What I found interesting by myself is that we can imagine both a cylinder which rotates left to right (or right to left) *and* a cylinder which rotates top to bottom (or otherwise)! For some frequency ratios it’s easier than for others. And that’s because we can have both cylinders with a wave drawn on them as different projections of one wave drawn on a single 4d torus {(x, y, z, w) : x^2 + y^2 = z^2 + w^2 = 1} where (x, y) = (cos(ω1 t + ϕ1), sin(ω1 t + ϕ1)) trace the first circular motion with angular velocity ω1 (and initial phase ϕ1), and (z, w) likewise trace the second one with ω2, ϕ2. Then we can discard either w or y coordinate to get differently oriented cylinders in x-z-y or x-z-w space, then discard the remaining y or w to get a Lissajous curve in the x-z plane.
Interesting! I read there is an other way to find the torus projection : lissajous Curves are also billiard trajectory, and if you unfold its square shape in x and y direction so that it represent the mirror trajectory, you end up with 4 square that ensure continuity on up & down border and right&left, and these limit condition referred as a torus surface
Amazing video, which really simplified my understanding of the curves! I do wonder how you were able to visualize the curves rotating about the y axis though, specifically at around 5:10 . I was only ever able to do so by using cylindrical sine waves with the parametric equations of {x=sin(t),y=sin(tk+l),z=cos(t)}, but this would only ever represent Lissajous figures with ratios of 1/n or n. When I changed the frequency of the first sin(t), the overall motion of the 3D curve would instead be something a lot more complicated than a simple rotation.
Thank you, i'm glad it has helped you ! Do you have plot enough points ? because your equation seems right Here is the code i used in python : import numpy as np from matplotlib import pyplot as plt from matplotlib import animation from mpl_toolkits.mplot3d import Axes3D # Lines to plot in 3D w=10 n=360*w*20 t=np.linspace(0,60*w/360,n) ratio = np.sqrt(2) xx, yy, zz = np.sin(t), np.cos(t), np.sin(t*ratio) # Create a figure and a 3D Axes fig = plt.figure() ax = Axes3D(fig) # Hide grid lines ax.grid(False) # Hide axes ticks ax.set_xticks([]) ax.set_yticks([]) ax.set_zticks([]) ax.set_axis_off() def init(): ax.plot3D(xx, yy, zz, c="blue", alpha=0.6) return fig, def animate(i): t=np.linspace(0,60*w*50/360,n) xx, yy, zz = np.sin(t), np.cos(t), np.sin(t*ratio) ax.plot3D(xx, yy, zz, c="blue", alpha=0.6) ax.view_init(elev=10, azim=i) return fig, # Animate anim = animation.FuncAnimation(fig, animate, init_func=init, frames=360, interval=25, blit=True) # Save as gif writergif = animation.PillowWriter(fps=30) anim.save(filename="anim.gif", writer=writergif)
Very good, I like how personal the style is. I had encountered all of the single topics on their own but the way you brought them together is really intriguing. Well done!
i've seen chaotic systems as described as pool tables with some form of curved obstruction somewhere on the board... and now i'm watching this and wondering if some wave (or maybe noise, in fact i fear that it would have to be noise 😨) can define a curve that best reflects the chaotic motion of such systems...
What a BRILLIANT video and critical research! Well done! AMAZING and Hats Off!!!
Thank you so much !!
Beautiful video
What I found interesting by myself is that we can imagine both a cylinder which rotates left to right (or right to left) *and* a cylinder which rotates top to bottom (or otherwise)! For some frequency ratios it’s easier than for others.
And that’s because we can have both cylinders with a wave drawn on them as different projections of one wave drawn on a single 4d torus {(x, y, z, w) : x^2 + y^2 = z^2 + w^2 = 1} where (x, y) = (cos(ω1 t + ϕ1), sin(ω1 t + ϕ1)) trace the first circular motion with angular velocity ω1 (and initial phase ϕ1), and (z, w) likewise trace the second one with ω2, ϕ2. Then we can discard either w or y coordinate to get differently oriented cylinders in x-z-y or x-z-w space, then discard the remaining y or w to get a Lissajous curve in the x-z plane.
Interesting! I read there is an other way to find the torus projection : lissajous Curves are also billiard trajectory, and if you unfold its square shape in x and y direction so that it represent the mirror trajectory, you end up with 4 square that ensure continuity on up & down border and right&left, and these limit condition referred as a torus surface
Amazing video, which really simplified my understanding of the curves! I do wonder how you were able to visualize the curves rotating about the y axis though, specifically at around 5:10 . I was only ever able to do so by using cylindrical sine waves with the parametric equations of {x=sin(t),y=sin(tk+l),z=cos(t)}, but this would only ever represent Lissajous figures with ratios of 1/n or n. When I changed the frequency of the first sin(t), the overall motion of the 3D curve would instead be something a lot more complicated than a simple rotation.
Thank you, i'm glad it has helped you !
Do you have plot enough points ? because your equation seems right
Here is the code i used in python :
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import animation
from mpl_toolkits.mplot3d import Axes3D
# Lines to plot in 3D
w=10
n=360*w*20
t=np.linspace(0,60*w/360,n)
ratio = np.sqrt(2)
xx, yy, zz = np.sin(t), np.cos(t), np.sin(t*ratio)
# Create a figure and a 3D Axes
fig = plt.figure()
ax = Axes3D(fig)
# Hide grid lines
ax.grid(False)
# Hide axes ticks
ax.set_xticks([])
ax.set_yticks([])
ax.set_zticks([])
ax.set_axis_off()
def init():
ax.plot3D(xx, yy, zz, c="blue", alpha=0.6)
return fig,
def animate(i):
t=np.linspace(0,60*w*50/360,n)
xx, yy, zz = np.sin(t), np.cos(t), np.sin(t*ratio)
ax.plot3D(xx, yy, zz, c="blue", alpha=0.6)
ax.view_init(elev=10, azim=i)
return fig,
# Animate
anim = animation.FuncAnimation(fig, animate, init_func=init,
frames=360, interval=25, blit=True)
# Save as gif
writergif = animation.PillowWriter(fps=30)
anim.save(filename="anim.gif", writer=writergif)
This was an amazingly produced video! Really well structured and animated, please make some more! You deserve far more attention..
Thank you very much ! That is so motivating ! Don't worry new content is on the way ;)
Nice stuff, thanks for breaking this down! I now have a greater appreciation for these curves :)
Thank you ! I'm glad you enjoy it !
Fascinating video. Really enjoy maths applied to music. This video should have more views!
Thank you so much! 🙏☺️
Very good, I like how personal the style is. I had encountered all of the single topics on their own but the way you brought them together is really intriguing. Well done!
Thank you very much! ☺️
Man, this was so good!
Thank you, it is so motivating! 😄
good video!
thank you !
i've seen chaotic systems as described as pool tables with some form of curved obstruction somewhere on the board... and now i'm watching this and wondering if some wave (or maybe noise, in fact i fear that it would have to be noise 😨) can define a curve that best reflects the chaotic motion of such systems...
Meta
Didn't get it immediately, that's right!!