Time Step
0
Grid Size (m × n)
20 × 20
Tiling Mode
Square
State
⏸ Paused
Hover/Draw:
Cell (0, 0) = 1
🎨 Click or drag on the grid to paint with the active palette color. Fully functional during simulation.
Color Palette (Values 1 to 10)
Shortcuts: Keys 1–9, 0
2D Euclidean Tiling
4 Neighbors
Evolution Rule
Speed
10 fps
States (K)
10
Boundary Condition
Value Succession
Rows (m) & Cols (n)
Overlay Numbers
Configuration Presets:
📐 Tilings, Wolfram Rules & Mathematical Specifications
- Regular Euclidean Tilings:
- Square ($\{4, 4\}$): Each cell shares edges with 4 orthogonal neighbors (Up, Down, Left, Right).
- Hexagonal ($\{6, 3\}$): Pointy-topped hexagons with 6 edge-sharing neighbors (odd-r offset layout).
- Triangular ($\{3, 6\}$): Alternating Up ($\Delta$) and Down ($\nabla$) equilateral triangles sharing 3 edges with adjacent triangles.
- Update Rule Formulation: A cell in state \(v\) transitions to its predecessor \(\text{pred}(v)\) at time \(t+1\) if and only if its neighborhood satisfies the active rule predicate for value \(\text{pred}(v)\).
- Wolfram Parity (Rule 90/150 style): Target converts if the count of predecessor neighbors is odd (\(C_P \bmod 2 = 1\)). Even counts cancel out, generating self-similar fractal carpets.
- Directional Alignments: Constrains updates along physical axes. Vertical requires both Up & Down neighbors to match; Horizontal requires Left & Right.
- Simultaneous Execution: Transitions are strictly synchronous (\(A^{(t)} \mapsto A^{(t+1)}\)) using double buffering. Scan order does not bias wave propagation.
Simulation loaded.