Falling pentagons pretending to be bestagons
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- čas přidán 10. 07. 2024
- After the falling triangles in the video • Falling triangles , and the falling squares in the video • Falling squares , here come the falling pentagons. In case you wonder how many other regular polygons I am going to try, I plan to stop at hexagons, and then add some coagulation reactions, as well as other shapes.
One observation one can make here is that despite not being hexagons, the pentagons favor a hexagonal closed-packing lattice.
To compute the force and torque of pentagon j on pentagon i, the code computes the distance of each vertex of pentagon j to the faces of pentagon i. If this distance is smaller than a threshold, the force increases linearly with a large spring constant. In addition, radial forces between the vertices of the pentagons have been added, whenever a vertex of pentagon j is not on a perpendicular to a face of pentagon i. This is important, because otherwise pentagons can approach each other from the vertices, and when one vertex moves sideways, it is suddenly strongly accelerated, causing numerical instability. A weak Lennard-Jones interaction between pentagons has been added, as it seems to increase numerical stability.
The temperature is controlled by a thermostat with constant temperature. There is a constant gravitational force directed downward.
This simulation has two parts, showing the evolution with two different color gradients:
Orientation: 0:00
Kinetic energy: 1:27
In the first part, the particles' color depends on their orientation modulo 72 degrees. In the second part, it depends on their kinetic energy, averaged over a sliding time window.
To save on computation time, particles are placed into a "hash grid", each cell of which contains between 3 and 10 particles. Then only the influence of other particles in the same or neighboring cells is taken into account for each particle.
The temperature is controlled by a thermostat, implemented here with the "Nosé-Hoover-Langevin" algorithm introduced by Ben Leimkuhler, Emad Noorizadeh and Florian Theil, see reference below. The idea of the algorithm is to couple the momenta of the system to a single random process, which fluctuates around a temperature-dependent mean value. Lower temperatures lead to lower mean values.
The Lennard-Jones potential is strongly repulsive at short distance, and mildly attracting at long distance. It is widely used as a simple yet realistic model for the motion of electrically neutral molecules. The force results from the repulsion between electrons due to Pauli's exclusion principle, while the attractive part is a more subtle effect appearing in a multipole expansion. For more details, see en.wikipedia.org/wiki/Lennard...
Render time: 49 minutes 6 seconds
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Part 2 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-wor...
Music: "City By Nght" by ELPHNT@ELPHNT
Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261-277 (2009). doi.org/10.1007/s10955-009-97...
www.maths.warwick.ac.uk/~theil...
Current version of the C code used to make these animations:
github.com/nilsberglund-orlea...
www.idpoisson.fr/berglund/sof...
Some outreach articles on mathematics:
images.math.cnrs.fr/auteurs/n...
(in French, some with a Spanish translation)
#molecular_dynamics #polygon - Věda a technologie
writhing in pentagony
It was cool watching the "streaks" of light and dark as the pentagons forced each other to rotate.
Yes. You have defined cool Well. Also rightly so in past tense. Next cool thing is hexagons pretending to Be sexagons.
I will say, for a video on this channel, the physics of this simulation were less stable than I've come to expect
Contact forces are hard to model.
tbf they say to ignore friction in physics problems
It does seem like this somewhat models increasing pressure the further down a liquid you go, though
best title
Pentagons are the next-to-bestagons?
So it what seem. Although, tbh, all regular polygons with at least five sides may behave similarly.
I just realized I am a pentagon in a society of hexagons. It explains why I'm always bouncing off the walls and never quite occupy the space cut out for me in this close packed lattice we all live in.
pretentious and self-centered nonsense.
@@CombineWatermelon actually, that's my point.
@@erich1394 okay then, shouldn't have posted it
@@CombineWatermelon nah I posted because its funny to sound overdramatic and self-important sometimes. I'm glad I posted it. I'm funny.
@@erich1394I, for one, quite enjoyed it.
what would falling spectre monotiles do? They aren't convex so it's very unlikely that they'd "figure out" that they actually tile
I might try that at some point. But my code will indeed have to evolve some more, to be able to handle more general, non-convex shapes.
I was curious, are the interactions of these shapes supposed to emulate any type of material? rubber surface? steel? glass? android's bottom?
I'm not sure exactly which materials this is an adequate model for, but it is supposed to represent a rather rigid material. Numerically, it is easier to have a harmonic interaction with a large spring constant than a perfectly solid object. But in practice, even the hardest objects have some large but finite Young module.
Why pretend when they already are?
Being firstagons doesn't make them bestagons.
Can you do another with random shapes?
Perhaps later. So far, my code works when all objects have the same shape. I may fix that once I have coded the pairwise interactions between a few more shapes.
Interesting..
HOW DARE YOU TALK SMACK ABOUT THE BESTAGONS?!
It's not me, it's the pentagons.