Chapter 8. Fractals

“Pathological monsters! cried the terrified mathematician

Every one of them a splinter in my eye

I hate the Peano Space and the Koch Curve

I fear the Cantor Ternary Set

The Sierpinski Gasket makes me wanna cry

And a million miles away a butterfly flapped its wings

On a cold November day a man named Benoit Mandelbrot was born”

Once upon a time, I took a course in high school called “Geometry.” Perhaps you did too. You learned about shapes in one dimension, two dimensions, and maybe even three. What is the circumference of a circle? The area of a rectangle? The distance between a point and a line? Come to think of it, we’ve been studying geometry all along in this book, using vectors to describe the motion of bodies in Cartesian space. This sort of geometry is generally referred to as Euclidean geometry, after the Greek mathematician Euclid.

For us nature coders, we have to ask the question: Can we describe our world with Euclidean geometry? The LCD screen I’m staring at right now sure looks like a rectangle. And the plum I ate this morning is circular. But what if I were to look further, and consider the trees that line the street, the leaves that hang off those trees, the lightning from last night’s thunderstorm, the cauliflower I ate for dinner, the blood vessels in my body, and the mountains and coastlines that cover land beyond New York City? Most of the stuff you find in nature cannot be described by the idealized geometrical forms of Euclidean geometry. So if we want to start building computational designs with patterns beyond the simple shapes ellipse(), rect(), and line(), it’s time for us to learn about the concepts behind and techniques for simulating the geometry of nature: fractals.

8.1 What Is a Fractal?

The term fractal (from the Latin fractus, meaning “broken”) was coined by the mathematician Benoit Mandelbrot in 1975. In his seminal work “The Fractal Geometry of Nature,” he defines a fractal as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.”

Let’s illustrate this definition with two simple examples. First, let’s think about a tree branching structure (for which we’ll write the code later):

Notice how the tree in Figure 8.3 has a single root with two branches connected at its end. Each one of those branches has two branches at its end and those branches have two branches and so on and so forth. What if we were to pluck one branch from the tree and examine it on its own?

Looking closely at a given section of the tree, we find that the shape of this branch resembles the tree itself. This is known as self-similarity; as Mandelbrot stated, each part is a “reduced-size copy of the whole.”

The above tree is perfectly symmetrical and the parts are, in fact, exact replicas of the whole. However, fractals do not have to be perfectly self-similar. Let’s take a look at a graph of the stock market (adapted from actual Apple stock data).

And one more.

In these graphs, the x-axis is time and the y-axis is the stock’s value. It’s not an accident that I omitted the labels, however. Graphs of stock market data are examples of fractals because they look the same at any scale. Are these graphs of the stock over one year? One day? One hour? There’s no way for you to know without a label. (Incidentally, graph A shows six months’ worth of data and graph B zooms into a tiny part of graph A, showing six hours.)

This is an example of a stochastic fractal, meaning that it is built out of probabilities and randomness. Unlike the deterministic tree-branching structure, it is statistically self-similar. As we go through the examples in this chapter, we will look at both deterministic and stochastic techniques for generating fractal patterns.

While self-similarity is a key trait of fractals, it’s important to realize that self-similarity alone does not make a fractal. After all, a straight line is self-similar. A straight line looks the same at any scale, and can be thought of as comprising lots of little lines. But it’s not a fractal. Fractals are characterized by having a fine structure at small scales (keep zooming into the stock market graph and you’ll continue to find fluctuations) and cannot be described with Euclidean geometry. If you can say “It’s a line!” then it’s not a fractal.

Another fundamental component of fractal geometry is recursion. Fractals all have a recursive definition. We’ll start with recursion before developing techniques and code examples for building fractal patterns in p5.js.

8.2 Recursion

Let’s begin our discussion of recursion by examining the first appearance of fractals in modern mathematics. In 1883, German mathematician George Cantor developed simple rules to generate an infinite set:

There is a feedback loop at work here. Take a single line and break it into two. Then return to those two lines and apply the same rule, breaking each line into two, and now we’re left with four. Then return to those four lines and apply the rule. Now you’ve got eight. This process is known as recursion: the repeated application of a rule to successive results. Cantor was interested in what happens when you apply these rules an infinite number of times. We, however, are working in a finite pixel space and can mostly ignore the questions and paradoxes that arise from infinite recursion. We will instead construct our code in such a way that we do not apply the rules forever (which would cause our program to freeze).

