Getting a better look at how the "Universal replicator" rule-set works by applying it to a single live cell. Notice that the central cell with zero live neighbors dies immediately, while the 6 immediately surrounding empty cells with only one live neighbor (the center cell) become alive. These are the first clues about the details of this rule-set.
The same run, but with totalistic color-coding instead, is shown in "Clearer Explication":
• Hex Automata: "Cleare... ,
Here the rule is implemented on the edge-wrapped hexagonally-shaped grid of hexagonally-spaced cells. Since the topology of the closed 2-d edge-wrapped surface is a 2-torus with a 180 degree twist, cells patterns with an added twist are often seen, regardless of the rule-set used.
2-Dimensional cellular automata, hexagonal array,
Color-coding of cells age/life-status:
All colored cells are alive except blue-colored cells.
yellow = just born (state = 1),
red = alive 2 or more time-steps (state = 1),
blue = fading "ghost" of cell that died (state = 0),
black = empty space (state = 0),
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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: radius begins at 32, then grows in increments , reaching radius 60 on time-step 52, and then remains constant.
Periodic boundary conditions: a hexagon-shaped grid has 3 pairs of opposite edges wrap to give a finite closed continuous surface equivalent to a 2-torus (the surface of a standard 3-d ring donut), but containing a 180 degree twist.
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 1, reaches a maximum of 6072 on time-step 63, and finishes with 6 (6 copies of the 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.
11 июл 2024
