Pentominoes

The Pentominoes

After examining the five different tetrominoes, it's time to put things up a notch with the twelve pentominoes - 18 if you count reflections (like Jed/Led or Saw/Zaw) as distinct. Here are the twelve pentominoes:

There are several ways one can name and remember them. The most common is with letters of the alphabet they look like. There's some variation on how that is done, and here's my favorite way to memorize them:

This just uses the word "flips" and then the last few letters of the alphabet T U V W X Y Z - I find it quite amusing how some of the pentominoes can be arranged such that they resemble the last few letters of the alphabet.

Sbiis Saibian does not have any names for those pentominoes on his polyominos site - he abandoned it because, according to communication, he somehow lost all his progress in his page on pentominoes. He said that he remembers that he named the "V" pentomino "Veeb" but that's it.

But in any case we can start examining the pentominos' behavior in the game of life.

The Dying Pentominoes

Remember how the monomino and the domino both turned into dead cells? Well the same thing happens to five of the twelve pentominoes. Let's look at each of them and how they "die".

The P pentomino is the first example of a dying pentomino. In the game of life it evolves like so:

This is interesting because this is the first polyomino we've seen in a while that dies like that - the only other ones were the very basic monomino and domino, and none of the trominoes or tetrominoes ended with all dead cells.

The second dying pentomino is the S pentomino. Here's what happens to the S pentomino in the game of life:

Once again, by the game of life's mechanics, the S pentomino decays into a domino and dies.

The U Pentomino also dies:

Note that the U pentomino dies very much like the P pentomino does.

The Y pentomino dies like so:

The Y pentomino's death arises largely from the live cells having too many neighbors - John Conway himself commented on that type of death, saying that live cells dying from too many neighbors is meant to resemble overpopulation.

Finally, here's how the Z pentomino dies:

The Z pentomino, as you can see, simply becomes two monominoes and dies.

As you can see, the dying pentominoes die in several different ways: three of then decay into a domino, one into four monominoes, one into two monominoes. However, the other types of polyominoes are more interesting - up next are two pentominoes with both evolve into a still life known as the loaf.

The Loaf Pentominoes

The first of the loaf pentominoes is the V pentomino, the one Sbiis Saibian named "Veeb". In Conway's game of life it stabilizes in only 3 steps:

The still life on the far right is the third most common still life, known as the loaf. LittlePeng9 of Googology Wiki says that he, for some reason, dislikes seeing the loaf crop up.

Note that on the first step, the V pentomino itself turns into a W pentomino. Therefore the W pentomino's evolution in the game of life is just part of the V pentomino's evolution, and the V and W pentominoes both evolve into a loaf in the game of life.

So the V and W pentominoes both simply turn into a loaf. We still have five more pentominoes to go, four of which all evolve similarly in the game of life.

The Oscillator Pentominoes

The next pentomino for us to look at is the L pentomino, which is the first of the oscillator pentominoes, pentominoes that evolve into one or more oscillators. Here's how the L pentomino evolves in Conway's game of life:

The L pentomino's behavior is very similar to the Tad tetromino's - they both take exactly nine steps to evolve into the traffic light (the set of four blinkers), and starting from the second step they evolve in exactly the same way.

Also closely related to the L pentomino in behavior is the I pentomino, which is the fifth "tower polyomino" continuing from the monomino, domino, Tropel, and Tetril. The I pentomino evolves like so:

The I pentomino evolves very similarly to the L pentomino, but a little faster (takes only 6 steps instead of 9).

The T pentomino is yet another pentomino that becomes a set of four blinkers. Here's how it evolves:

This pentomino turns into our old friend the Tad tetromino after the first step, and it turns into a traffic light after exactly ten steps.

The last of the oscillator pentominoes is the X pentomino, which evolves like this:

Like the I pentomino, the X pentomino takes only six steps to evolve into a traffic light.

Of all the pentominoes we reviewed, not one of them exhibited unique behavior - five died, two became loaves, four became traffic lights. However, of the twelve pentominoes we have one left, and it is by FAR the most interesting.

The F Pentomino (usually called R pentomino)

The F pentomino is by a landslide the most interesting of the pentominoes in Conway's game of life. "F pentomino" is actually a nonstandard name for this pentomino, as in the Game of Life community it's almost always called the R pentomino. Therefore from here on out we'll instead call it the R pentomino This pentomino does not die, it does not turn into a single still life, and it does not just turn into a simple set of oscillators. It is the simplest example of a methuselah, a pattern in Conway's game of life that is small but takes a long time to stabilize.

So how many steps does it take for the R pentomino to stabilize? Let's look at the first ten steps:

This is interesting so far - ten steps is the most steps it took any of the other 11 pentominos to stabilize, and it looks like it'll be a long time till this pentomino stabilizes.

After 20 steps the R pentomino looks like this:

After 30 steps it looks like this:

40 steps:

50 steps:

Note that there's a block, and a newly formed traffic light that's about to get eaten up by another traffic light in formation. Ten more steps brings us stage 60 of the R pentomino:

It looks like the only parts that will stay stable for long are the block in the middle and the vertical blinker to the bottom right of it. In twenty more steps (step 80) things get really interesting:

There are three fairly isolated blocks, and the blinker seems quite stable. But notice the structure highlighted in red. That structure, released in step 69 is known as a glider, and it's important because unlike many other structures this one endlessly moves in the game of life, like so:

Gliders can move diagonally in four different directions, depending on their orientation. Structures that move in a regular pattern are known as spaceships, and the glider is by far the most common spaceship. It was discovered by Richard K. Guy, a mathematician who has often collaborated with Conway, in 1970 when tracking how the R pentomino evolves, and this particular glider is the first one ever observed. Gliders are interestingly a common occcurrence in the game of life as we will see.