Before I implement the Cantor set, let’s take a look at what it means to have recursion in code. Here’s something you are used to doing all the time—calling a function inside another function.

function someFunction() {

  background(0);

}

What would happen if you called the function you defined within the function itself? Can someFunction() call someFunction()?

function someFunction() {
  someFunction();
}

In fact, this is not only allowed, but it’s quite common (and essential to how I will implement the Cantor set). Functions that call themselves are recursive and good for solving certain problems. For example, certain mathematical calculations are implemented recursively; the most common example is factorial.

The factorial of any number n, usually written as n!, is defined as:

Here I'll write a function in p5.js that uses a for loop to calculate factorial:

function factorial(n) {
  let f = 1;

  for (let i = 0; i < n; i++) {
    f = f * (i+1);
  }

  return f;
}

Upon close examination, you’ll notice something interesting about how factorial works. Let’s look at 4! and 3!

therefore. . .

In more general terms, for any positive integer n:

Written out:

The factorial of n is defined as n times the factorial of n-1.

The definition of factorial includes factorial?! It’s kind of like defining “tired" as “the feeling you get when you are tired.” This concept of self-reference in functions is an example of recursion. And you can use it to write a factorial function that calls itself.

function factorial(n) {
  if (n == 1) {
    return 1;
  } else {
    return n * factorial(n-1);
  }
}

It may look crazy, but it works. Here are the steps that happen when factorial(4) is called.

You can apply the same principle to graphics with interesting results, as you will see in many examples throughout this chapter. Take a look at this recursive function.

Example 8.1: Recursive Circles I

https://editor.p5js.org/embed/SyjffpmOl

function drawCircle(x, y, radius) {
  ellipse(x, y, radius);
  if(radius > 2) {
    radius *= 0.75f;

    drawCircle(x, y, radius);

  }
}

drawCircle() draws an ellipse based on a set of parameters that it receives as arguments. It then calls itself with those same parameters, adjusting them slightly. The result is a series of circles, each of which is drawn inside the previous circle.

Notice that the above function only recursively calls itself if the radius is greater than 2. This is a crucial point. As with iteration, all recursive functions must have an exit condition! You likely are already aware that all for and while loops must include a Boolean expression that eventually evaluates to false, thus exiting the loop. Without one, the program would crash, caught inside of an infinite loop. The same can be said about recursion. If a recursive function calls itself forever and ever, you’ll be most likely be treated to a nice frozen screen.

This circles example is rather trivial; it could easily be achieved through simple iteration. However, for scenarios in which a function calls itself more than once, recursion becomes wonderfully elegant.

Let’s make drawCircle() a bit more complex. For every circle displayed, draw a circle half its size to the left and right of that circle.

Example 8.2: Recursion twice

https://editor.p5js.org/embed/S1NQfaQOg

function setup() {
  createCanvas(640, 360);
}

function draw() {
  background(255);
  drawCircle(width/2, height/2, 200);
}

function drawCircle(x, y, radius) {
  stroke(0);
  noFill();
  ellipse(x, y, radius);
  if(radius > 2) {

    drawCircle(x + radius/2, y, radius/2);
    drawCircle(x - radius/2, y, radius/2);

  }
}

With just a little more code, we could also add a circle above and below each circle.

Example 8.3: Recursion four times

https://editor.p5js.org/embed/r1-HfaXux

function drawCircle(x, y, radius) {
  ellipse(x, y, radius);
  if(radius > 8) {
    drawCircle(x + radius/2, y, radius/2);
    drawCircle(x - radius/2, y, radius/2);
    drawCircle(x, y + radius/2, radius/2);
    drawCircle(x, y - radius/2, radius/2);
  }
}

Try reproducing this sketch with iteration instead of recursion—I dare you!

8.3 The Cantor Set with a Recursive Function

Now I'm ready to visualize the Cantor set in p5.js using a recursive function. Where do I begin? Well, I know that the Cantor set begins with a line. So I will start there and write a function that draws a line.

function cantor(x, y, len) {
  line(x, y, x+len, y);
}

The above cantor() function draws a line that starts at pixel coordinate (x,y) with a length of len. (The line is drawn horizontally here, but this is an arbitrary decision.) So if I called that function, saying:

cantor(10, 20, width-20);

we’d get the following:

Now, the Cantor rule tells us to erase the middle third of that line, which leaves us with two lines, one from the beginning of the line to the one-third mark, and one from the two-thirds mark to the end of the line.

