Uploaded December 2021 | Updated September 2026, 2 weeks ago
In lack of own ideas I steal others. Recently I've seen several cool videos showing waves propagating in a von Koch snowflake fractal, e.g. by @Maciej Matyka and @Nils Berglund. Plus a blizzard (in local measures) hit my geographical region today.
The real space domain at which the snowflake fractal is defined has a resolution of 2000x2200. Herein a localised wave packet evolves.
For the first half, a quite odd wave packet is used. It's a Gaussian clipped to be identically zero outside a specific radius. Upon evolving in time, this wave packet pushes out some high momentum states associated with the abrupt edge of the wave packet. Further, this part is plotted at a log scale.
In the second half, an ordinary Gaussian wave packet evolves similarly.
Music by @gpcbass. The song has been used in another video on this channel. The song is a reproduction of a song called Caramba which originates back to 1991, then performed by the eminent Swedish orchestra Svef Tarkel in which @gpcbass played the bass.
In lack of own ideas I steal others. Recently I've seen several cool videos showing waves propagating in a von Koch snowflake fractal, e.g. by @Maciej Matyka and @Nils Berglund. Plus a blizzard (in local measures) hit my geographical region today.
The real space domain at which the snowflake fractal is defined has a resolution of 2000x2200. Herein a localised wave packet evolves.
For the first half, a quite odd wave packet is used. It's a Gaussian clipped to be identically zero outside a specific radius. Upon evolving in time, this wave packet pushes out some high momentum states associated with the abrupt edge of the wave packet. Further, this part is plotted at a log scale.
In the second half, an ordinary Gaussian wave packet evolves similarly.
Music by @gpcbass. The song has been used in another video on this channel. The song is a reproduction of a song called Caramba which originates back to 1991, then performed by the eminent Swedish orchestra Svef Tarkel in which @gpcbass played the bass.
![Butterfly-effect simulation using 501 double pendulums [1440p]](https://i.ytimg.com/vi/k3TaCGmJZ9I/mqdefault.jpg "Butterfly-effect simulation using 501 double pendulums [1440p]
This is a slightly different 1440p version of one of my previously posted videos. They say 1440p resolution are rendered with a significantly higher bitrate.
Below is essentially a copy-paste of the previous video:
Simulation of 501 double pendulums having initial start positions ranging between 2.2995 radians and 2.3 radians spanned from the vertical line. That is, the difference in starting angle between each adjacent pair of double pendulums is one millionth of a radian. Hence, the outermost apices of all the double pendulums sweeps only 1/2000 of a radian at the very start.
Despite such a tiny difference between the 501 initial conditions, the outermost apices of the pendulums rapidly spread out homogeneously all over their energetically allowed range, as the double pendulum is a chaotic system for certain criteria of the initial conditions, meaning that it is ridiculously sensitive to initial conditions. This is sometimes referred to as the Butterfly effect.
CPU time was ~3 minutes for calculating all positions of the pendulums for all 3200 frames on a mid-2012 MacBook Air. Plotting/saving all frames took ~1.5 hour in total.
Here is a good example page to start out playing with one double pendulum in Python: https://scipython.com/blog/the-double-pendulum/. My scripts need a thorough clean-up before sharing.
Music by @gpcbass!
Done in Python and ffmpeg.")
