You can support the coding train and watch this video ad-free on Nebula! nebula.tv/videos/codingtrain-coding-challenge-buffons-needle Find the code and share your own to the Passenger Showcase: thecodingtrain.com/challenges/176-buffon-needle
To specify rotation of the toothpick without relying on Pi in the code, you could generate two random points! The first random point would be inside the canvas, and would define the location of the center of the toothpick. The second point could fall anywhere, even outside the canvas. The second point is used to define a vector, which will guide the rotation of the toothpick. Draw a construction line connecting the center of the toothpick with the second random point, then draw the toothpick with the appropriate length along that construction line. This should get you random orientations without relying on already knowing the value of pi!
This would not give a random angle. You can assign the first point (the centre of the toothpick) as (0, 0). And if you pick a second point randomly (aka uniform distribution in x and y), the angle of the toothpick will be arctan(x/y). The maths gets a bit hairy here, but as you can easily imagine, if x and y have a uniform distribution, the angle arctan(x/y) will not
@@SellusionStar Well, this is a very common misconception. It would be random, yes, but not uniform. It would be clumped up. There are many different randomness. There is a uniform, where every value is equally likely. But there are different ones as well. The most known another one is Gaussian distribution - a bell like shape. While both of them can provide a random distribution, they will have different results. For example, if we try and draw a person, we can choose a random height for it. If we choose a uniform distribution, then it will be equally likely it will be 1 meter tall, as it will be 1.8 meters tall. While still random, it is not useful in this scenario, since human height is not uniformly distributed. If we wanted a more accurate random value for human population, we would use Gaussian distribution, where a height of 1.7 meters is much more likely than 2 meters. It does not mean that a random chosen height from such distribution will be 1.7 meters, but it will result in the heights being more clumped up around 1.7 meters
@@gediminasusevicius thanks, got you! That's exactly what I was thinking, when I imagined how one would through the needles in real life while making sure they are evenly/uniformly distributed. Because I thought they would probably clump in some places...
16:06 - I believe that if you used `angleMode(DEGREES);` in your setup, you could specify the angles with -180 to 180 and thereby removing the dependency on PI!
Unfortunately PI sneaks in still. I was actually thinking the sine function itself doesn’t need PI to be computed. But PI gets inserted when he’s talking about that integration with PI in the limits. But when you say to express it in degrees, it would seem to avoid that completely. But when you switch between radians and degrees, calculators don’t tell you that the specific sine function changes. When you take the sine in degrees, it’s actually applying a scaling transformation on the degrees first - to get it back to radians - then applying the standard sine function. If 180 degrees is PI, then to do that conversion, you multiply by PI/180 to get it to radians. In this way, PI is sneakily still present!
The toothpick challenge was one of my favorite and i tweeted my output to you a few days after that video. One of the single greatest influences to code again. Thanks for your videos!!!
just awesome. I was actually randomly searching for buffon's needle experiments today and you did the actual thing I wanted to to see. Love and respect
Shiffman, this message that I'm sending doesn't have much to do with what's in the video, I apologize for that, but I just want to thank you for everything you've done in my life by being a great teacher. In 2018, I started my journey as a developer and I was really lucky to find your channel and playlists teaching javascript in the unique way that you have to teach. Today I have a career in the field, a passion for what I do, and I just wanted to express my gratitude for being an amazing mentor (even though you have no idea who I am hahah). Thank you very much, Shiffman. Your work here is awesome! I wanted to say some more beautiful things but since English is not my native language, I can't express myself the best way possible. If I could talk in Portuguese you probably would cry for a month. 😂
Watching these toothpicks appear on the board reminds me of when I trim my beard into the sink XD I'm wanting to blow the screen every so often. Loving these videos, Dan! From a fellow Dan.
To get a uniform angle in (0,2pi) you could generate uniform x,y in (0,1) until x^2+y^2=1 you essentially create a uniform distribution on the circle, and there the angle is uniformly distributed.
14:17 That seems like the best choice, and not because it would look more interesting, but because you wouldn't eschew the results. If you limit to just 90º the toothpicks will have "one" chance to be at an angle 0º < θ < 90º, one chance to land vertically and one chance to land horizontally, if you expand to 180º that changes to 2 chances "at an angle", two chances horizontally and one chance vertically, but since they can in reality land at whatever angle between 0º and 360º, the chances are actually four times "at an angle", two times horizontally and two times vertically. It shouldn't, I believe, make much of a difference, but since it's the same to use a full circle as it is to use just 1/4 of one, go for the more accurate simulation.
