Exploring cellular automata 2 — my lovely maggot children

In some other post I linked to the Wolfram “atlas” of 1-dimensional elementary cellular automata, the list of 256 rules that can apply to a row of “on” or “off” pixels to determine a successive row, in which a pixel’s successive value is only dependent on its own and those of its two neighbors. There are only 256 of these because of this limitation.

We can extend this to a 2-dimensional space. This seems to be easier to follow what’s going on — maybe just by the nature of looking at a computer screen. It’s the space that the Game of Life lives in. A grid of pixels can be on or off, and a successor grid follows, with each pixel’s state determined by its initial state and those of its neighbors. In order to visualize this, you could lay each successor on top of its parent grid to make a 3D object (analogous to forming a 2D image for a 1D array of pixels). But this would be really hard to look through. Instead, just play each image after its successor as an animation.

A puffer in Game of Life. Each frame of the animation is a new epoch. Needless to say, this isn’t an obviously possible construction given the GoL rules. Wikipedia
A GoL breeder. This configuration only creates guns, which create glider spaceships. In the last frame, the breeder is colored red, the guns green, and the gliders blue. Bickibeepia

A ruleset can be described with a series of images, like the 1-D automaton, but again, you’d have to either stack the start example on top of the resulting center pixel, or place them side by side. In addition, there are going to be many more images necessary for a rule, as there are more configurations of on-and-off neighboring cells in 2 dimensions.

This also introduces the uncertainty in what a neighborhood is in 2D. We could consider the 4 cells in the cardinal directions as the only neighbors (the von Neumann neighborhood), or we could tack on the corners, too (the Moore neighborhood).

The von Neumann neighborhood. Pink cells show an extended version.
The Moore neighborhood. Wikibleedia

For the von Neumann neighborhood, each rule alone would require 2^5 = 32 images. With a choice of on or off for the resulting cell for each image, there are 1024 elementary cellular automata. Not a bewildering number, but still a lot.


Two examples of rule icons for an elementary cellular automaton using the von Neumann neighborhood. For a single ruleset, one would need 32 of these bad boys. These read a bit differently than the 1-D rule icons — the square on the right replaces the center square in the image on the left.

The Moore neighborhood would have 9 squares, replacing the center square, and so one would need 2^9 = 512 icons for each rule. There are 512^2 = 262144 rules of this type. Yikes! There are the same number of squares in the “extended” von Neumann neighborhood, so these numbers apply there as well.

A single icon of 512 from one of the elementary automata using the Moore neighborhood regime.

Both of these neighborhoods are probably characterized completely — tracking a few of the properties of 263000 systems doesn’t seem to be out of the realm of possibility, especially when considering that this number can be reduced by a factor of due to horizontal-mirroring, vertical-mirroring, centerpoint-mirroring, and color-inverting of each of these. So, the 512 in the 5-cell neighborhood reduces to only 64 with unique behaviors, and the 9-cell neighborhood reduces to 32,768.

What seemed exciting was to look at larger systems. A Super von Neumann neighborhood of 21 squares will have 4,398,046,511,104 rulesets of 2^{21} = 2,097,152 icons each, and the Super Moore neighborhood of 25 squares has 1,125,899,906,842,624 rulesets of 2^25 = 33,554,432 icons each. Now we’re having fun.

The Super von Neumann neighborhood of 21 squares.
The Super Moore neighborhood.

It felt like there was some chance of picking something weird out of here.

A ruleset based on an oscillator

One contruction I didn’t mention in the last post was an oscillator — a repeating, unmoving pattern. Some oscillators can be seen in the wakes of the dirty puffers.

Some period-2 oscillators in GoL. Bumbobedia.
A larger oscillator, with period 8. Bunklebeepia.

Oscillators are emergent in GoL; there isn’t anything about the rule that explicitly reflects an intent of repetition. I was interested in making a rule that was based on an oscillator, and to see if something like spaceships would emerge.


The simplest oscillator I could think of that seemed interesting was a clockwise spinning line. 3 activated cells, with a diagonal between the first two, would appear to spin around in a circle. If there were three cells in this configuration, the next epoch would deactivate the cell in the back, and activate a new one in the next step around a circle. This resulted in an 8-period oscillator. I originally called the idea Spinworld, but it soon seemed appropriate to call it Maggotworld.

Four base oscillators (maggots) in Maggotworld with different phases. White pixels are considered activated. The base oscillator has a period of 8 epochs.

