A growing seed must eventually screen-wrap & bump into itself. Afterwards, wild and interesting patterns can result. And the precise way that this "self-collision" occurs for a given rule/seed combination is largely determined by the size of the grid & the grid's screen-wrapped topology.
The square field of cells, as used in this video, screen-wraps to give a the closed 2-d surface of a 3-d torus. By contrast, the hexagonal field screen-wraps to give the same torus, but with a 180 degree twist which is need for a closed surface:
math.stackexch...,
(NOTE: I have been using the term "rectangular grid" when I should have said "square grid". The type of array used in this video looks rectangular (width greater than height), only because the aspect-ratio of a regular hexagon is about 1.15 : 1 (column width : row height). But the horizontal & vertical directions contain the same number of cells, so it is definitely "square". Descriptions of past videos will be corrected.).
Prior to self-collision, seed development will initially be identical for the different grid topologies. But once collision occurs, the forms deviate significantly based on topology. Compare this video to the same seed, rule-set, and grid size, but using the hexagonally-shaped field of cells in the companion video "Wakka Wakka":
• Hex Automata: "Wakka ... ,
This video uses the "Universal Replicator" rule, but the discussion above applies in general.
In this video, the usual "Age-based" color-coding is used:
YELLOW cells are just born,
RED cells have been alive 2 or more consecutive time-steps,
BLUE cells are already deceased, but their "ghost" lingers for 1 time-step.
The same rule-set, but with a different seed, also appears in the video "Universal Replicator" :
• Hex Automata: "Univer... ,
2-Dimensional cellular automata, hexagonal array,
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General Procedure:
STEP 1). Make a 2-dimensional grid (array) of "cells" which can each have a value of 0 (off/dead) or 1 (on/alive). Conway's famous "Game of Life" cellular automaton uses a square grid, but here we use a hexagonal grid (chicken-wire or honeycomb). Initialize the grid by filling it with all zeros. This is the "main grid".
STEP 2). Add a starting "seed" pattern to the main grid by changing some of the cell values to "1" (on/alive). Sometimes specific compact seeds are used, alternatively sometimes they are a random unstructured spread of ones that II call "primordial soup".
STEP 3). The program then looks at every cell in the entire main grid, one-by-one. When examining each cell, the total number of live neighbor cells is counted among its 6 immediately adjacent neighbor cells (if using "totalistic" rules). The program then consults the rule-set to decide if the central cell will be alive (1, on) or dead (0, off) in the next time-step. In order to not disturb the cell pattern that is being updating, all of these new values are accumulated on a separate "temporary grid".
STEP 4). After every cell is updated on the temporary grid, the main grid is re-initialized to all zeros, and then the temporary grid is copied to the main grid
STEP 5). Repeat Steps 3 & 4 for hundreds or thousands of iterations. The result of each iteration serves as the input for the next iteration. The grid is finite, so the live cell pattern will eventually go repeat or go extinct, although this could take thousands of time-steps.
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Note: this "Hexagon-Multiverse" (HMCA) cellular automaton is similar to Conway's famous "Game of Life" in the sense that both are 2-dimensional, have binary cell states, and are synchronous and deterministic. But the Game of Life uses a square grid, while the HMCA uses a more natural (common in nature) and more symmetrical hexagonal grid. Additionally, the HMCA achieves interesting results using a variety of rule-sets, whereas the Game of Life is limited to a single rule-set.
Hexagonal Cell Array: size remains constant at 120 columns x 120 rows.
Periodic boundary conditions: horizontal & vertical dimensions wrap across opposite edges, giving a topology equivalent to the 2-dimensional surface of a 3-dimensional torus.
Neighborhood: semi-totalistic (details to be published at a future date),
Rule-set found by random search. Details to be published later.
Time: 256 steps (display rate 5 fps). The first & final frames are shown for 1 & 2 seconds, respectively.
Live cell population: starts at 121, fluctuates wildly, reaches a maximum of 7602 on time-step 123, and ends with 726 (6 copies of the original seed) on the final time-step 256.
Resolution: 2578 screen pixels per cell,
Program: "Hexagon-Multiverse 1.0" (unpublished), PHP language.
Platform: MacBook Pro (M1), Sonoma 14.1.1 OS, Safari 17.1 browser.
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