Vladimir Bulatov(bulatov.org),
Gathering 4 Gardner 15,
Summary
Natural Patterns
Patterns in nature are described by physical laws. They rarely are symmetric
Symmetric Patterns
Symmetric patterns are usually man made
Natural Patterns Generator
Gray-Scott reaction-diffusion equations
`(del u)/(del t) = D_u Delta u -uv^2 + f (1-u)`
`(del v)/(del v) = D_v Delta v + uv^2 - (f + k) v `
The equations describe the process that consists of reaction and diffusion of reagents `U` and `V`.
`U + 2V -> 3V` (`U` and `V` react and produce more `V`)
`V -> P` (chemical `V` is transformed into inert chemical `P` which is removed)
`u(x,y;t)` and `v(x,y;t)` are concentrations of chemicals `U` and `V`
`D_u` and `D_v` are diffuse coefficients.
`f` (feed) and `k` (kill) are parameters which control the pattern formation
Gray-Scott unstable loops
The initial perturbations is transformed into mostly static patterns
Gray-Scott straight worms
The fast growing worms avoid each other
Gray-Scott parallel waves
fast moving parallel waves with some instability
Gray-Scott mazes
Stable compicated mazes pattern
Gray-Scott static spots
Fast forming static spots
Gray-Scott gliders
Stable u-shaped gliders discovered by Robert Munafo
Gray-Scott spiral waves
Nice spiral waves
Pattern → symmetric pattern?
How can we enfoce the pattern to be symmetric?
Simple Symmetrization
Select a symmetry group ✶442 (generated by 3 mirrors)
Simple Symmetrization
Reflections in the mirrors form a symmetric tiling ✶442
Simple Symmetrization
Place the image of the pattern into each tile
Simple Symmetrization
Remove the tiles' boundaries
Simple Symmetrization
Final symmetric pattern ✶442
Simple Symmetrization
Running simulation looks good 🙂
Simple Symmetrization Test
Select a symmetry group 442 (generated by 3 rotations)
Simple Symmetrization Test
The fundamental domain of 442 is twice the size of ✶442
Simple Symmetrization Test
rotations form a symmetric tiling 442
Simple Symmetrization Test
Copy the image of the fundamental domain into each tile
Simple Symmetrization Test
remove tiles' boundaries
Simple Symmetrization Test
Final 442 symmetric pattern. Not so good 😕
Simple Symmetrization Test
Running simulation. It's not getting better 😕
Simple Symmetrization Analisys 😕
Simple Symmetrization works only for 4 out of 17 wallpaper groups
"Good" groups
✶2222, ✶632,
✶333, ✶442
kaleidoscopic groups are generated by reflections only
"Bad" groups
632,
333,
442,
2222, 3✶3,
4✶2,
22✶,
2✶22,
✶✶,
✶X, 22X,
XX,
O
The remaining groups' generators include
rotations,
glide reflections and
translations as well.
Simple Symmetrization
The diagram of Simple Symmetrization
Symmetric Simulation (SymSim)
The diagram of SymSim
SymSim test
symmetry 442 - step 0
SymSim test
symmetry 442 - step 1
SymSim test
symmetry 442 - step 10
SymSim test
symmetry 442 - step 20
SymSim test
symmetry 442 - step 30
SymSim test
symmetry 442 - step 40
SymSim test
symmetry 442 - step 50
SymSim test
symmetry 442 - step 60
SymSim test
symmetry 442 - step 70
SymSim test
symmetry 442 - step 80
SymSim test
symmetry 442 - step 90
SymSim test
symmetry 442 - step 100
SymSim animation
SymSim for pattern 442
SymSim examples
Spiral waves with symmetry XX
SymSim examples
U-gliders with symmetry 333
SymSim examples
Periodic windmill with symmetry 4✶2
SymSim examples
Spirals with symmetry 4✶2
SymSim examples
Oscillating tentackes 333
SymSim examples
Spinning worms 333
SymSim examples
Animated symmetric maze 632
SymSim examples
Complex Ginzburg-Landau pattern O
SymSim properties
SymSim can be used in wide range of geometries: Euclidean, Hyperbolic, Spherical, Inversive, etc.
SymSim can be used for wide array of symmetries in those geometries.
The solutions obtained using SymSim procedure are true solutions of the original equations.
The symmetrization procedure effect is to selects symmetrical solution from the wider class of arbitrary solutions.
SymSim computations add very little cost to the original simulation.
In practice, the symmetrization step can be used only once per 1000 simulation steps.
SymSim was initially implemented for Gray-Scott Reaction Diffusion equation.
Now it works for Complex Ginzburg-Landau equation (superconductivity phase transitions).
The Navier–Stokes equations (fluid dynamics) support is under development
SymSim is implemented as online applicaition using HTML, JavaScript and WebGL-2.
It runs on any platform which supports these technologies (desktop, tablet, cellphone).