Instead of matching the Super Moore neighborhood around every pixel with one of the 33.5 million in the ruleset, the maggotworld propagator just checks if there are three activated pixels in the right configuration nearby to turn on. There are eight ways this could happen — corresponding to the eight squares that are activated throughout the cycle. This doesn’t mean there are only eight icons of the 33.5 million that lead to an activated central pixel. The pixels toggle; a dead pixel toggles on if there are one, or three maggots approaching it. It toggles off if there are two or four approaching it. The same is likewise true for the tail ends of a maggot — two maggots with the same tail will leave the tail point on, as it is toggled twice. So there are many ways pixels could be left on or off depending on the situation. This seems ripe for emergent behavior!

I ended up trying a bunch of different setups to see how they behaved way back in April, and wrote most of the above then, but haven’t gotten around to this showcase until now.

Random starts

I began by just slapping random starting configurations down and seeing what happened.

The seed…
…and the result. My optimistic dream of seeing spaceships shoot out into the black shattered.
A high density of activated squares…
…is quickly lost after the first epoch, leaving just a few maggots wallowing in their filth. Some of these are neat.

After some initial discouragement, some neat patterns did emerge. In the last animation you can spot a few figure 8s, figure triple-Os(?), and maggots passing straight through each other. Not a spaceship, but at least something a bit more complicated.


Not so random starts

If I couldn’t get a spaceship out of this, I could at least look for something cool. Time to get systematic. I placed maggots at varying relative phases and positions to start and let them go.

Two maggots 180 degrees out of phase at varying relative starting positions.
And the results.
All the same with different starting phases.

This was a bit discouraging — none of the ones that touched made any lasting patterns beyond a single oscillator. Anything interesting would have to be made of more than 2 of these, or have some stationary gunk waiting around to push them around a bit.

In one of the random setups, I saw this little friend, which I’ll call a slingshot oscillator.

Slingshot oscillator.
Slingshot has a period of 14.

It’s got a period of 14. Having a different period is… useful? How is any of this useful? I dunno whatever.

Another neat result would be to get a pattern that turns on cells outside of the normal oscillator radius. Here’s one. It looks like a really low resolution hand-style cursor. Let’s call it the flinger.

Flinger start!
Flinger go! It turns on a cell in the upper right well outside of the original oscillator’s domain.

The last thing of this type is a little u. I like this because, while it seems like these rules easily reduce the number of lit cells, this generates an extra sixth cell by the end of its process.

U-thing go!

Behavior in grids

If I couldn’t get a spaceship flying away in empty space, maybe I could get something neat with stationary patterns placed ahead of time. I started by plopping oscillators in a lattice of lit points. It’s real easy to get some ongoing domino action this way.

Diagonal motion.
A couple more.
Vertical motion. There’s a lot of trash left behind, but the key to see here is that only one or two lit squares next to a lit bar drives the motion. The lit side determines which way it travels.

Trams and fuses

The last one looked neat. I tried to make more “trams” following a track. Here are a bunch. There are also configurations that rip the track apart. Let’s call em fuses.


Some of the styles are repeated with a hole in the track to show how they deal with that. Holes are also placed at different locations along the track for the same style of tram to show off how they travel: trams hop two spaces at a time, not one. An “oddly” placed hole will force it to reflect or scatter differently than an “evenly” placed one.

I was taken with the self-repairing track on the bottom.

Self-repairing tramtrack.

This seemed like a way to create a caterpillar-like spaceship of indefinite length that scootches forward a couple squares each cycle. I wasn’t able to figure how to get it to repeat, thought.

Another fuse had an end that, after some acrobatics, sent another tram that flipped the orientation of the fuse.

A single lit square between the tracks flips the sides of the track: two lit on top and one on the bottom become one on the top and two on the bottom.


One last thing and I’ll let you go, I swear. A row of lit squares can be used as a timer of indefinite length. This could go up above with the other oscillators, but the fact that this can be stretched to as long as you need it puts it in its own class.

A timer. This repeats. It takes some time to do that so I called it a timer.

These can also take a turn. These turns could be used to stop the motion, or to initiate the timer in the first place:

Two timers. The top timer is initiated after the maggot turns the corner. The bottom stops after the path has been traversed.

Maybe these are more appropriately called fuses? But I already used that word.

Showing these little tricks and behaviors is interesting enough alone for me, but it would be really swell if we could figure out how to produce oscillators / timers of truly any period. I’m also still wondering if there is an elusory spaceship, one without a grid or track to support it, hiding in these rules. The holy grail after that is to get these things to do binary logic, but let’s save that for another day.

