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Virtually Passed
Registrace 30. 10. 2012
This channel is dedicated to teaching math, physics and engineering. Want a topic taught? Leave a comment :)
How To Make a Computer Create Something Beautiful: String Art #SoMEpi
This is the second video I've made about string art; a technique to make an artwork from a single thread wrapped around nails along a circular canvas. The first video was received very well. I even had many people throw a bunch of really creative ideas at me. I would have never thought about the radon transform if it wasn't for such an amazing community! Thanks so much!
Chapters:
0:00 Last Time
0:20 New Method
0:50 Intro
0:56 Rules
1:24 Radon Transform
3:29 First attempt
4:36 Radon Transform of Line
5:20 Faaaiiiilllurrreee!
6:26 Improvement 1
7:30 Improvement 2
8:16 Color!
8:59 Outro
9:11 Patreon
Music: Go Season by Milano
--
Patreon: www.patreon.com/VirtuallyPassed
Instagram: virtuallypassed
---
Want to buy a custom black & white circular String Artwork? Check out my website:
sites.google.com/view/virtuallypassed/home?authuser=0
For a custom String Artwork, email me at virtuallypassed@gmail.com with an image you'd like turned into string art. I can only make a few artworks each month due to limited time and resources.
---
Chapters:
0:00 Last Time
0:20 New Method
0:50 Intro
0:56 Rules
1:24 Radon Transform
3:29 First attempt
4:36 Radon Transform of Line
5:20 Faaaiiiilllurrreee!
6:26 Improvement 1
7:30 Improvement 2
8:16 Color!
8:59 Outro
9:11 Patreon
Music: Go Season by Milano
--
Patreon: www.patreon.com/VirtuallyPassed
Instagram: virtuallypassed
---
Want to buy a custom black & white circular String Artwork? Check out my website:
sites.google.com/view/virtuallypassed/home?authuser=0
For a custom String Artwork, email me at virtuallypassed@gmail.com with an image you'd like turned into string art. I can only make a few artworks each month due to limited time and resources.
---
zhlédnutí: 10 419
Video
The Mathematics of String Art
zhlédnutí 539KPřed rokem
String Art recreates an image using string. An algorithm is used to calculate the order in which the string needs to be wrapped around the nails. This is something different from what I usually do - it's my entrance to the #some3 contest. Chapters: 0:00 Problem statement 0:39 Intro 0:51 Rules 1:13 Image model 2:05 Lines model 2:50 Ax = b 4:58 pinv(A) 6:17 Yuck! 6:30 Improved Lines model 8:13 Co...
5.0 A better way to understand Differential Equations | Nonlinear Dynamics | Bendixson's Criterion
zhlédnutí 6KPřed rokem
Series1: Part 1: czcams.com/video/ySfs8YVMY7Q/video.html Part 2: czcams.com/video/I9UEBRya4X0/video.html Part 3: czcams.com/video/6mLCFyEv3Z0/video.html Series 2: Part 4: czcams.com/video/wZvFKcQ_3Rc/video.html Bendixson's criterion is another method used to disprove the existence of closed orbits. A periodic solution is a type of closed orbit. This theorem only holds for simply connected regio...
4.0 A better way to understand Differential Equations | Nonlinear Dynamics | Index Theory
zhlédnutí 114KPřed rokem
Series1: Part 1: czcams.com/video/ySfs8YVMY7Q/video.html Part 2: czcams.com/video/I9UEBRya4X0/video.html Part 3: czcams.com/video/6mLCFyEv3Z0/video.html Index theory is a method used to gain global information about a nonlinear differential equation. One powerful insight is that closed orbits (periodic solutions) must have at least one fixed point inside of the curve. In fact, for a periodic or...
Least Squares Formula PROOF
zhlédnutí 9KPřed 2 lety
First video: czcams.com/video/6eLJSzOHdc8/video.html Linear least squares is a method commonly used to fit curves to data. The equation used for least squares here is derived using the column space of A. The 'best' value of X is the one that minimizes the distance between the vector b and [A]X. Most math textbooks stop at the result: A^T A X = A b and use different algorithms to find the best X...
Fitting a Line using Least Squares #SoME2
zhlédnutí 62KPřed 2 lety
Interactive links: 1DOF: cryptic-spinach.github.io/1DoF/ 2DOF: cryptic-spinach.github.io/2DoF/ 2DOF mobile: cryptic-spinach.github.io/least_squares_2DoF/ Collaboration links: I worked with Chase Reynolds for the #SoME2 contest. Check out his website and Twitter here. chasereynolds.me/ crypticspinach The Least Squares method is widely used to fit curves to data. This is called Least ...