We can now add two more lines of code to draw the second pair of lines, moving the y-position down a bunch of pixels so that we can see the result below the original line.

function cantor(x, y, len) {
  line(x,y,x+len,y);

  y += 20;

  line(x, y, x + len / 3, y);

  line(x + len * 2/3, y, x + len, y);

}

While this is a fine start, such a manual approach of calling line() for each line is not what I want. It will get unwieldy very quickly, as I'd need four, then eight, then sixteen calls to line(). Yes, a for loop is the usual way around such a problem, but give that a try and you’ll see that working out the math for each iteration quickly proves inordinately complicated. Here is where recursion comes to the rescue.

Take a look at where I draw that first line from the start to the one-third mark.

   line(x, y, x + len / 3, y);

Instead of calling the line() function directly, I can simply call the cantor() function itself. After all, what does the cantor() function do? It draws a line at an (x,y) position with a given length! And so:

   line(x, y, x + len/3, y);           becomes ----> cantor(x, y, len/3);

And for the second line:

   line(x + len * 2/3, y, x + len, y); becomes ----> cantor(x + len * 2/3, y, len / 3);

Leaving us with:

function cantor(x, y, len) {
  line(x, y, x + len, y);

  y += 20;

  cantor(x, y, len/3);
  cantor(x + len * 2/3, y, len/3);
}

And since the cantor() function is called recursively, the same rule will be applied to the next lines and to the next and to the next as cantor() calls itself again and again! Now, don’t go and run this code yet. The sketch missing that crucial element: an exit condition. You'll want to make sure to stop at some point—for example, if the length of the line ever is less than 1 pixel.

Example 8.4: Cantor set

https://editor.p5js.org/embed/rkj_zTXde

function cantor(x, y, len) {

  if (len >= 1) {

    line(x, y, x + len, y);
    y += 20;
    cantor(x, y, len/3);
    cantor(x + len * 2/3, y, len/3);
  }
}

Exercise 8.1

Using drawCircle() and the Cantor set as models, generate your own pattern with recursion. Here is a screenshot of one that uses lines.

8.4 The Koch Curve

Writing a function that recursively calls itself is one technique for generating a fractal pattern on screen. However, what if you wanted the lines in the above Cantor set to exist as individual objects that could be moved independently? The recursive function is simple and elegant, but it does not allow you to do much besides simply generating the pattern itself. However, there is another way you can apply recursion in combination with an ArrayList that will allow you to not only generate a fractal pattern, but keep track of all its individual parts as objects.

To demonstrate this technique, I'll look at another famous fractal pattern, discovered in 1904 by Swedish mathematician Helge von Koch. Here are the rules. (Note that it starts the same way as the Cantor set, with a single line.)

The result looks like:

I could proceed in the same manner as I did with the Cantor set, and write a recursive function that iteratively applies the Koch rules over and over. Nevertheless, I am going to tackle this problem in a different manner by treating each segment of the Koch curve as an individual object. This will open up some design possibilities. For example, if each segment is an object, I could allow each segment to move independently from its original position and participate in a physics simulation. In addition, I could use a random color, line thickness, etc. to display each segment differently.

In order to accomplish this goal of treating each segment as an individual object, I must first decide what this object should be in the first place. What data should it store? What functions should it have?

The Koch curve is a series of connected lines, and so I will think of each segment as a “KochLine.” Each KochLine object has a start point (“a”) and an end point (“b”). These points are p5.Vector objects, and the line is drawn with p5.js’s line() function.

class KochLine {

  constructor(a, b) {

    this.start = a.copy();
    this.end = b.copy();
  }

  display() {
    stroke(0);

    line(this.start.x, this.start.y, this.end.x, this.end.y);

  }
}

Now that I have my KochLine class, I can get started on the main program. I'll need a data structure to keep track of what will eventually become many KochLine objects, and an array (see Chapter 4 for a review of arrays) will do just fine.

let lines;

In setup(), I'll want to create the array and add the first line segment to it, a line that stretches from 0 to the width of the sketch.

function setup() {
  createCanvas(600, 300);

  lines = [];

  let start = createVector(0, 200);

  let end   = createVector(width, 200);

  lines.add(new KochLine(start, end));

}

Then in draw(), all KochLine objects (just one right now) can be displayed in a loop.

function draw() {
  background(255);
  for (let l in lines) {
    l.display();
  }
}

This is our foundation. Here's what I have so far:

• KochLine class: A class to keep track of a line from point A to B.
• Array: A list of all KochLine objects.

With the above elements, how and where should I apply Koch rules and principles of recursion?