I Lied

I said I had only one last thing but I totally lied to you. You must be feeling pretty silly right now! Just one more bit. I promise. ūü§ě

Before finishing up with the maggotworld animations, I had tried a lot of different rulesets to see how they’d act. One of them was simple: A cell would turn on if it had an odd number of neighbors (in the Moore neighborhood), and off otherwise.

Fredkin’s Replicator.

What a pretty pattern! I was pleased to learn later that I had recreated something called Fredkin’s Replicator. You can see some explanation in Numberphile’s Terrific Toothpick Patterns (starting at 14:20). A lecture also by Neil Sloane gives more insight into analyzing this and other “odd-ruled” cellular automata. Do yourself a favor and check out at least the Numberphile vid.

The Fredkin’s replicator having reached the 63rd generation. What a beaut! You can see the same repeating structures, and the same shapes repeated at different scales.

Visualizing text as paths

I was reminded recently that I actually have a blog. I have a Cellular Automata Part II post that’s been sitting around since April, but here’s something quick before I get around to polishing that up.

During a prescribed Art Night, I decided to come up with a visualization of long texts. I grabbed a few examples of literature from¬†Project Gutenberg¬†as plain text files, and converted to binary using each letter’s¬†ASCII character code.

a\qquad\to\qquad 97\qquad\to\qquad \begin{array}{ccccccc} 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ \end{array} 

An entire word will then create a string of ones and zeroes 7 times longer. The visualization comes in by turning string into something like the output of a Lindenmayer system. A line is drawn, and at each step it reads the next character. A one instructs the drawing to take a right turn, and a zero instructs it to take a left turn.

\text{baloney} \\ \strut\quad\downarrow\\ 98, 97, 108, 111, 110, 101, 121 \\ \strut\quad\downarrow \\ 110001011000011101100110111111011101 \\ \strut\quad\downarrow \\ RRLLLRLRRLLLLRRRLRRLLRRLRRRRRRLRRRLR

Starting from the bottom of an image, drawing upward, and taking these turns one at a time gives a blocky figure.

“baloney”. There are 49 turns, from each of baloney’s 7 characters’ 7 bits.

This was pretty quick to implement. It initially took about 10 or 15 minutes to finish for single-MB size books using repeated AppendTos in Mathematica to construct a list of successive points along the path. A rewrite using Reap/Sow made these finish in less than a minute. I have no learning about optimization, except that the latter is supposed to be fast in M-ca. I suspect that Append rewrites the entire list each time, while Reap/Sow does not. There are likely 800 other better languages in which to write this kind of thing, but hey! Heyo! Woo!

Gutenberg adds in header and footer to each text file, saying some standard stuff about where it was downloaded and when it was written, etc. For each of these, I removed everything except the nitty gritty, proper text. This would only serve to tack on a bit at the beginnings and ends, ultimately not changing much to the shapes. They would have been rotated, though.

Here are the results for 4 books. Here’s a zoomable pdf of the Frankenstein one. In each of these, a green point marks the beginning, and a red point the end. Frankenstein’s green point, for example, is in the lower right, and its red is on the ear of the evil head in the upper left. Boo!

Frankenstein; Or, The Modern Prometheus. Mary Wollstonecraft (Godwin) Shelley.



The King James Bible.
Moby Dick; Or, The Whale. Herman Melville.
Ulysses. James Joyce.

It was really fun to get these results. I acknowledge that there is probably little meaning in these images from a stylometric point of view, but it is interesting to observe the differences. I’m sure these are very sensitive to format, the words themselves, and the ASCII codes for English letters. The Bible is loaded with many paragraph breaks and colons for each verse, in contrast to the long paragraphs of Ulysses. Ensuring that each character begins with a right turn (1) guarantees that each image is going to have¬†some kind of curliness. A single typo wouldn’t change the shape of the subsequent curve, but it would lead to a rotation about that point.

So, is it just happenstance that Ulysses is so much more unfolded than Moby Dick? Is there something fundamentally different about the second half of Moby Dick that makes it hover around the end point?

An issue with the visualizations themselves is that they do not show any overlap as a line covers regions it has already followed. A way around this would be to assign an increasing height to each point, and show these paths as text tunnels in space. These could be seen as a 3D model or as projected images from different directions. Maybe I’ll try this out.

The full image files showing these walks are about 10x the size of the original text files. A text file storing the list of ordered pairs of the path is about 100x the original text file size. So, a forseeable application of this technique could be to make text file sizes much larger, as well as difficult to read and process.

As a final thought, it was easy to call these random walks. Clearly these aren’t random in that they carry the meaning of the texts, but perhaps English letters appear random after being converted in this way. A way to test this is by looking at the distance from the starting point — A true random walk of N steps¬†ought to be a distance of roughly \sqrt{N} from the starting point.





