you pump in some controllable variables and a state pops out.
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!
@@RonWolfHowl And it's quite common with programming. Simulating a result can often be easier than going through all the equations, because you only need to think about a realistic setup for the system.
Thanks! and yes, is now - link's in the description. If you know any golfers, computer scientists, or anybody else who might want to help with that view count... let them know!
Man, you make very cool videos, i especially am entertained by the visual gags you do (not many in this vid, admittedly) to put your point across. Very creative, very scientific, love your videos!
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.
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
@@satibel More than that, cooking at too high a temperature will cook the outside of the dish, but not the inside- It can be both burned, and not safe to eat. Similarly, many dishes rely on being cooked at too high of a temperature to get a "crust", or like for a Turkey, to cook the skin to a crisp, yet to just get the inside of it cooked far enough to not lose all the moisture. Some other dishes, such as ones that depend on rising agents, have complex curves as well, either in time, or in temperature, where you might need to hold one temperature for a time, then bring it up to a different temperature for another time, then lower it again to a third temperature for a time, to achieve the result you wanted.
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.
@@asailijhijr More like bottom left? Also, since he puts SLOW and not the quantity I don't know which is the latest speed (probably 0, but It doesn't have much computational sense) Edit: I've looked his codes and in the ones I've looked there were no 0 velocities. I am probably mistaken. Correct me if it is so.
"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
I’m glad RU-vid 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
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.
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.
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!
@@DiamondSane well also air resistance. Thermal dynamic influence. Rigidity and age of the gold ball. Strength of gravity at that particular part of planet. And God knows how many other variables that are too small to calculate.
@@liggerstuxin1 The entire point of the phase space is that it's meant to cover the variables we *can* control. Unless you know some way to control the local gravity? If you can't control it, then it belongs in the simulation, but not in the phase space.
@@QwertyuiopThePie I was replying to a comment where someone stated variables that you can’t control but still “matter” (heh, I will pretend I did that on purpose). I suppose I went off on a tangent there, but I was also mentioning that there are so many variables that we cannot calculate that also matter. Many more than we even know obviously. But sooooo many we know we can’t calculate.
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!
@@AuxenceFI think you'd just end up with a one dimensional phase diagram per technique. But since there's no continuous division, i think you'd really want to plot it as a third dimension to every other comparison
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.
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.
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 👌🏻
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.)
indeed and someone took it one step further and used deep-learning AI to simulate Trackmania racing, figuring out absolute most efficient racing lines.
You also have control of the time you choose strike it at, which can have different results depending on the wind strength and direction at the time of the strike.
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!
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.
Haha I just watched that Scott Manley video! And ABSOLUTELY it’s a phase space or parameter space. I find stuff literally all the time where I feel myself walking through some crazy high dimensional space without realizing it at first. It’s a fun way of looking at problems.
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.
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.
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)
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.
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.
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
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 was awesome. It be interesting to see how the results differ as certain variables grow - E.g. going from a completely flat green, then increasing the left to right break, then increasing the uphill / downhill break.
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
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.
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.
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"
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.
I wonder if we can write down a functional mapping between green space and phase space. As some of your more exotic simulations hint at, it must be frightening in the general case (if it exists), but I'm especially curious if we restrict our search to smooth rolling hills that a golfer is likely to encounter. Great video as always!
Oh man that sounds really interesting. Doing it in reverse, with a smooth function, you’d basically have to check for the existence for a reverse path back to the tee... it’d be easy to make IVT kind of arguments but because energy lost to drag is nonlinear that gets super messy. Fascinating!
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 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.
Huh. Phase space is actually _exactly_ how I used to think about putting, when I was able to golf. Specifically, I'd try to visualize the edge cases (fast and straight but not hopping over vs. slow and curvy but not too far into chaos) and then narrow down in my mind what path would be in the "center" from a statistical standpoint. I.e. not the geometric center, but sort of... in engineering terms, the minima where the gradient of change in actual ball path vs. the two input variables I had control over was lowest, to give twitchy muscles, shaky hands, breezes, and stray bees the least chance of screwing it up.
Launch angle and grass are included in the simulation, and weather isnt something he can control, so simulating weather for a rudimentary phase space experiment isn't really useful
Yes, there are lots of factors but the whole point is that you generate the phase space based on the factors you can control. You can't control the grass type. You can't control the temperature or moisture. The club face material factors into the speed. So each of these factors might affect the eventual phase space but not in any meaningful way because where your ball ends on that particular green on that particular day with that particular humidity and that particular temperature, is still a function of ONLY the direction in which you hit it and how hard you hit it.
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.
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.
I think a great video that could build off of this would be to do basketball instead of golf. I’m sure the simulation would be very similar, if not slightly more complex. It would also be cool to use your findings to compare with free throw or 3 pointer stats of some players to put their skills in context. I love your videos and keep em coming!
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.
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!
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.
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).
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..
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.
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.
This video was pretty cool and gives me an idea for creating a hole in one golfing robot. If phase space could be accurately mapped to represent the real green, it could be possible to build a robot that would outplay a human. Of course it's impossible to perfectly recreate every blade of grass that is constantly growing and changing, but with precision coordinate based mapping and wind speed sensory input the computer could calculate a take the shot with much better accuracy and get a hole in a lot less shots than any human ever has or ever will. A game of nothing less than eagles would be a sight to see. Anyway. Nice video and it's fascinating to think about and play with.
I'm a plastics technician and i was stuck on how to make a simple to read document on setting up the injection moulding presses, this might just be it ! There is so many parameters you can set, injection speed and pressures, backpresure and screw speed, it might be very useful !
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 😂
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!