Compare automata on square, rectangular, triangular, cubic, and tetrahedral cells.
All five geometries advance together. Switch shapes or presets to compare the same generation; switching pauses playback. Geometry supplies the neighbors; a discrete local rule supplies the next state.
Step advances one generation. Shape and preset changes preserve the current generation and history. Reset, rule, threshold, or starting-field changes begin again at generation 0.
Mint: A. Amber: B. Equal cell volumes within each field make this fraction also an area or volume fraction. Overlapping lines indicate equal occupancy, not necessarily equal arrangements.
| Cell | Shared boundary | Interior neighbors | Construction |
|---|---|---|---|
| Square | Edge | 4 | 18 x 18 grid |
| Rectangle | Edge | 4 | Same grid, horizontal stretch 1.6 |
| Triangle | Edge | 3 | Each square split into 2 right triangles |
| Cube | Square face | 6 | 5 x 5 x 5 grid |
| Tetrahedron | Triangular face | 4 | Each cube split into 6 tetrahedra |
States live on the highest-dimensional cells. Triangles and tetrahedra, together with all their faces, form simplicial complexes. Squares, rectangles, and cubes form cell complexes. This automaton uses their cell-adjacency graphs; it does not evolve separate vertex and edge states or add higher-order interactions.
All boundaries are open: missing neighbors are omitted from the denominator. Updates are synchronous. Initial states are assigned per parent square or cube, so its triangles or tetrahedra begin with the same state. The initial spatial field matches within a dimension, but cell counts and graph distances differ. These are controlled illustrations, not equal-resolution benchmarks.
Squares and rectangles have identical adjacency and initial labels, so their states must agree at every generation. Geometric stretching changes their appearance, not this unweighted rule. Triangulating changes the neighborhood and the set of possible neighbor fractions.
q = count(neighbors in target state) / count(neighbors)q = 0 when there are no neighbors
Excitable wave: 1 becomes 2; 2 becomes 0; 0 becomes 1 if q >= threshold, otherwise stays 0. Target state is 1.
Cyclic pursuit: target = (current + 1) % 3. Adopt the target if q >= threshold; otherwise keep the current state.
Binary threshold: next = 1 if q >= threshold, otherwise 0. Target state is 1. This is a threshold rule, not Conway's Life.
The original Neato Burrito CA Lab retains Life, HighLife, Seeds, Day & Night, Brian's Brain, cyclic CA, Wireworld, and Rule 110. Its 2D rules use eight surrounding square-grid sites; this comparison uses shared boundaries. The existing 3D simplicial page explores relaxation instead.
3D cells are projected with depth sorting. The all-active view hides state 0 and is an illustrative view, not a volume measurement. The middle-layer filter selects cells by their parent cube's z index; it is not a geometric cross-section.
Construction reference: Edelsbrunner & Kerber, Dual Complexes of Cubical Subdivisions (Freudenthal triangulation). These tetrahedra are not regular, and the triangles are not equilateral.