Minkowski cellular automata simulator

About the program

The purpose of this simulator is to experiment with cellular automata (analogs of Game of Life), living on the Minkowski plane: an analog of the Euclidean plane governed by the pseudo-euclidean distance (note the minus sign):

$$d((x_1,y_1),(x_2,y_2)) = (x_1-x_2)^2-(y_1-y_2)^2.$$
Animated image of rotations in Euclidean and Minkowsky spaces
Rotations in Euclidean and Minkowsky spaces


Lattice matrix

Cellular automata in this simulator live on a lattice of regularly spaced points with a certain rotational symmetry. In the Euclidean plane, there are two such lattices: square lattice having 4-fold rotational symmetry and hexagonal lattice having 6-fold symmetry. However, in the Minkowski plane there is infinite number of such lattices.

It might be not obvious, but such lattices (both Euclidean and Minkowskian) are identified by the integer matrices of determinant 1, also known as the elements of the special linear group \(SL(2,\mathbb{Z})\).

$$M=\left(\begin{matrix}a && b\\c && d\end{matrix}\right), |M|=ad-bc=1,$$ $$a,b,c,d \in \mathbb{Z}.$$

This, however, is not a one-to-one relation, since different matrices could define the same lattice. Particularly, congruent matrices generate the same lattice. The trace of the matrix (the sum of its diagonal elements \(a+c\)) determines lattice type: if \(|tr(M)|<2\), then the lattice is Euclidean, and if \(|tr(M)|>2\) then it is Minkowskian. The case when \(tr(M)=\pm 2\) is degenerate and not supported by the simulator.

In the simulator, Lattice matrix field defines the lattice. It must contain four integers separated by spaces, in row-major order: "a b c d".

Some matrices and their lattices:

Matrix Lattice
\(\left(\begin{matrix}0 && -1\\1 && 0\end{matrix}\right)\) Euclidean square lattice
\(\left(\begin{matrix}1 && 1\\-1 && 0\end{matrix}\right)\) Euclidean hexagonal lattice
\(\left(\begin{matrix}2 && 1\\1 && 1\end{matrix}\right)\) The simplest Minkowskian lattice


In cellular automata, cell state is affected by the states of its neighbors. In this simulator, all cells at the certain distance are considered neighbors. It is also possible to specify several distances. Instead of specifying the distance directly (which proved to be inconvenient), a list of sample neighbor coordinates is given. The format of the list is "x1 y1;x2 y2;..."

Cell coordinates are not obvious, to see them enable Draw tool and hover the mouse above the grid. Text in the bottom left corner of the view displays cell coordinates. Alternatively, use Cue tool and select some cell. Cell coordinates are displayed at the same place


Some tools on the top toolbar are grouped in pairs. When using mouse, left mouse button activates the selected tool, and right mouse button activates its pair.

Editing cells

Enable Draw tool and use mouse do toggle cells. When using custom rules with more than two states, additional buttons allowing to select state are shown. Draw tool hotkey is D.

Running the automaton

To run the simulation for one step, use the Step button. Its hotkey is N. If the calculation takes long time, a button allowing to cancel the simulation appears.

To undo the last simulation step, press Z.

To run the simulation continuously, click Go button.

Random configuration

To generate random configuration of cells, use controls in the "Randomize" section of the left panel. Enter the size of the random block, and the density of "alive" cells, then click Random fill. The hotkey is A.

Using cue mark

Use Cue mark tool to select a cell and see its neighbors. Neighbor cells are shown with empty circles; curves of equal distance are shown by dashed lines. Cue cool hotkey is U. Alternatively, use Draw tool and right mouse button to put cue mark. To remove cue mark, put it on the same place, or click Hide button on the top toolbar.

Copying and pasting

To copy a configuration of cells, activate Copy tool on the top toolbar, and drag mouse over it. Selected configuration of cells is put into the "buffer" text area on the left control panel. Hotkey is C.

To paste a copied configuration, Paste tool and click where to paste it. Hotkey is P.

Use Pan tool to drag the view using the mouse. Alternatively, hold Ctrl key and drag mouse - this method works when any other tool is active

Use Rotate tool or drag mouse left and right holding Shift to rotate the view around the center. When Euclidean lattice matrix is used, this tool does a familiar Euclidean rotation. When the matrix is Minkowskian, it does pseudo-rotation (squeeze mapping).

To zoom the view in and out, either use mouse wheel or buttons Zoom In/Out on the left control panel.

"Navigator" section of the left control panel shows current position and rotation of the view. Rotation angle is in degrees; when Minkowskian lattice is active, rotation angle could be arbitrarily high or low: unlike Euclidean rotation, pseudo-rotation does not loops after 360°.

Adjusting the display

Several buttons on the left panel control the appearance of the view:

The slider above them controls the size of the colored circle of the cell image.

Exporting data

There are two ways to export data from the simulator. The first one is to use Copy tool and then copy the text from the "Buffer" text area. This text can be stored, then pasted back to the buffer.

The other way is to export current simulator state to the URL using the Save to url... button. It generates an URL that encodes current world state, rule and view position.

Custom rules

Custom rules allow to run custom automata with any number of states and user-defined transition rules. They need not be totalistic. To enable custom rule, click Custom rule... button and edit the code. Click Basic sample / Complex sample buttons to load sample code.

Custom rule is defined by a JavaScript object having following fields:

Note that the order of neighbor summation is not defined. Also, only cells with nonzero state are summed.

Shortcuts overview

NNext step
ZUndo last step
GRun/stop simulation
ARandom fill
EClear the world
Mouse tools
DEnable drawing mode
UEnable cue mark mode
MEnable panning mode
REnable rotation mode
CEnable select and copy mode
PEnable paste mode
Viewing and navigation
QHide cue mark if shown
HHavigate to origin
[Zoom in
]Zoom out