I love the way you teach! You simplify things so much and keep refining the solution from a dummy start to the ultimate solution. I wish you could make a program where one can visulaize all the data structures. Thanks for doing things the way you do. ❤❤ from India
A possible way to pick a random angle without referencing pi whatsover: use a uniform distribution over a HUGE interval (or take the uniform distribution on (0,1) and map it to a large interval). I haven't directly tested this way of rng, but here's why I think it should work. Let's write the total width of our interval in the form z = 2*pi*N + x, where N is a large enough natural number, and x any real number between 0 and 2*pi. If we had x=0, then we are picking the angles on an interval of with 2*pi*N, which for the purposes of rotating the toothpicks is indistingushable from picking from an interval of width just 2*pi. When x is not zero, the probability of picking the angle in the "wrong" part of the interval is equal to x / (2*pi*N + x). This number is vanishing small, the larger we pick N -- we don't directly pick N, but a number very large number z = 2*pi*N+x to make sure tha N is really, really big.
I love pi. It's such a cool thing like you can find it anywhere and the math here done is not that difficult. It's just basic highschool math and as you made a graph I was like let's an take an integral under the curve.
11:30 No, it's not the top half of an ellipse. it's a section of a sine wave, specifically the graph of y=(1/2)*sine(x) which people can drop into the online Desmos graphing calculator and visualize.
First of all: Happy pi day from Spain. I want to propose a problem. How about an extension to 3D of this problem. Suppose we place random needles on the space at random points with some length and two angles defining the orientation. What is the probability of a needle to cross two parallel planes at unit distance one from each other. What if instead of two planes we take a cube? What is the probability of a needle crossing one side of the unit cube. Hint: Does that probability involve pi? The result may be surprising 😉
What could be done to remove π from the code is to take a rough approximation of π (like π=3), run the code and use the new and better value of π to run the code again
Hi Daniel, Instead of using TWO_PI what you can do is pick two random points and lerp a point between those two with distance of l and using that you would get a random angle without using rotate, TWO_PI. For getting X from that you can just do x2 - x1. Hope it helps.
you can’t use trigonometry functions, instead you should pick a random gradient. So a gradient of 0 would be horizontal and 1 would be vertical. This would circumvent the need to use PI anywhere in your code (and implicitly when using trig functions)
Maybe you can get around the random angle chicken and egg problem by choosing a point in a disk at random. Pick a random x coordinate between 0 and 1, and pick a random y coordinate between 0 and 1. Check if x^2+y^2
Since you're intersecting a vertical line, you could draw the line using x0,y0-x1,y1. Then you simply divide x0 and x1 by t. If they're different, they intersect. If (x0/t != x1/t) intersect++; Something like that?
And you have pi/2 million subs, nice! Also, to not be forced to use PI in your simulation you can make a rectangle with random height and width, then choose any two diagonally opposite corners and get the angles from that.
@@GradientAscent_ 2 random numbers will produce a rectangle with random height and width. If all possible values are equally likely the resulting angle between opposite corners will not be evenly distributed.
int x0 = (int)(x - sin(rot) * l); int x1 = (int)(x + sin(rot) * l); bool hit = x0 != x1; Assuming that t = 1 and l = 0.5 and type of x is double/float (C#)
I'm wondering, since the lines are verticle, the Y-axis information is actually irrelevant. I feel like this could be simplified into a 1 dimensional exercise and instead of angles, you're really just looking at a random placement of a line on the X-axis who's length is between 1/2t and 1
Could you change the angle to degrees, select randomly from 360, then it should still shake out to approximate pi without ever writing it in the code? Would that work?
i'm not saying you're wrong. but if toothpick length is 1. and width of the gap is 1. then if angle is 45, toothpick width is half of 1. and that means the distance of 1/2 of 1/2 which is 1/4? not .335....@ 9:50
I have an irrelevant issue I really need someone to help me with. I am using Processing and trying to import JavaFX from the contribution manager, I put the files in the correct location, I know this because it pops up in the "Sketch" folder. I am trying to make a GUI, so far there is no video's from Daniel here on how to make a GUI yet. When I use it, it imports import processing.javafx.*; I am trying to get it to import all of these things import javafx.application.Application; import javafx.scene.Scene; import javafx.scene.control.Label; import javafx.scene.layout.StackPane; import javafx.stage.Stage; Can someone please help me with this? I can't find source for this and if their is a different way to access it it would be great.
