Monday, 19 February 2024

Polyominoes and Conway's Game of Life

I've been reading a free ebook (title shown above) that I downloaded from this site. To quote from a footnote in the book:

A polyomino is a pattern made up of orthogonally connected live cells, and a tetromino is a polyomino with 4 live cells. More generally, polyominoes with 2, 3, 4, ..., 8 live cells are called dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, and octominoes.

Once we exceed eight cells, the polyominoes are referred to as 9-ominoes, 10-ominoes, 11-ominoes etc. This is a formidable but comprehensive book of 494 pages that I'll try to make my way through gradually. It was published in 2022 with the dedication:


I've mentioned John Conway before in posts about:

More recently I've been mentioning him in posts about his Game of Life:

In this and probably subsequent posts I'll be pursuing the Game of Life from the perspective of polyominoes formed by the decimal digits when using a 5 high x 3 wide grid of pixels. See Figure 1:


Figure 1
In my post titled Diurnal Age Meets Conway's Game Of Life I looked at the sort of "ash" that was produced by a "soup" of pixels representing my diurnal age. The terms "soup" and "ash" are used in the book mentioned at the start of this post. However, I thought it would also be interesting to examine how the digits from 0 to 9 behave when not in close proximity to other digits. To this end, I'll begin with the digit 0.

In a 5 high x 3 wide grid of pixels, the digit 0 occupies 12 of those pixels. See Figure 2:


Figure 2: a 12-omino in the shape of the digit 0

The zero quickly becomes the famous pulsar. See Figure 3.


Figure 3: source

The digit 1 forms a pentomino and it quickly morphs into what is called a blinker. See Figure 4 and Figure 5.


Figure 4: pentomino in shape of the digit 1


Figure 5: the digit 1 becomes a blinker

The digit 2 is an 11-omino and it quickly transforms into two boats. A boat is an instance of a still-life. See Figures 6, 7, 8 and 9.


Figure 6


Figure 7: first step on the way to two boats


Figure 8: second step on the way to two boats


Figure 9: two boats (unchanging)

I'll look at the other digits in future posts. That's enough for now.

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