What is this?

It's a simulator of the Single Rotation cellular automaton, that tracks positions of the cells and uses Lanczos interpolation to add an intermediate frames. Its primary purpose is a nice visual effect, a better tool for experiments with reversible cellular automata is the online simulator.

Single Rotation rule

This cellular automaton acts on a grid of cells, where each cell is either dead or alive. On each generation, the grid is divided into 2x2 blocks, and if the block contains exactly one alive cell, the block is rotated by 90°. Thus "single rotation". After each generation, the 2x2 subdivision is alternated. This scheme is known as Margolus neighborhood.

Despite the simplicity, complex emergent behavior appears in this rule, resembling the iconic Game of Life. One important difference is that "Single Rotation" is time-reversible, that is: for every state it is always possible (and easy) to calculate previous state. You can play with this rule in the online simulator.

Cell tracking

"Single Rotation" rule is conservative: it preserves total number of alive cells. Moreover, from the definition of the rule it is clear that it is possible to track position of the every individual alive cell. The application does this idea.

Interpolation

Cell coordinates are discrete, alive cells are "jumping" by 1 unit in some direction. To make animation more visually pleasing, this simulator interpolates intermediate positions, using Lanczos resampling.

Oscillation filtering

One can easily check that in the "Single Rotation" rule a single cell would constantly rotate with period 4. To make this less annoying, a possibility to filter higher frequencies of the cell movement is added. For example, when "filter" setting is 4, then every oscillation with period 4 or less is replaced by the constant average value, and rotating cells visually stand still. Higher filter setting allow to cease oscillations of higher period. This is also done by Lanczos filter.

2D version

Using the simulator

Scene description language

Both 3d and 2d simulators use a simple language to describe loaded scenes, that is superset of the RLE (i.e. bare RLE code for the pattern is a valid scene description). The language makes it simpler to combine multiple patterns and assign colors to cells.

Scene is described by a sequence of statements, separated by newlines or ";". Statement names are case-sensitive. Four statements are supported:

at X Y

Sets current origin to the cell with coordinates X, Y. Used by the following rle commands.

-- TEXT

Defines description text. Several lines may be defined. Text can not contain semicolons (;).

PATTERN_RLE

Puts given RLE code at the current coorinates. Rle code is a standard format, used by many cellular automata simulators.

b encodes empty cell,
o - alive cell,
$ - new line,
numbers encode repeat count of the following character.

colors COLOR-LIST

Sets current color palette. When adding patterns, new cell colors are taken from this pattern. COLOR-LIST can be either a ":"-separated list of any CSS colors, or the text "default". In the latter case, pattern is returned to the default. Example: colors red:#00ff00:rgb(0,0,255).

Effect of the at and colors statements lasts until they are redefined. Additional spaces and empty lines are ignored.

3D version

The 3D version visualizes world lines of cells, mapping time to 3rd dimension.

URL arguments

3D application accepts a range of advanced configuration parameters available via URL arguments. Argument names are case-sensitive.

antialias=true
Enables anti-aliasing, making smoother images but reducing performance.
visibility=distance
where distance is a number in the range 1000 ~ 50'000. It defines, how far the world lines are seen. Big distance significantly decreases FPS. Here are some sample links:
- visibility=3000 short distance, should improve FPS on slow machines.
- visibility=5000 a bit shorter distance, then default.
- visibility=40000 very long distance, if you don't need much FPS or have a fast machine.
Mesh generation parameters and their values:
- chunkSize=500
  number of steps in one mesh chunk
- skipSteps=1
  Generate tube section every n'th step (1 - every step)
- boardSize=100
  Size of the square board, must be even
- lanczosOrder=3
  Lanczos interpolation order, 1 ... 10. 1 - linear interpolation, 3 - smooth interpolation
- interpSteps=1
  How many mesh steps are there between 2 generation. integer, 1 ... 4
- smoothingPeriod=4
  Low-pass filter, removing oscillations with period bigger than this. integer, 1 ... 100. 1 - no filtering.
- timeScale=0.1
  "speed of light". z-axis length of one generation
- tubeRadius=0.1
  Radius of a single rube
- tubeSides=3
  Number of sides in the tube cross-section (2...10)

More information

Here are some my pages, where you can find more information about reversible cellular automata and Single Rotation rule.

Online simulator, uglier but more suited for experiments tool, where you can try your own patterns and rules.
Single rotation: remarkably simple and rich reversible cellular automaton - a blog post about this rule, with some thoughts on its properties.
A table of the Single Rotation spaceships - result of the automatic search for the spaceships (patterns that travel along the grid). Lists more than 300 different patterns.
And a blog post about the above table.

Source code and license

Source code is available on Github, under the permissive MIT license. It is written in CoffeeScript.