Heeey mr. Coding Train. Have you tried using arduino with other softwares through serial communication? I've been working with touchdesign and an arduino but would like to See other creative posibilites. Oh, and btw touchdesigner comps use python
I object using Pi while trying to determine it. This should be solved graphically: You know the position of the vertical lines thus no need to use Pi. Happy Pi day!
You used radians in random function and you didn't like it. Why not just switch to degrees. Would that create any issues. I'm not into p5.js but I don't think that should be any issues when you're telling the software that you're using degree values. Afterall a graph doesn't need π
well, assume you have N nodes, then you have N-filters and N-1 two-way signal splitters to the other N-1 nodes, and maybe signal pipes, optical fibers, between the node signal splitters, then let it bounce
Nice video! However, I do no like that you used pi and sine in the program. Instead of picking an angle at random, you should pick a second point (x,y) and make the toothpick lie along the resulting line segment.
Awesome explanation. Your pacing and thoroughness is really good. But please stop calling the horizontal axis (or the theta axis in this case) the x axis, especially when you're plotting x on the vertical (not y) axis.
You can support the coding train and watch this video ad-free on Nebula! nebula.tv/videos/codingtrain-coding-challenge-buffons-needle
Find the code and share your own to the Passenger Showcase: thecodingtrain.com/challenges/176-buffon-needle
One way to avoid using pi in the code would be to use a random number (float) between 0 and like 100000000, so all angles are roughly equally likely
you are video make people interesting and easy understand what your share thanks sir
Just when the world needed him most, he returned...
Which data type
LOL
I gotta say these new whiteboard animations you have going on are really impressive
The most underrated teacher i’ve ever learned from, Thank you for all the time you spent through out the years on this amazing content!
To specify rotation of the toothpick without relying on Pi in the code, you could generate two random points! The first random point would be inside the canvas, and would define the location of the center of the toothpick. The second point could fall anywhere, even outside the canvas. The second point is used to define a vector, which will guide the rotation of the toothpick. Draw a construction line connecting the center of the toothpick with the second random point, then draw the toothpick with the appropriate length along that construction line. This should get you random orientations without relying on already knowing the value of pi!
This would not give a random angle.
You can assign the first point (the centre of the toothpick) as (0, 0). And if you pick a second point randomly (aka uniform distribution in x and y), the angle of the toothpick will be arctan(x/y). The maths gets a bit hairy here, but as you can easily imagine, if x and y have a uniform distribution, the angle arctan(x/y) will not
@@gediminasusevicius You could choose a random point in a surrounding circle using Monte Carlo rejection and that would work
@@gediminasusevicius I don't get why it doesn't yield a randomly angled toothpick. Could you please elaborate a bit more? 🙂
@@SellusionStar Well, this is a very common misconception. It would be random, yes, but not uniform. It would be clumped up.
There are many different randomness. There is a uniform, where every value is equally likely. But there are different ones as well. The most known another one is Gaussian distribution - a bell like shape. While both of them can provide a random distribution, they will have different results.
For example, if we try and draw a person, we can choose a random height for it. If we choose a uniform distribution, then it will be equally likely it will be 1 meter tall, as it will be 1.8 meters tall. While still random, it is not useful in this scenario, since human height is not uniformly distributed. If we wanted a more accurate random value for human population, we would use Gaussian distribution, where a height of 1.7 meters is much more likely than 2 meters. It does not mean that a random chosen height from such distribution will be 1.7 meters, but it will result in the heights being more clumped up around 1.7 meters
@@gediminasusevicius thanks, got you! That's exactly what I was thinking, when I imagined how one would through the needles in real life while making sure they are evenly/uniformly distributed. Because I thought they would probably clump in some places...
I love this tradition of you calculating Pi in many different ways!
You might like to check out Matt Parker on the Standup Maths channel where he has been doing similar things over the years. Some really creative ones.
7:25 I've heard of falling angels, but falling angles are new to me :)
Excellent video!
12:00 it is an image, but there is no greenscreen... and the image is between him and board which he interacts with
Runway in action! runwayml.com/
That looked so natural, I didn't even notice this. Very impressive.
Love your videos. Youre an absolute treasure.
