Gems: Long Live the Oscillators, and the Slowest Glider


The parent of this page is: Cool Universes in the Cosmos of Cellular Automata

We call Gems a universe among the 2-D totalistic cellular automata, defined by S4568/B3457, where most random patterns below a certain size (around 12 x 12) turn into oscillators of very long periods. The figure below shows examples of such beautiful gems. All of them started as little blobs of random cells. Click on the figure to see it in action.

Figure 1. Oscillators in the universe of Gems, and one surprise! (Click on the figure)

(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. Most oscillators that form naturally (from random blobs) have periods ranging from 2 to more than 100. Some periods seem to be more common than others, though, and some seem to be extremely rare. For example, we have not been able to produce any oscillator with an odd period , nor a still life (it would appear with blue color in our figure). Is this a coincidence, or do the mathematics of this universe preclude odd periods?

The legend of figure 1 promises a surprise, which is that one of the "gems" in figure 1 is not an oscillator, but a surprizingly slow glider. In fact, it's the slowest glider known (as of 2002), across all universes! We are talking about the largest blob close to the center. This fellow performs one step downwards after 5648 epochs! You'd have to exhaust your patience to observe it moving downwards through the "aisle" formed by the nearby oscillators.

What happens to random blobs that are larger than around 12 x 12? The answer is that they tend to expand to infinity, but extremely slowly, and always taking a diamond-like shape, or rather like a rectangle standing on one of its vertices. The latter are almost always cut off. Very weird.

Time now for some hands-on activity. Below, we present our program 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 anywhere in the black space, click and hold it clicked down, then drag it down and right for a short distance (e.g., around 10 pixels), 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. Click on this button. You'll see the selection rectangle being filled randomly with red cells. Finally, click on the "play" button (), and watch what happens to your little random blob.

If your blob is not too big (around 12 x 12 pixels or smaller), chances are that it will turn into an oscillator. You can measure its period by noticing one of its "faces" (that you could recognize later) and using the "step" button () to see after how many steps it comes back to the face that you recognize. You may want to construct several such small random blobs before you set them in action. It is not at all uncommon to find oscillators of periods measuring 100 epochs or more. Sometimes your little random blob will disappear. At other times, especially if it is larger than what we suggested above, it will very slowly expand to infinity. This extremely slow expansion is another rare characteristic of the universe of Gems.

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