None of these seem to be plots of distance = sqrt steps.

Parameterizing a circle with the intersection point of two perpendicular lines.

I’ve been really taken with Desmos, an online calculator and easy to use graphing tool. My students have been using it for some time, and I’m especially happy with the “slider” tool that it offers. Whenever you put a letter into a function while graphing, it suggests a value to assign it, and lets you tune that value with the slider. This tool is similar to Mathematica’s Manipulate or Animate functions, which I’ve had success using in previous classes to show how a function depends on its parameters.

My year-long teacher’s Mathematica license recently expired, making it a bit tougher to install on a new device. While I do have access to an unsupported copy, Desmos has more than replaced M-ca for any of my presentation needs.

In a recent class, we were playing around with linear systems and intersecting lines. To show that a negative reciprocal slope leads to a perpendicular line, I assigned a slider to the value m, and made two linear equations with slopes of m and -1/m:

y=mx \qquad\text{and}\qquad y=-\cfrac{1}{m}\ x

The slider has the nice effect of letting you rotate the lines to see that they’re always perpendicular.

desmos-graph (1).png
Play with it yourself, why don’t ya.¬†You can animate or adjust the slope with the¬†m slider on the left.

The kids were delighted by the pinwheel spinning of the lines as the slope was adjusted. To show that we weren’t limited to lines that passed through the origin, I tacked on a¬†y-intercept to both of the equations, and asked the students, what do you think happens when I adjust the slope now?

My point was to show that the lines remain perpendicular. I would have been pleased to hear that the students could also predict that the point of intersection of the two lines would now move around, instead of be fixed at the origin.

One student went further, however: he was able to predict that the point of intersection of the two lines will always be fixed to a circle.

desmos-graph (3)
You can adjust the slope once again, as well as the points the line are fixed to pass through using the sliders. Only adjusting the slope m keeps the point of intersection on a circle. You can also adjust the points the lines are forced through with the sliders below.


The student had seen a connection to his geometry class from the previous year. An inscribed angle is half the measure of the intercepted arc. An angle inscribing half the circle must then be a right angle. What this student had realized was the converse of this statement: that a right angle, formed by two perpendicular lines each forced to pass through particular points, must lie on a circle, and those two points are the endpoints of a diameter of that circle. I thought this was awfully insightful!

I figured it would be neat to try to show that this must be true on my own. Solving the system

\left\{ \begin{array}{c} y =m (x-a)+c\\\\ y=-\frac{1}{m}\ (x-b)+d  \end{array} \right.

gives the point

\left(\frac{a\thinspace m^2+\left(d-c\right)m+b}{m^2+1},\frac{d\thinspace m^2+\left(b-a\right)m+c}{m^2+1}\right)

I thought this was really neat. We haven’t shown that this point lies on a circle yet, but assuming it does, it shows a way to parameterize a circle with¬†m as the ratio of quadratics. Maybe this is something a mathematician would immediately recognize, but it’s new to me!

To show this does lie on a circle, I need to find an appropriate transformation that turns the above into the more familiar

\left( R\cos\theta + x_1 , R\sin\theta + y_1\right)

for a circle of radius R and center (x_1,y_1) . The obvious choice is to connect the slope of one of the lines to the angle on the circle:

m \rightarrow \tan\theta

The parameterization becomes

\left(\frac{a\thinspace \tan^2\theta+\left(d-c\right)\tan\theta+b}{\tan^2\theta+1},\frac{d\thinspace \tan^2\theta+\left(b-a\right)\tan\theta+c}{\tan^2\theta+1}\right)

This is where all your trig identities pay off. Those denominators become squared secants, letting you get rid of the fractions altogether.

\left(\frac{a\thinspace \tan^2\theta+\left(d-c\right)\tan\theta+b}{\sec^2\theta},\frac{d\thinspace \tan^2\theta+\left(b-a\right)\tan\theta+c}{\sec^2\theta}\right)

\bigg(\enspace a\thinspace \sin^2\theta+\left(d-c\right)\sin\theta\cos\theta+b\cos^2\theta\quad,\quad d\thinspace \sin^2\theta+\left(b-a\right)\sin\theta\cos^2\theta+c\cos^2\theta\enspace\bigg)

I’m having a bit of difficulty with formatting here. I’ll have to just write it like so:

\begin{array}{c} x=a\thinspace \sin^2\theta+\left(d-c\right)\sin\theta\cos\theta+b\cos^2\theta \\\\y=d\thinspace \sin^2\theta+\left(b-a\right)\sin\theta\cos\theta+c\cos^2\theta\end{array}