I just found your channel a week ago and it’s been helping me with python. So glad you are still active on RUclips.
Nice to see you again, Welcome Back!
Your enthusiasm and thorough explanations are unlike anything I've seen before. Thank you so much!
16:06 - I believe that if you used `angleMode(DEGREES);` in your setup, you could specify the angles with -180 to 180 and thereby removing the dependency on PI!
Same thought
Was lookin for this comment.
Unfortunately PI sneaks in still. I was actually thinking the sine function itself doesn’t need PI to be computed. But PI gets inserted when he’s talking about that integration with PI in the limits.
But when you say to express it in degrees, it would seem to avoid that completely. But when you switch between radians and degrees, calculators don’t tell you that the specific sine function changes.
When you take the sine in degrees, it’s actually applying a scaling transformation on the degrees first - to get it back to radians - then applying the standard sine function. If 180 degrees is PI, then to do that conversion, you multiply by PI/180 to get it to radians.
In this way, PI is sneakily still present!
The toothpick challenge was one of my favorite and i tweeted my output to you a few days after that video. One of the single greatest influences to code again. Thanks for your videos!!!
Loved the video! Thanks for featuring my sketch!!
just awesome. I was actually randomly searching for buffon's needle experiments today and you did the actual thing I wanted to to see. Love and respect
The struggle with drawing a decent looking graph is real! Yours is great!
I love this channel. Super interesting stuff and excellent delivery. Congrats!!!!
Shiffman, this message that I'm sending doesn't have much to do with what's in the video, I apologize for that, but I just want to thank you for everything you've done in my life by being a great teacher. In 2018, I started my journey as a developer and I was really lucky to find your channel and playlists teaching javascript in the unique way that you have to teach. Today I have a career in the field, a passion for what I do, and I just wanted to express my gratitude for being an amazing mentor (even though you have no idea who I am hahah). Thank you very much, Shiffman. Your work here is awesome! I wanted to say some more beautiful things but since English is not my native language, I can't express myself the best way possible. If I could talk in Portuguese you probably would cry for a month. 😂
Thank you for sharing this wonderful story!
Man, I missed you a lot! thanks for comming in this amazing Pi day
I can only grasp half of this, but I'm enjoying myself anyway. What enthusiasm!
I always look forward to your videos and you never disappoint! Thanks for being so inspiring!
Dan! I love how you tackle hard problems with this energy and didactics! You´re awesome! Hope you know that!
So glad to see a new coding challenge, i almost finished the playlist !
It is always a pleasure and inspiration watch your videos !
Absolutely amazing, I love your work and your natural and logical way of explaining things!
I am so here for that art direction on the background !
I just saw it was march 13 th and I instantly came to your channel to see what you had
Watching these toothpicks appear on the board reminds me of when I trim my beard into the sink XD I'm wanting to blow the screen every so often.
Loving these videos, Dan!
From a fellow Dan.
To get a uniform angle in (0,2pi) you could generate uniform x,y in (0,1) until x^2+y^2=1 you essentially create a uniform distribution on the circle, and there the angle is uniformly distributed.
If you can calculate atan(x/y), then you might as well do 4*atan(1)
14:17 That seems like the best choice, and not because it would look more interesting, but because you wouldn't eschew the results.
If you limit to just 90º the toothpicks will have "one" chance to be at an angle 0º < θ < 90º, one chance to land vertically and one chance to land horizontally, if you expand to 180º that changes to 2 chances "at an angle", two chances horizontally and one chance vertically, but since they can in reality land at whatever angle between 0º and 360º, the chances are actually four times "at an angle", two times horizontally and two times vertically.
It shouldn't, I believe, make much of a difference, but since it's the same to use a full circle as it is to use just 1/4 of one, go for the more accurate simulation.
19:07 there it is, near the center of the canvas: PI! Challenge completed!
wohoo!! this 25 min is gonna be fantastic! your're amazing. i have learned a lot from you.
I love the way you teach! You simplify things so much and keep refining the solution from a dummy start to the ultimate solution. I wish you could make a program where one can visulaize all the data structures. Thanks for doing things the way you do. ❤❤ from India
Thank you!
If only I had your programming skills, my life ideas would come to life in a matter of days
A possible way to pick a random angle without referencing pi whatsover: use a uniform distribution over a HUGE interval (or take the uniform distribution on (0,1) and map it to a large interval).