3.1 Linearization PROOF | Nonlinear Dynamics
zhlédnutí 4,3KPřed 2 lety
3.1 Linearization PROOF | Nonlinear Dynamics
1.1 Stability of Fixed Points PROOF | Nonlinear Dynamics
zhlédnutí 9KPřed 2 lety
1.1 Stability of Fixed Points PROOF | Nonlinear Dynamics
3.0 A better way to understand Differential Equations | Nonlinear Dynamics | Linearization
zhlédnutí 15KPřed 2 lety
3.0 A better way to understand Differential Equations | Nonlinear Dynamics | Linearization
2.0 A better way to understand Differential Equations | Nonlinear Dynamics | 2D Linear Diff Eqns
zhlédnutí 19KPřed 2 lety
2.0 A better way to understand Differential Equations | Nonlinear Dynamics | 2D Linear Diff Eqns
1.0 A better way to understand Differential Equations | Nonlinear Dynamics | 1D Linear Diff Eqns
zhlédnutí 39KPřed 2 lety
1.0 A better way to understand Differential Equations | Nonlinear Dynamics | 1D Linear Diff Eqns
Radius of Curvature Proof - approximating a curve with a circle!
zhlédnutí 18KPřed 2 lety
Radius of Curvature Proof - approximating a curve with a circle!
Can YOU find the offset of this circle? Math Puzzle
zhlédnutí 4,3KPřed 2 lety
Can YOU find the offset of this circle? Math Puzzle
Energy analysis of forced spring mass damper system
zhlédnutí 6KPřed 3 lety
Energy analysis of forced spring mass damper system
Can you change vibration speed? #VeritasiumContest
zhlédnutí 2,1KPřed 3 lety
Can you change vibration speed? #VeritasiumContest
Mathematics of Vibration [Part 2: Harmonic Excitation]
zhlédnutí 6KPřed 3 lety
Mathematics of Vibration [Part 2: Harmonic Excitation]
Bayes' theorem alternative proof (using algebra)
zhlédnutí 2,5KPřed 3 lety
Bayes' theorem alternative proof (using algebra)
Optimal path to swim across river in least time - Math Puzzle (HARD)
zhlédnutí 12KPřed 3 lety
Optimal path to swim across river in least time - Math Puzzle (HARD)
Time to swim across river with variable speed - Math Puzzle
zhlédnutí 1,6KPřed 3 lety
Time to swim across river with variable speed - Math Puzzle
Optimal path to rescue friend - Math Puzzle
zhlédnutí 2KPřed 3 lety
Optimal path to rescue friend - Math Puzzle
Optimal Path Around Quicksand - Math Puzzle (HARD)
zhlédnutí 4,8KPřed 3 lety
Optimal Path Around Quicksand - Math Puzzle (HARD)
Conservation of Mechanical Energy Proof (2DoF)
zhlédnutí 957Před 5 lety
Conservation of Mechanical Energy Proof (2DoF)
Conservation of Mechanical Energy Proof (1DoF)
zhlédnutí 2,6KPřed 5 lety
Conservation of Mechanical Energy Proof (1DoF)
Harmonic Force Excitation Steady State Amplitude and Phase Proof
zhlédnutí 13KPřed 5 lety
Harmonic Force Excitation Steady State Amplitude and Phase Proof
Math Puzzle - How many circles wrap around a circle?
zhlédnutí 6KPřed 6 lety
Math Puzzle - How many circles wrap around a circle?
Second Moment of Area Example 4 (I beam)
zhlédnutí 164KPřed 6 lety
Second Moment of Area Example 4 (I beam)
Useless video
Very helpful, thank you.
ahh i was trying to solve this exact simulation soem time ago. i tried using some learning methods but i couldnt do anything significant. the greedy idea was really smart
Thanks! I hope this doesn't make you give up though! I'm sure there are many different ways to approach this problem :)
Le nom d'algorithme svp. Merci
Way Kewl. Occurred to me that you could take it to the next level by doing glowing "EL Wire" art, Assuming it would be an easy inversion, like just starting with a negative of the image or pattern. Controlling the point to point connections with a micro controller could generate lots of interesting visual effects. Scale might need to really large though given the added diameter of the wire.
Simply amazing.
Glad you think so!
this is beautiful, I actually have to proof this theorem is true for complex numbers, in my complex analysis class.
Oh interesting! I personally haven't even thought about proving this for complex numbers! Good luck
nice
Thanks 🙏
So, is the machine available for purchase?
Unfortunately not. I disassembled it to make other robots :(
@@virtually_passed Do you realize people would buy that that machine on amazon? It like a machine that prints money.
