Nils Berglund | The Rock-Paper-Scissors equation on the sphere, with improved behavior at the poles @NilsBerglund | Uploaded April 2024 | Updated October 2024, 3 minutes ago.
This is a remake of videos such as youtu.be/YaoqL0d573w , showing a simulation of the Rock-Paper-Scissors reaction-diffusion equation on the sphere. The difference is that the control of the behavior near the poles has been improved. The previous videos used a rectangular simulation grid that became very dense near the poles, requiring a regularization procedure that slowed down the dynamics. Here the grid points have been reduced near the poles by a blocking procedure, so that almost no regularization is required.
The reaction-diffusion equation is defined as follows. At each point in space and time, there are three concentrations u, v, and w of chemicals, which we may call Red, Blue and Green. Denoting by rho = u + v + w the total concentration, the system of equations is given by
d_t u = D*Delta(u) + u*(1 - rho - a*v)
d_t v = D*Delta(v) + v*(1 - rho - a*w)
d_t w = D*Delta(w) + w*(1 - rho - a*u)
where Delta denotes the Laplace operator, which performs a local average, and the parameter a is equal here to 0.75 while D is constant, equal to 0.2. The terms proportional to a*v, a*w and a*u denote reaction terms, in which Red is beaten by Blue, Blue is beaten be Green, and Green is beaten by Red. The situation is thus similar to the Rock-Paper-Scissors game (see en.wikipedia.org/wiki/Rock_paper_scissors ), and there exist simpler cellular automata with similar properties, see for instance softologyblog.wordpress.com/2018/03/23/rock-paper-scissors-cellular-automata
The video has four parts, showing the same simulation with two different color gradients, and two different visualizations:
First chemical, 3D: 0:00
Predominant chemical, 3D: 1:44
First chemical, 2D: 3:33
Predominant chemical, 2D: 5:18
In parts 1 and 3, the color hue and radial coordinate depend on the concentration of the first chemical. In parts 2 and 4, the color depends on the most abundant chemical, though the other two chemicals can still be present, which is why some color patches disappear or appear out of the blue. Parts 3 and 4 show an equirectangular representation of the sphere.
The equation is solved by finite differences, where the Laplacian is computed in spherical coordinates. Some smoothing has been used at the poles, where the Laplacian becomes singular in these coordinates. The color hue and radial coordinate show the value of the concentration u of one of the chemicals. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude.
Render time: Parts 1 and 2 - 1 hour 7 minutes
Parts 3 and 4 - 58 minutes 16 seconds
Compression: crf 23
Color scheme: HSL/Jet
Music: "Waltz of the Flowers" by Tchaikovsky
See also https://images.math.cnrs.fr/Qu-est-ce-qu-une-Equation-aux-Derivees-Partielles-Stochastique.html for more explanations (in French) on the Allen-Cahn equation.
The simulation solves a reaction-diffusion equation by discretization. The algorithm is adapted from the paper hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
#reaction-diffusion #spirals
This is a remake of videos such as youtu.be/YaoqL0d573w , showing a simulation of the Rock-Paper-Scissors reaction-diffusion equation on the sphere. The difference is that the control of the behavior near the poles has been improved. The previous videos used a rectangular simulation grid that became very dense near the poles, requiring a regularization procedure that slowed down the dynamics. Here the grid points have been reduced near the poles by a blocking procedure, so that almost no regularization is required.
The reaction-diffusion equation is defined as follows. At each point in space and time, there are three concentrations u, v, and w of chemicals, which we may call Red, Blue and Green. Denoting by rho = u + v + w the total concentration, the system of equations is given by
d_t u = D*Delta(u) + u*(1 - rho - a*v)
d_t v = D*Delta(v) + v*(1 - rho - a*w)
d_t w = D*Delta(w) + w*(1 - rho - a*u)
where Delta denotes the Laplace operator, which performs a local average, and the parameter a is equal here to 0.75 while D is constant, equal to 0.2. The terms proportional to a*v, a*w and a*u denote reaction terms, in which Red is beaten by Blue, Blue is beaten be Green, and Green is beaten by Red. The situation is thus similar to the Rock-Paper-Scissors game (see en.wikipedia.org/wiki/Rock_paper_scissors ), and there exist simpler cellular automata with similar properties, see for instance softologyblog.wordpress.com/2018/03/23/rock-paper-scissors-cellular-automata
The video has four parts, showing the same simulation with two different color gradients, and two different visualizations:
First chemical, 3D: 0:00
Predominant chemical, 3D: 1:44
First chemical, 2D: 3:33
Predominant chemical, 2D: 5:18
In parts 1 and 3, the color hue and radial coordinate depend on the concentration of the first chemical. In parts 2 and 4, the color depends on the most abundant chemical, though the other two chemicals can still be present, which is why some color patches disappear or appear out of the blue. Parts 3 and 4 show an equirectangular representation of the sphere.
The equation is solved by finite differences, where the Laplacian is computed in spherical coordinates. Some smoothing has been used at the poles, where the Laplacian becomes singular in these coordinates. The color hue and radial coordinate show the value of the concentration u of one of the chemicals. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude.
Render time: Parts 1 and 2 - 1 hour 7 minutes
Parts 3 and 4 - 58 minutes 16 seconds
Compression: crf 23
Color scheme: HSL/Jet
Music: "Waltz of the Flowers" by Tchaikovsky
See also https://images.math.cnrs.fr/Qu-est-ce-qu-une-Equation-aux-Derivees-Partielles-Stochastique.html for more explanations (in French) on the Allen-Cahn equation.
The simulation solves a reaction-diffusion equation by discretization. The algorithm is adapted from the paper hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
#reaction-diffusion #spirals
![[Flash warning] A planar wave crossing a gradient index lens at an angle](https://i.ytimg.com/vi/vZ_pKzjFXvo/hqdefault.jpg "[Flash warning] A planar wave crossing a gradient index lens at an angle
The first part of this video shows some flashing due to the formation of standing waves.
This is a variant of the simulation https://youtu.be/CdpTq3dSrXA , in which the incoming planar wave makes a small angle with the axis of symmetry of the lens. The videos are inspired by Huygens Optics recent short https://youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens.
Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens.In this simulation, the index decreases like sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming planar wave being focused at a point in the focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 1:03
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window.
There are periodic boundary conditions between the top and bottom boundaries, absorbing boundary conditions on the right boundary, and a time-periodic signal is imposed on the left boundary. The display at the right shows the signal along the focal plane, which is indicated by a vertical line.
Render time: 30 minutes 1 second
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight
Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap
Music: Chicago by Joe Bagale
See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!
#wave #lens #gradient_index")