I haven't directly tested this way of rng, but here's why I think it should work. Let's write the total width of our interval in the form z = 2*pi*N + x, where N is a large enough natural number, and x any real number between 0 and 2*pi. If we had x=0, then we are picking the angles on an interval of with 2*pi*N, which for the purposes of rotating the toothpicks is indistingushable from picking from an interval of width just 2*pi.
When x is not zero, the probability of picking the angle in the "wrong" part of the interval is equal to x / (2*pi*N + x). This number is vanishing small, the larger we pick N -- we don't directly pick N, but a number very large number z = 2*pi*N+x to make sure tha N is really, really big.
Next year, I want to see Matt Parker drop a thousand toothpicks and just see him spend the entire video picking them up to calculate pi!!
GENUINELY THANK YOU FOR UPLOADING ❤️
I love pi. It's such a cool thing like you can find it anywhere and the math here done is not that difficult. It's just basic highschool math and as you made a graph I was like let's an take an integral under the curve.
to celebrate pi day, I am not going to watch this video when its release but exactly pi days after!
Loved ur enthusiasm, very interesting challenge. Thank you for this 😄
Welcome back!
11:30 No, it's not the top half of an ellipse. it's a section of a sine wave, specifically the graph of y=(1/2)*sine(x) which people can drop into the online Desmos graphing calculator and visualize.
ayyyyy a new coding train video
I was missing your videos!
We're late by 1 day, but let's wait a full year again
Awesome!!! Thank you!
I really would like to have you back for the coding challenges with p5js.
Happy PI day to all! So magical!
First of all: Happy pi day from Spain. I want to propose a problem. How about an extension to 3D of this problem. Suppose we place random needles on the space at random points with some length and two angles defining the orientation. What is the probability of a needle to cross two parallel planes at unit distance one from each other. What if instead of two planes we take a cube? What is the probability of a needle crossing one side of the unit cube. Hint: Does that probability involve pi? The result may be surprising 😉
FINALLY BACK!
@17:00 you could have just used the modulo of the column width to the toothpick X. No need for searching for closest
Omg he left his cryogenic tank just for us! Welcome to the future!
What could be done to remove π from the code is to take a rough approximation of π (like π=3), run the code and use the new and better value of π to run the code again
11:31 *not actually an ellipse either (it's a sine wave).
Hi Daniel, Instead of using TWO_PI what you can do is pick two random points and lerp a point between those two with distance of l and using that you would get a random angle without using rotate, TWO_PI. For getting X from that you can just do x2 - x1. Hope it helps.
I do not think each angle has an equal possibility.
you can’t use trigonometry functions, instead you should pick a random gradient. So a gradient of 0 would be horizontal and 1 would be vertical. This would circumvent the need to use PI anywhere in your code (and implicitly when using trig functions)
Amazing video
Badhiya video, guru g. Aur batao, sab badhiya ☺️
Maybe you can get around the random angle chicken and egg problem by choosing a point in a disk at random. Pick a random x coordinate between 0 and 1, and pick a random y coordinate between 0 and 1. Check if x^2+y^2
Edit: I meant pick the x and y coordinated randomly between -1 and 1 and then check if x^2+y^2
Missed you 🥲
Awesome 👌 👍 😍
Did you use p5js to create the animated illustrations, too? For example at 14:09.
These I actually created in p5, but then they were redrawn by an animator! It's a little bit of both throughout the video!
Since you're intersecting a vertical line, you could draw the line using x0,y0-x1,y1. Then you simply divide x0 and x1 by t. If they're different, they intersect.
If (x0/t != x1/t) intersect++;
Something like that?
And you have pi/2 million subs, nice! Also, to not be forced to use PI in your simulation you can make a rectangle with random height and width, then choose any two diagonally opposite corners and get the angles from that.
That would make angles closer to ±45° more likely than angles closer to 0° or ±90°.
@@nagualdesign It does not have to be a rectangle, two points would suffice!
@@GradientAscent_ 2 random numbers will produce a rectangle with random height and width. If all possible values are equally likely the resulting angle between opposite corners will not be evenly distributed.
Is this connected to 3B1B?
Only in the sense that they’re both Gods among us.
int x0 = (int)(x - sin(rot) * l);
int x1 = (int)(x + sin(rot) * l);
bool hit = x0 != x1;
Assuming that t = 1 and l = 0.5 and type of x is double/float (C#)
I'm wondering, since the lines are verticle, the Y-axis information is actually irrelevant.