Compliments to the quality and content of the video. I have some questions: 1. How do you approach the analysis of the steady-state response of a multi-degree-of-freedom system to harmonic excitation? 2. Please explain the phase angle in the steady state response of a system under harmonic excitation. How does the phase angle vary either frequency and what does it indicate about the system’s behaviour? Thanks in advance 😊!
Great questions! 1) you'll typically have a matrix FRF function in that case where each element of the FRF will relate one specific input to one specific output. An important caveat here is that this only applies for linear systems where the superposition principal can be applied.
In answer to your second question: there's a lot of information to unpack about phase but some key insights are: when resonance occurs you will be neither in phase nor out of phase with the input. It'll be at 90 degrees. After resonance you will be at 180 degrees phase which implies that you're fully out of sync with the input. Before resonance you will be at 0 degrees phase which implies you're fully in sync. If you have a multi degree of freedom system though then it becomes a bit more difficult. You can use the bode gain phase relationship to protect how phase relates to gain
You're pulling rabits out of a hat with no consequential or motivational thought whatsoever. This is robotic, mechanical and unintuitive.
Thanks for the feedback: in case you're curious I've made another video proving the same thing in a more intuitive way: czcams.com/video/vyBkvGnPwJk/video.html
@@virtually_passed Oh, thank you! That's what I meant, Very intuitive, clear and convincing! beatuful, thank you
Sorry I have a question about a topic (moment of inertia) you explained in a video How to contact you? Thanks @virtually_passed
Try emailing me: virtuallypassed@gmail.com
Can't you just "spawn" every string, color it, and sort it from darkest to lightest, and then only display some % of top ones?
It is my duty to thank you for the good content you have provided.
Thanks :) 🙏
Nicely explained 😊
Thanks a lot 😊
My guess was initially that you would use FFT for such a problem. FFTs are used similarly in other areas such as antenna/loudspeaker beam forming (pattern synthesis).
I agree, it seems like a natural progression. However formulating the problem as an fft problem is very difficult in my view :)
@@virtually_passed indeed! There might be some insight to gain in “simple pattern synthesis for complicated arrays” by RF get down here on CZcams - but I haven’t investigated too much into the usability of the method in your case
HAHA i was like wow, thats pretty cool. And then you were just casiaually like oh yah, theres also a completely different way with FOURIER TRANSFORMS??? whattt?
Mind = blown! 🤯 Yep, and there's another method using the radon transform which I talked about in my latest video :)
thank youuuu😘
Welcome 😁
Thank you the mathematics gods will spare you
🙌🙏
How do you prove that the derivative of (x+y)^n is n(x+y)^(n-1)? I'm coming from a Calculus 1 course where this is proved using the binomial expansion, looking for a proof for the binomial formula. 😅
This follows from the derivative: d/dx(f(x))^n = n f(x)^n-1 f'(x)
@@virtually_passed I see. Thank you! (I’ve just been presented with the piece I was missing here: the algebraic proof of the chain rule. 😊)
Very cool. I suspect a good optimization for color would be to lay down the thread lines in order of lowest average darkness to highest average darkness. (instead of laying down all the red, then green, then etc, in which case the last color laid down would dominate the image). Of course, this would mean the robot would need to support multiple threads concurrently; not sure how much of a challenge that would be.
This is a very interesting idea. I'll be honest with you that I haven't fully solved the color issue. It's very complicated since order highly affects the outcome in a very nonlinear way. So far I found CMYK works quite well in a set order but I'm sure it can be improved.
I like the way you say vector
Lol, how do you say it?
Shouldn't the radon transform of a line be a delta distribution or something similar? Or do they have thickness?
Good question. You can model the line as a Dirac delta function. In which case, the radon transform of this yields 1/abs(sin). However, in reality the line will have some width. I've tried modelling the line with a box profile, triangular profile and oval profile, but all of the radon transforms of these profiles are very complicated and slow to implement numerically. In the end I used a mathematical hack to make the Dirac delta function have thicknesses. Does that answer your question
@@virtually_passed kind of. I was thinking that given a line, there's exactly one value for theta 1 and theta 2 where it overlaps, and for all the others values of thetas, the function in integral to compute the radon transform is 0 at almost every point Do you compute the radon transform numerically? Thanks for the video btw!
Congratulations on all your work! It's stunning! I just bought the Matlab script for colored strings. How to obtain the sequence of chains to perform for each color?
Hey there, I'm glad you like it. At the moment the script does not output the precise list of nails. That's a more complicated process that I didn't include. However if this is something you're interested in then send me an email at virtuallypassed@gmail.com and I'll try to create a custom script for you that contains it all in one MATLAB file
Why can't you "divide" A transpose from both sides of that final equation?