Here is what the R pentomino turns into after 100 steps:

The glider highlighted in red is the same glider we noted in step 80. As you can see the glider has moved northeast, and it looks like it's not going to crash into anything. Also, note that there are several blocks here along with the blinker that hasn't changed much at all, and yet another still life known as the pond in the top right.

Here's stage 125 of the R pentomino:

This certainly looks more chaotic than stage 100. There are two blocks laying around that haven't been destroyed, and a blinker lurking around to the right. Also, along with the red glider two aditional gliders (orange and yellow-orange) have escaped.

Stage 150:

Here we have the same three gliders as previously - it looks like they're not going to get consumed by anything and just escape. The only other stable things are three blocks and a blinker on the top-left.

Stage 175:

A fourth glider (colored yellow) has now escaped. The left side of the grid looks quite stable, with a still life known as a ship to the left. The ship is the seventh most common of the still lives. Along with that is a block, two dominoes which will die in the next stage, and a Treeb which will turn into a block in the next step. The right side, on the other hand, is absolute chaos: it looks like the new yellow glider may not survive. Let's find out what happens to it by jumping another 25 steps to stage 200:

It looks like the yellow glider barely escaped the mess! Also, a fifth glider (olive green) has just escaped. In addition, there are some new still lives: three beehives, along with a 5-block still life (colored purple) known as a boat. The boat is the fourth most common still life behind the block, beehive, and loaf, and it's the only still life that is made of exactly five blocks. The left area and the center area both look quite stable, but the right is still a little messy. But hey, the mess isn't so bad anymore, so it wouldn't be long till the R pentomino stabilizes, right? Well, let's skip 50 steps to see what happens in step 250:

We have the same five gliders as previously, although two of them have escaped past the field of the simulator I'm using - they are indicated with two arrows I drew.

The area on the right in fact got more messy than it previously was, as the rightmost beehive from stage 200 was eaten and replaced with a pair of blinkers. The still lives to the left, however, haven't changed from 50 steps ago. Another fifty steps from this brings us to stage 300:

The chaos has spread even further than previously. In addition our third glider (the yellow-orange one) has escaped the field. Not much else to say, other than a new beehive and ship to the right. Another 100 steps brings us stage 400:

Two of our gliders (the yellow and olive green ones) have escaped, the one to the southeast and the other to the northwest. However there are two new gliders now, one colored dark green and the other colored teal. It looks like the dark green one will survive but the teal one won't. Let's see what happens in yet another 100 steps with step 500:

Neither of the two new gliders survived - they were both consumed by the chaos. The chaos seems a bit more confined to specific areas than previous as the area to the southeast stabilized into some blocks and blinkers, but with all the weird behavior, who knows what will happen. Let's jump 100 steps to step 600:

The chaos has gotten worse yet again! Now for step 700:

A little better, but not by very much. Now for step 800:

The most notable thing here is our eighth glilder (indigo, top), which looks like it'll be the sixth one to survive. That's about it. Now for step 900:

The eighth glider has escaped, and the chaos is dying down. Now for step 1000:

Yes that's right, a thousand steps and the R pentomino still hasn't stabilized! Though it has come close now. Let's slow down a bit and skip 50 steps to step 1050:

The only area where things are still going on is the south-center. Let's skip 25 steps to step 1075:

Almost stable now. Let's go to step 1100, one of the last steps before it stabilizes:

Step 1101:

Step 1102:

And finally, step 1103, the step where the R pentomino stabilizes:

So NOW we've reached the end of this polyomino's evolution. All this ends with:

Still Lives:

6 blocks

4 beehives

1 loaf

1 boat

1 ship

Oscillators:

4 blinkers

Spaceships:

6 gliders

This amounts to stabilizing at a count of 116 live cells. Now think about how much more complicated this is than any other polyomino we've encountered: the most steps any polyomino of order 5 or lower (other than the R pentomino) takes to stabilize is only ten in contrast to 1103 steps for this pentomino! So let's review all the polyominos we've covered.

Review

Like previously let's review all the pentominoes we've covered so far:

Polyomino Order End result # of steps to end result # of cells at end result

Monomino 1 dies 1 0

Domino 2 dies 1 0

Tropel 3 1 blinker 0 3

Treeb 3 1 block 1 4

Basil 4 1 block 0 4

Tetril 4 1 beehive 2 6

Led/Jed 4 1 beehive 3 6

Saw/Zaw 4 1 beehive 2 6

Tad 4 4 blinkers 9 12

P pentomino 5 dies 4 0

S pentomino 5 dies 4 0

U pentomino 5 dies 4 0

Y pentomino 5 dies 3 0

Z pentomino 5 dies 3 0

V pentomino 5 1 loaf 3 7

W pentomino 5 1 loaf 2 7

L pentomino 5 4 blinkers 9 12

I pentomino 5 4 blinkers 6 12

T pentomino 5 4 blinkers 10 12

X pentomino 5 4 blinkers 6 12

F (R) pentomino 5 8Blo, 6G, 4Bli, 4Bh, 1L, 1Bo, 1Sh* 1103 116

*Blo = block, Bh = beehive, L = loaf, Bo = boat, Sh = ship, Bli = blinker

As you can see, all but one of the polyominoes we've looked at so far has a simple pattern in how it evolves in Conway's game of life - the one that doesn't is famous because of its unique evolution. It introduces a whole new world of concepts in the game of life, and really brings out how fascinating Conway's game of life really is. Up next we'll look at the 35 hexominoes, the sixth order of polyominoes.

The Hexominoes (coming soon)