since the only activity
seems to be some repetitive and limited action on its border. The
figure below shows an example of such growth in Stains. Click on
it to see it in action.A universe with what seems to be a unique property in the
world of 2-D totalistic cellular automata is S235678/B3678, which
we call Stains. The property is that nearly every random
pattern grows only up to some point, acquiring usually an
irregular shape, and eventually turning into a huge oscillator since the only activity
seems to be some repetitive and limited action on its border. The
figure below shows an example of such growth in Stains. Click on
it to see it in action.
Figure 1. Random initial input in the universe of Stains (Click on it)
(If you are not already familiar with how to interact with our figures, please note that you can start/stop the figure by clicking once on it, and reset it to its original configuration by dragging with your mouse anywhere on its surface.)
The colors of the oscillators are indicative of their period. For a full explanation of our color-coding, read this page of ours. Generally, the activity at the border of stabilized patterns in this universe displays local periods of one ("still lives", blue color), two (cyan), three (green), and four (yellow). When such local periods coexist we get an oscillator, the period of which is the least common multiple of the local periods. Hence, if local periods of two and three coexist, we get an overall period of 2 x 3 = 6, which is shown with the light-orange color in our figures. Another possibility is 3 x 4 = 12 (dark orange, the pattern formed at the bottom-right corner in figure 1). We are not aware of the existence of higher "prime" periods (i.e., not formed by multiplication), such as 5, or even 7, but we expect that such periods exist.
Surprisingly, not everything is static in this universe. The glider, the well-known 3x3 pattern that moves diagonally in the Game of Life, glides here, too! Figure 2, below, shows some configurations that lead to the construction of gliders (second column of patterns), along with constructions of some of the most common oscillators of various periods.
Figure 2. Constructions for common oscillators and the 3x3 GOL-glider
Did you notice the four gliders sailing across the four diagonal directions in figure 2? Since this is the well-known 3x3 Game-of-Life glider, we'll use a capital G when referring to it. One of those Gliders, the south-east-bound one, passes close to some of the oscillators. When that happens, the oscillators glow red for a short time, having "sensed" the Glider's approach and becoming potentially "active patterns", in our program's terminology. When the "danger" is away, the oscillators take back their original colors. (All this is dictated by our implementation, and has nothing to do with cellular automata per se, but the attribution of higher-level descriptions such as sensing, danger, etc., seems nevertheless interesting to us.)
Are there any other "ships" that, like
the Gliders, sail through the Stains space? One wouldn't expect
much actually, since this seems to be a very static universe everything turns into an
amorphous blotch after a while. There is no shortage of
surprises, however, in the world of cellular automata. Currently,
five ships are known in this universe, all shown in figure 3
below, and we urge you to check this page
by Prof. Eppstein, to see if any new ones have been added since
we wrote this page. (The ones we show below are listed as: G114,
G3565, G5683, G12607, and G16481 in the mentioned reference.)
Figure 3. And yet, ships are known to exist in the world of Stains!
The top-most of the above four horizontally-moving ships glides with a speed of c/2. This means that, if by c we denote the maximum possible speed in this kind of 2-D cellular automata (alluding to the speed of light, c, in physics), then the first horizontal ship moves at one-half of the maximum speed (the latter being one cell per unit of time, in a given direction). The second ship has a speed of c/3, the third (the longish one) c/4, equal to the speed of the Glider, and finally the fourth one comes panting and gasping, at a speed of c/6.
Now it's time for some hands-on activity. Below, we present our program not as a "figure" like the ones you already encountered in this page, but as a full-featured applet with its normal interface. You won't click and drag on this applet to interact with it; instead, you can click the buttons on its toolbar (top), and look at its status-bar (bottom) for some information. Although we hope the interface is self-explanatory, this page of ours describes it in more detail. Now, what to do with it? Please read the information that follows.
Here is our suggestion for what to do with the
program, above. Move your mouse close to the upper-left corner in
the black space, click and hold it clicked down, then drag it
somewhere close to the bottom-right corner of the space, and
finally release the button of the mouse. You'll see a white
rectangle being formed while you drag the mouse. This is your
"selection rectangle". You can do a number of things
with this selection rectangle, all of which are possible if you
click once on it. Go ahead, click once anywhere within the
selection rectangle. You'll see a menu popping up, with one of
its buttons in the middle saying: "Fill with random
cells" and a percentage (50%) right next to it. Change the
number 50 (the suggested percentage) into 7. Then
click on the button "Fill with random cells". You'll
see the selection rectangle being filled sparsely with red cells.
Finally, click on the "play" button (
), sit back, and watch.
You'll see most of the lonely cells of the space disappearing, but a few of them that happen to be close together expand, forming stains. Each time you repeat this process a new formation of stains appears. Your task is to try form stains of higher periods (redder). The authors have tried several times, but haven't gotten any stain of period larger than 3 x 4 = 12 (dark orange), and even those are very rare. Will it turn out that you'll have better luck?