Good question. When dealing with matrices we can't divide anymore. We need to multiply both sides by the inverse matrix. And this operation is only defined for square matrices. A^T isn't square in general (there could be more rows than columns or visa versa). However, in the very unlikely case that A happens to be square (ie there are just as many unique data points as unknowns) then you can inverse A^T and the pseudo inverse will collapse into the regular inverse of A. Hope that makes sense
thanks man it really helps.
Glad to hear it!
bro just threw fourier in there at the end and then dipped
I've got another video followup. I've also got details about the Fourier method on my patreon. Spoiler: the radon transform method is better than the Fourier method in my opinion.
british + american accent combo lets go
And Australian :)
1:43, Where, Where? Ah, it's life, bummer
Wow, what an exciting application of math! When I saw the thumbnail, it reminded me of the math behind CT-imaging, and indeed the mathematical problem that you solve seems to be related... Keep on doing the good stuff!
Glad you enjoyed it. I made a follow up video where I ended up using the math from CT scanners!
Thank you!
You're welcome!
Just an amazing explanation.. Thanks
Glad it was helpful!
You are amazing. You are a guy who explains a difficult problem in a simple way.
Thanks :)
In my youth during the 70's I was surrounded by various wire-art framed pictures and even did a few (geometric) versions for myself... but this absolutely blows my non-mathematical mind... thankyou for showing us what's really possible with this form of art🥰 (PS... I now know that I _need_ that stormtrooper artwork!!!)
Thanks! Glad you like it! 😊
Very good explanation...
Thanks 👍
What program do you use to write?
Microsoft onenote 2016. Although you can use many others like krita or Photoshop
@@virtually_passed Thanks
Are oscillations getting bigger around equilibrium ?
Nope the amplitude of oscillation decays exponentially with a linear SMD system. The oscillations get smaller. :)
@@virtually_passed Thx for answering. Im beginner but it looks like amplitude of the oscillations gets smaller with tim, but frequencies seem to be bigger and bigger. I guess im wrong but visually that's what it looks like nah ?
I don't really comment on these kind of videos but damn this one was pretty well explained. English isn't my mother language so I've been trying to go for english videos explaining the stuff I'm going trhu at uni, and you pretty much made it so clear to me something my teacher wouldn't accomplish explaining during the whole unit. This actually rocks, great content I genuinely ended up being interested even tho I initially came here cause I got an exam soon.
Thanks 🙏👍
❤
may the force be with you.
Thank you padawan
Hello! Can you please inform about the damping ratio if there are two masses, two dampers and two springs?
What a great question. For a single dof system you can easily find the damping ratio using the formulas shown. However for a 2 dof system there will be 2 damping ratios corresponding to each mode. You'll have to use something called modal analysis to find these two modes by diagonalizing the equations of motion. Once you have the uncoupled equations of motion you can use the formula shown to each equation to find both damping ratio. Hope that makes sense
Amazing video
Glad you think so!
tysm, exactly what i was looking for !!!
No problem!
Like the music :)
Thanks! All because of your suggestion to change song 😉
Art work was super, What software do you use? Can you sell the software sir ?
Thanks! I coded up the algorithm myself with MATLAB. You can get access to the code via my Patreon :) patreon.com/VirtuallyPassed
Thank you
Sir different shapes also possible ?? Like semi circle, rectangle, ovel and other shapes hapes. ??
@@KeerthiRoyal25 Only circle and rectangle at the moment
You should diversify your song choices, they're not bad, but it's getting to be too much 😭
Thanks for the advice :) I'll choose a different song next time. Do you have any favorites?
@@virtually_passed not really, idk what the scene is for copyright free music. Main thing is to repeat songs less, since at least subscribers get a lot of shorts with the same audio. Doesn't help that they're enjoyable to watch multiple times, adding to the repetitiveness 😂
I guess the axis are position and velocity, or kinetic and potential energy?
x=position, y = velocity :)
Song?
Go season
So... I can find the brightest point on my transform but it doesn't correspond exactly to any possible thread. What do? I tried filtering my transform based on actually reachable threads, but then there's no need to calculate the radon of a single thread and subtract that (since the transform of the image is just a sparse matrix of points).
Once you have an ordered list of nails each described with (alpha,s) values, then you need to convert those values to Psi1,Psi2 values using the last formula I derived. Once you do that you'll find that your values will likely not correspond to a perfect equally spaced set of nails, so you'll have to do a linear interpolation. You'll also have to make an algorithm that converts all these individual disconnected lines, into something a single thread can reach. Another alternative is to perform the greedy algorithm in Psi1,Psi2 space.