Remember the Game of Life cellular automata? In that simulation, I always kept track of two generations: current and next. When I was finished computing the next generation, next became current and I moved on to computing the new next generation. I am going to apply a similar technique here. I have an array that keeps track of the current set of KochLine objects (at the start of the program, there is only one). I will need a second array (let’s call it “next”) where I will place all the new KochLine objects that are generated from applying the Koch rules. For every KochLine object in the current array, four new KochLine objects are added to the next array. When I'm done, the next array becomes the current one.

Here’s how the code will look:

function generate() {

  let next = [];

  for (let l in lines) {

    next.push(new KochLine(???,???));
    next.push(new KochLine(???,???));
    next.push(new KochLine(???,???));
    next.push(new KochLine(???,???));

  }

  lines = next;
}

By calling generate() over and over again (for example, each time the mouse is pressed), I recursively apply the Koch curve rules to the existing set of KochLine objects. Of course, the above omits the real “work” here, which is figuring out those rules. How do I break one line segment into four as described by the rules? While this can be accomplished with some simple arithmetic and trigonometry, since the KochLine object uses p5.Vector, this is a nice opportunity for me to practice my vector math. Let’s establish how many points I need to compute for each KochLine object.

As you can see from the above figure, I need five points (a, b, c, d, and e) to generate the new KochLine objects and make the new line segments (ab, cb, cd, and de).

    next.add(new KochLine(a, b));
    next.add(new KochLine(b, c));
    next.add(new KochLine(c, d));
    next.add(new KochLine(d, e));

Where do I get these points? Since I have a KochLine object, I can ask the KochLine object to compute all these points for me.

function generate() {
  let next = [];
  for (let l in lines) {

    let a = l.kochA();
    let b = l.kochB();
    let c = l.kochC();
    let d = l.kochD();
    let e = l.kochE();

    next.push(new KochLine(a, b));
    next.push(new KochLine(b, c));
    next.push(new KochLine(c, d));
    next.push(new KochLine(d, e));
  }

  lines = next;
}

Now I just need to write five new functions in the KochLine class, each one returning a p5.Vector according to Figure 8.16 above. Let’s knock off kochA() and kochE() first, which are simply the start and end points of the original line.

  kochA() {

    return this.start.copy();

  }

  kochE() {
    return this.end.copy();
  }

Now I will move on to points B and D. B is one-third of the way along the line segment and D is two-thirds. Here I can make a p5.Vector that points from start to end and shrink it to one-third the length for B and two-thirds the length for D to find these points.

  kochB() {

    let v = p5.Vector.sub(this.end, this.start);

    v.div(3);

    v.add(this.start);

    return v;
  }

  kochD() {
    let v = p5.Vector.sub(this.end, this.start);

    v.mult(2 / 3.0);

    v.add(this.start);
    return v;
  }

The last point, C, is the most difficult one to find. However, if you recall that the angles of an equilateral triangle are all sixty degrees, this makes it a little bit easier. If I know how to find point B with a p5.Vector one-third the length of the line, what if I were to rotate that same p5.Vector sixty degrees and move along that vector from point B? I'd be at point C!

  kochC() {

    let a = this.start.copy();

    let v = p5.Vector.sub(end, start);

    v.div(3);

    a.add(v);

    v.rotate(-radians(60));

    a.add(v);

    return a;
  }

Putting it all together, if we call generate() five times in setup(), we’ll see the following result.

Example 8.5: Koch curve

https://editor.p5js.org/embed/rJUB76mdl

let lines;

function setup() {
  createCanvas(600, 300);
  background(255);
  lines = [];
  let start = createVector(0, 200);
  let end   = createVector(width, 200);
  lines.push(new KochLine(start, end));

  for (int i = 0; i < 5; i++) {
    generate();
  }

}

Exercise 8.2

Draw the Koch snowflake (or some other variation of the Koch curve).

Exercise 8.5

Draw the Sierpiński triangle (as seen in Wolfram elementary CA) using recursion.

8.5 Trees

The fractals I've shown in this chapter so far are deterministic, meaning they have no randomness and will always produce the identical outcome each time they are run. They are excellent demonstrations of classic fractals and the programming techniques behind drawing them, but are too precise to feel natural. In this next part of the chapter, I want to examine some techniques behind generating a stochastic (or non-deterministic) fractal. The example I'll use is a branching tree. I'll first walk through the steps to create a deterministic version. Here are our production rules:

Again, I have a nice fractal with a recursive definition: A branch is a line with two branches connected to it.

The part that is a bit more difficult than our previous fractals lies in the use of the word rotate in the fractal’s rules. Each new branch must rotate relative to the previous branch, which is rotated relative to all its previous branches. Luckily for us, p5.js has a mechanism to keep track of rotations for you—the transformation matrix. If you aren’t familiar with the functions push() and pop(), I suggest you read the online p5.js tutorial 2D Transformations, which will cover the concepts you’ll need for this particular example.

