Tetrominoes
The Tetrominoes
Up next to discuss are the tetrominoes, which are probably the most familar of all the polyominoes because they're used in the game Tetris.
Almost everyone knows what Tetris is - it's a game well-known for being addictive, and it makes use of the tetrominoes. It looks like this (although there are many variations in appearance):
and in the game the goal is to clear as many lines as you can by stacking tetrominoes in the right way. Tetris's appeal and addictiveness lies in the balance between simplicity and diversity of the tiles - using the lower-order polyominoes would make the game too easy, and the higher-order ones would make the game too difficult.
There are seven different blocks in Tetris and they are:
Note the pairs that are just reflections of each other - the blue one and the orange one are twins, as are the red one and the green one. For our purposes those are equivalent, since flipping the entire grid doesn't really make a different on how the game of life goes. Therefore for our purposes there are five tetrominoes:
We should give names for each of these tetrominoes. Sbiis Saibian, on his now inactive polyominoes site, gives the polyominoes in the picture above names: from left to right his names for the polyominoes are Tetril, Saw, Tad, Led, and Basil. In addition, he names the reflection of Saw "Zaw" and the reflection of Led "Jed". The name Tetril comes from tetra + tail, the name Basil is adapted from "box", and the rest of the names come from the letters of the alphabet that they look like (Saw, Tad, Led, Zaw, Jed).
So what are we waiting for? Let's examine the behaviors of each of them, from least to most interesting.
The Block Tetrominoes
The first tetromino we'll examine is Basil. This one's behavior is straightforward - you start with the grid:
That's the "block" still life we encountered with the tromino Treeb. Since it's a still life, it stays that way forever, and that's that.
Not much else to say about the Basil's behavior, other than that it's a rare example of a polyomino that is itself a still life in the game of life.
With the next few tetrominoes, we'll encounter our second still-life.
The Beehive Tetrominoes
The first of the "beehive tetrominoes" as I call them is Tetril, a "tower" tetromino which is four monominoes stacked onto each other. You'll see why I call them beehive tetrominoes shortly. Tetril is the fourth tower tetromino after the monomino, domino, and Tropel. Here's how the Tetril evolves in the game of life:
That form above is our second still life - it's the second most common one and it's commonly known as a beehive. Like the block, the beehive is commonly encountered in the game of life.
Two other tetrominoes become beehives in the game of life - Led and Saw. First let's look at Led:
In the first step Led becomes Zaw, which is just the flipped version of Saw:
In the second step we encounter a familiar form from looking at Tetril:
And in the third step this form turns into the beehive:
and that's that with Led's behavior. Jed's behavior is almost exactly like Led's, only flipped, so that the Led becomes Saw which then becomes the 3x2 grid and then the beehive.
Saw's behavior is closely related to Led's. It begins like so:
then we yet again encounter the 3x2 grid:
which becomes the beehive:
It's interesting that three of the five tetrominoes turn into beehives - of course Zaw's behavior is Saw's but flipped. But the behavior of our last tetromino, called Tad, is the most interesting.
The Oscillator Tetrominoes
Tad's behavior is the most interesting among the tetrominoes. It takes nine steps for Tad to stabilize to a repeating pattern, and the pattern itself is quite cool. Finding how Tad progresses in the game of life is tricky to do by hand, but there are many programs that you can use to run the game of life. I like to use this one, and with it it's easy to see the pattern. Here is how the Tad evolves in Conway's game of life:
As you can see, after nine steps the Tad becomes a group of four blinkers - those are the Tropels that flip back and forth between horizontal and vertical. Groups of four oscillators or still lives are notably common in the game of life - the most common group of four is this one, the group of four blinkers that is commonly known as a traffic light.
Conclusion
As with the trominoes, the tetrominoes have behavior that varies a lot more than the lower orders, and introduce some notable concepts in the game of life. Here's a review of the polyominoes we've covered:
Polyomino Order End result # of steps to end result
Monomino 1 dies 1
Domino 2 dies 1
Tropel 3 blinker 0
Treeb 3 block 1
Basil 4 block 0
Tetril 4 beehive 2
Led/Jed 4 beehive 3
Saw/Zaw 4 beehive 2
Tad 4 4 blinkers 9
So as you can see, the behavior has had some variance now. Up next we'll look at the pentominoes.