The middle bits of these should pop out: a sine times a cosine is a part of one of the double angle formulas:

2\sin\theta\cos\theta = \sin2\theta

While we’re tossing in sine of a double angle, we might as well introduce the cosine of the double angle as well. This shows up from the squares:

\sin^2\theta = \cfrac{1-\cos2\theta}{2} \qquad \text{and} \qquad \cos^2\theta=\cfrac{1+\cos2\theta}{2}

Our parameterization becomes

\begin{array}{c} x=a\thinspace\left(\cfrac{1-\cos2\theta}{2}\right) +\left(\cfrac{d-c}{2}\right)\sin 2\theta+b\left(\cfrac{1+\cos2\theta}{2}\right) \\\\y=d\thinspace \left(\cfrac{1-\cos2\theta}{2}\right)+\left(\cfrac{b-a}{2}\right)\sin2\theta+c\left(\cfrac{1+\cos2\theta}{2}\right)\end{array}

What’s neat about this is that the center of the circle now falls out as a constant term at the end, and we’ve maintained some kind of symmetry with the sines and cosines.

\begin{array}{c} x= \left(\cfrac{b-a}{2}\right)\cos2\theta +  \left(\cfrac{d-c}{2}\right)\sin2\theta +  \left(\cfrac{a+b}{2}\right)\\\\ y= - \left(\cfrac{d-c}{2}\right)\cos2\theta +  \left(\cfrac{b-a}{2}\right)\sin2\theta + \left(\cfrac{c+d}{2}\right)\end{array}

Here’s where my trig knowledge stopped. The sine and cosines can be combined, though: a linear combination of sine and cosine should leave a single sine curve, but with a phase angle tossed in.

w \cos\theta + u\sin\theta = \sqrt{w^2+u^2}\enspace \sin\left(\theta+\arctan\frac{w}{u}\right)

This is great! We’ve got a way to combine the¬†a, b, c, ds to get something looking like a radius.

\begin{array}{c} x = R \sin\left(2\theta + \arctan\left(\cfrac{a-b}{d-c}\right)\right) +\left(\cfrac{a+b}{2}\right) \\\\ y =R \sin\left(2\theta + \arctan\left(\cfrac{c-d}{a-b}\right)\right) +\left(\cfrac{c+d}{2}\right)\end{array}


R = \sqrt{\left(\cfrac{b-a}{2}\right)^2+\left(\cfrac{d-c}{2}\right)^2}

At this stage, we need to do is turn that first sine into a cosine (using \sin\theta = \cos\left(\theta-\frac{\pi}{2}\right).

\begin{array}{c} x = R \cos\left(2\theta + \arctan\left(\cfrac{a-b}{d-c}\right)-\cfrac{\pi}{2}\right) +\left(\cfrac{a+b}{2}\right) \\\\ y =R \sin\left(2\theta + \arctan\left(\cfrac{c-d}{a-b}\right)\right) +\left(\cfrac{c+d}{2}\right)\end{array}

We’re left with one remaining question: are the phase angles the same?

\arctan\left(\cfrac{a-b}{d-c}\right)-\cfrac{\pi}{2} \quad \stackrel{?}{=}\quad \arctan\left(\cfrac{c-d}{a-b}\right)

A couple more identities that I don’t have memorized clears this up:

\arctan(-x) = -\arctan(x) \qquad\text{and}\qquad \arctan\left(\cfrac{1}{x}\right) - \cfrac{\pi}{2} = -\arctan(x)

\longrightarrow \arctan(-x) = \arctan\left(\cfrac{1}{x}\right) - \cfrac{\pi}{2}

This answers the question above: yes! Our parameterization is

\begin{array}{c} x = R \cos\left(2\theta + \arctan\left(\cfrac{c-d}{a-b}\right)\right) +\left(\cfrac{a+b}{2}\right) \\\\ y =R \sin\left(2\theta + \arctan\left(\cfrac{c-d}{a-b}\right)\right) +\left(\cfrac{c+d}{2}\right)\end{array}

If we wanted to make it a bit nicer, replace:

\phi = 2\theta + \arctan\left(\cfrac{c-d}{a-b}\right)

and we get a nice

\begin{array}{c} x = R \cos\phi +\left(\cfrac{a+b}{2}\right) \\\\ y =R \sin\phi +\left(\cfrac{c+d}{2}\right),\end{array}

a circle centered at x=(a+b)/2 and y=(c+d)/2. Woof! Bark bark! Woof woof bark!