I feel like this could be simplified into a 1 dimensional exercise and instead of angles, you're really just looking at a random placement of a line on the X-axis who's length is between 1/2t and 1
It uses sinus to get PI, so sinus is needed while finding PI, and can not be replaced with a linear random number
@@Iuigi_t got it, thanks for clarifying
missed you!
Do you have any plans to maybe cover some other programming languages in the future?
Could you change the angle to degrees, select randomly from 360, then it should still shake out to approximate pi without ever writing it in the code?
Would that work?
Will you do shader programming again? Really enjoy your videos, learn so much things!
you could replace 2pi with high number, that would't be precise but random enough to work as well
i'm not saying you're wrong. but if toothpick length is 1. and width of the gap is 1.
then if angle is 45, toothpick width is half of 1. and that means the distance of 1/2 of 1/2 which is 1/4?
not .335....@ 9:50
Hey just ran into your content and I'm new to programming. What Playlist of yours would recommend I start with
This is the beginner one: ruclips.net/p/PLRqwX-V7Uu6Zy51Q-x9tMWIv9cueOFTFA
I also have things organized on my website: thecodingtrain.com/
I love pi day; all my favorite math RUclipsrs have a bunch of cool videos that I can watch while eating pizza (pi-zza)
This is magic :0
18:06 Reality shattered
nice video
fun fact : pi day is the same day as the day of the gifted
happy PI day bro
I have an irrelevant issue I really need someone to help me with.
I am using Processing and trying to import JavaFX from the contribution manager, I put the files in the correct location, I know this because it pops up in the "Sketch" folder.
I am trying to make a GUI, so far there is no video's from Daniel here on how to make a GUI yet.
When I use it, it imports
import processing.javafx.*;
I am trying to get it to import all of these things
import javafx.application.Application;
import javafx.scene.Scene;
import javafx.scene.control.Label;
import javafx.scene.layout.StackPane;
import javafx.stage.Stage;
Can someone please help me with this? I can't find source for this and if their is a different way to access it it would be great.
HOO HAPPY PI DAY!! Guys wish u luck and a lot of success!
I also recommend watching the movie, "Pi"
I really thought you were giong to drop a box of toothpicks onto the floor
Heeey mr. Coding Train. Have you tried using arduino with other softwares through serial communication? I've been working with touchdesign and an arduino but would like to See other creative posibilites. Oh, and btw touchdesigner comps use python
Oh, I have worked with Arduino! Maybe someday I'll get to make a video with it.
happy pi day everyone!
I object using Pi while trying to determine it. This should be solved graphically: You know the position of the vertical lines thus no need to use Pi. Happy Pi day!
You used radians in random function and you didn't like it. Why not just switch to degrees. Would that create any issues. I'm not into p5.js but I don't think that should be any issues when you're telling the software that you're using degree values. Afterall a graph doesn't need π
You could have used degrees for the angles so its 0-180 instead of 0-pi
solve travelling salesman problem with physical photon bounces (ray tracing, radar n-bounce problem)
well, assume you have N nodes, then you have N-filters and N-1 two-way signal splitters to the other N-1 nodes, and maybe signal pipes, optical fibers, between the node signal splitters, then let it bounce
fastest full signature signal to arrive to start wins (phase of signal arrival packets matter)
acute distress
Nice video! However, I do no like that you used pi and sine in the program. Instead of picking an angle at random, you should pick a second point (x,y) and make the toothpick lie along the resulting line segment.
I would find it interesting how someone in the real world would make sure that the toothpicks fall evenly distributed.
They don't have to fall evenly, they have to fall randomly.
@@nagualdesign how would you throw them to achieve this? 😄
@@SellusionStar Drop them one by one, so they don't interfere with each other. As long as they land somewhere on the table it'll be fine.
I like your watch
i missed you
I thought you'd post today
Awesome explanation. Your pacing and thoroughness is really good.
But please stop calling the horizontal axis (or the theta axis in this case) the x axis, especially when you're plotting x on the vertical (not y) axis.
Apologies, this is very good feedback, I will try to be more clear next time!
It should have been have been pieces of fruit falling on a pie batter
Need to go back and look at Image to Text models, Large Language Models chatGPT GPT- 4 etc
Gene Kogan Where is your follow up on these .... 🧐
TWO_PI has a name, it’s actually called TAU (τ).