Trominoes

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The Trominoes

We now reach the third level of polyominoes - the trominoes. Unlike with the monominoes and dominoes, the trominoes come in two different forms - if you count rotations as distinct (which are the same for our purposes) there are six different forms of trominoes. In comparison to the previous mundane polyominoes the possibilities for trominoes seem quite endless!

The first of the trominoes is a tromino Sbiis Saibian gave the name Tropel (source). Here's what "Tropel" looks like:

Tropel is simply three squares stacked on top of each other, and it's a "tower" polyomino. As we will see later it's a polyomino that, in Conway's game of life, will switch back and forth between two states indefinitely.

The other tromino is named by Sbiis Saibian "Treeb". Treeb looks like this:

Treeb is a "bent" polyomino, and it has diagonal symmetry, one of the less common polyomino symmetries. In Conway's game of life it ends at a stable set of four tiles, probably the simplest such set of all - we'll discuss that in a bit.

Trominoes' Behavior

How would this duo of trominoes work in Conway's game of life? Let's take a look.

First let's start a board, and fill in three squares like a Tropel like so:

Here are the rules to recap:

1. A live cell with 0 or 1 live neighbors becomes a dead cell.

2. A live cell with 2 or 3 live neighbors stays alive.

3. A live cell with 4 or more live neighbors becomes a dead cell.

4. A dead cell with exactly three live neighbors becomes alive.

5. Other dead cells remain dead.

Of the live cells, the top and bottom ones become dead, since they have only one alive neighbor. The middle one, on the other hand, remains alive since it neighbors two alive cells.

What about the dead cells? Remember that a dead cell becomes alive if and only if it neighbors (horizontally, vertically, or diagonally) exactly three live cells. In this case, there are in fact two dead cells that fit this requirement - those are the ones to the left and right of the middle cell. Therefore the position changes a little bit and we end up with this position:

Now that's just the starting tromino, but rotated ninety degrees! That's not strange at all, and once again we can follow the rules to turn the left and right cells into dead cells, and the ones above and below the middle one alive, giving us:

Now we're back to wehere we started! Hell, we can continue this process indefinitely, flipping back and forth between the horizontal and vertical Tropel.

What we just encountered was our first oscillator - an oscillator, in Conway's game of life, is just a pattern that flips back and forth between two or more states indefinitely, so it would indefinitely go ABABABAB... or ABCABCABC..., etc. This specific oscillator is called the blinker and it's by far the most common oscillator - the next most common oscillator is about 100 times less common than the blinker. We'll encounter the blinker again and again when examining the higher polyominoes.

Now to examine how the Treeb behaves in the game of life. Of course we start with a Treeb in the board:

Then, we look at each of the rules. Each of the three live cells neighbors exactly two live cells, so all three live cells remains alive.

Of the dead cells, exactly one of them neighbors exactly three live cells, so that becomes alive. Nothing happens to the rest of the cells.

So now the grid looks like this:

In the next step, something cool happens: all the live cells remain alive since they each neighbor exactly three live ccells. And none of the dead cells become live, so in the next step the grid looks the same:

We just encountered our first still life. In Conway's game of life, a still life is a group of cells that, by the rules, stays the same in each step. This still life we enountered from Treeb is known as a block, and it's the most common still life, though not by a landslide like the blinker among the oscillators - there are other still lives that are quite close to as common as the block is.

Conclusion

The trominoes, unlike the previous two orders of the polyominoes, exhibit some variation in their behavior in the game, and they also introduce some important concepts within the game of life. Their behavior sets the base for examining the varied behavior of the higher polyominoes, starting with the tetrominoes.

The Tetrominoes