` Symmetry on orbifolds

Exploration of Symmetrical Simulation on Orbifolds

Vladimir Bulatov (bulatov.org),
Joint Mathematics Meeting, January 9, 2025, Seattle

Summary

Our goal is to create process for generation of natural animated symmetric patterns

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 Waves

Parallel unstable waves

Gray-Scott Spiral Waves

Spiral Waves

Gray-Scott Mazes

Mazes

Gray-Scott Gliders

Gliders

Gray-Scott Voronoi Cells

Volonoi Cells

Periodic Boundary Conditions 1

Periodic Boundary Conditions

Periodic Boundary Conditions 2

Periodic Boundary Conditions

Periodic Boundary Conditions 3

Periodic Boundary Conditions

Periodic Boundary Conditions 4

Periodic Boundary Conditions

Periodic Boundary Conditions 5

Simulation is effectively performed on a 2D torus (orbifold notation O)

Periodic Boundary Conditions 6

Simulation is performed on a finite grid. Typical grid dimension [512x512]

Periodic Boundary Conditions 7

Simulation is performed on a finite grid. Typical grid dimension [512x512]
Points outside of the grid are mapped to the interior points

Periodic Boundary Conditions 8

Simulation is performed on a finite grid. Typical grid dimension [512x512]
Points outside of the grid are mapped to the interior points

Periodic Boundary Conditions 9

Simulation is performed on a finite grid. Typical grid dimension [512x512]
Points outside of the grid are mapped to the interior points

Periodic Boundary Conditions 10

Simulation is performed on a finite grid. Typical grid dimension [512x512]
Points outside of the grid are mapped to the interior points

General Orbifold

What to do for a general orbifold?

General Orbifold

What to do for a general orbifold?
Orbifold's fundamental domain may be of arbitrary shape

General Orbifold

Exterior points are located in arbitrary positions relative to the orbifold's interior
How to map those external points into interior?

Reverse Pixel Mapping

We use Reverse Pixel Mapping
It is efficient method to map arbitrary point in the plane into corresponding point inside of the fundamental domain

Simple Symmetrization

Points may be mapped into arbitrary positions between grid points

Simple Symmetrization

We use linear or quadratic interpolation to calculate the corresponding value

Simple Symmetrization

Original simulation

Simple Symmetrization

Original simulation overlayed by orbifold tiles

Simple Symmetrization

Symmetrized simulation

Simple Symmetrization

Symmetrized simulation - a lot of significant discontinuities

Simple Symmetrization

The simulation with Simple Symmetrization

Symmetric Simulation (SymSim)

We replace Simple Symmetrization with Symmetric Simulation
Result of the symmetrizaton step is fed back into the simulation chain

SymSim test on (333) orbifold

Simple Symmetrization. SymSim is not used
Persistant discontinuities on the boundaries of the fundamental domain

SymSim test on (333) orbifold

SymSim is used
Discontinuities are instantly disappearing

SymSim test on (333) orbifold

SymSim in slow motion

SymSim test on (✶333) orbifold

Simple symmetrization. SymSim is not used.
Discontinuities of derivatives on the boundaries of the fundamental domain.

SymSim test on (✶333) orbifold

SymSim is used
Pattern is smoothed out rather quicky

SymSim test on (✶333) orbifold

SymSim in slow motion

SymSim properties

SymSim can be used in wide range of geometries: Euclidean, Hyperbolic, Spherical, Inversive, etc.
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.
On practice the symmetrization step can be used only once per 1000 simulation steps.
SymSim is implemented for Gray-Scott Reaction Diffusion equation and Complex Ginzburg-Landau equation (superconductivity phase transitions).
SymSim is implemented as online applicaition using HTML, JavaScript and WebGL2.
- It runs on any platform which supports these technologies (desktop, tablet, cellphone).
- The application perform up to 100,000 simulation steps per second on 512x512 grid.
- The animations are rendered in 4K resolution in real time and can be exported to a video.