Let’s begin by drawing a single branch, the trunk of the tree. Since I am going to involve the rotate() function, I'll need to make sure I am continuously translating along the branches while we draw the tree. And since the root starts at the bottom of the window (see above), the first step requires translating to that spot:

translate(width/2,height);

…followed by drawing a line upwards (Figure 8.20):

line(0, 0, 0, -100);

Once I've finished the root, I need to translate to the end and rotate in order to draw the next branch. (Eventually, I'm going to need to package up what I'm doing right now into a recursive function, but I'll sort out the steps first.)

Remember, when you rotate in p5.js, you are always rotating around the point of origin, so here the point of origin must always be translated to the end of our current branch.

translate(0, -100);
rotate(PI / 6);
line(0, 0, 0, -100);

Now that we have a branch going to the right, I need one going to the left. I can use push() to save the transformation state before I rotate, letting me call pop() to restore that state and draw the branch to the left. Let’s look at all the code together.

translate(width/2,height);

line(0, 0, 0, -100);

translate(0, -100);

push();
rotate(PI / 6);
line(0, 0, 0, -100);
pop();

rotate(-PI / 6);
line(0, 0, 0, -100);

If you think of each call to the function line() as a “branch,” you can see from the code above that I have implemented a definition of branching as a line that has two lines connected to its end. I could keep adding more and more calls to line() for more and more branches, but just as with the Cantor set and Koch curve, my code would become incredibly complicated and unwieldy. Instead, I can use the above logic as a foundation for writing a recursive function, replacing the direct calls to line() with my own function called branch(). Let’s take a look.

function branch() {

  line(0, 0, 0, -100);

  translate(0, -100);

  push();

  rotate(PI / 6);
  branch();

  pop();

  push();

  rotate(-PI/6);
  branch();

  pop();
}

Notice how in the above code I use push() and pop() around each subsequent call to branch(). This is one of those elegant code solutions that feels almost like magic. Each call to branch() takes a moment to remember the position of that particular branch. If you turn yourself into p5.js for a moment and try to follow the recursive function with pencil and paper, you’ll notice that it draws all of the branches to the right first. When it gets to the end, pop() will pop me back along all of the branches I've drawn and start sending branches out to the left.

Exercise 8.6

Emulate the p5.js code in Example 8.6 and number the branches in the above diagram in the order that p5.js would actually draw each one.

You may have noticed that the recursive function I just wrote would not actually draw the above tree. After all, it has no exit condition and would get stuck in infinite recursive calls to itself. You’ll also probably notice that the branches of the tree get shorter at each level. Let’s look at how we can shrink the length of the lines as the tree is drawn, and stop branching once the lines have become too short.

function branch(len) {

  line(0, 0, 0, -len);
  translate(0, -len);

  len *= 0.66;

  if (len > 2) {
    push();
    rotate(theta);

    branch(len);

    pop();

    push();
    rotate(-theta);
    branch(len);
    pop();
  }
}

I've also included a variable for theta that allows me, when writing the rest of the code in setup() and draw(), to vary the branching angle according to, say, the mouseX position.

Example 8.7: Recursive tree

{p5 sketch link}

let theta;

function setup() {
  createCanvas(300, 200);
}

function draw() {
  background(255);

  theta = map(mouseX, 0, width, 0, PI/2);

  translate(width/2, height);

  stroke(0);
  branch(60);
}

Exercise 8.7

Vary the strokeWeight() for each branch. Make the root thick and each subsequent branch thinner.

The recursive tree fractal is a nice example of a scenario in which adding a little bit of randomness can make the tree look more natural. Take a look outside and you’ll notice that branch lengths and angles vary from branch to branch, not to mention the fact that branches don’t all have exactly the same number of smaller branches. First, let’s see what happens when we simply vary the angle and length. This is a pretty easy one, given that we can just ask p5.js for a random number each time we draw the tree.

function branch(len) {

  let theta = random(0, PI / 3);

  line(0, 0, 0, -len);
  translate(0, -len);
  len *= 0.66;
  if (len > 2) {
    push();
    rotate(theta);
    branch(len);
    pop();
    push();
    rotate(-theta);
    branch(len);
    pop();
  }
}

In the above function, I always call branch() twice. But why not pick a random number of branches and call branch() that number of times?

Example 8.8: Stochastic tree

https://editor.p5js.org/embed/H1QwVaX_x

function branch(len) {

  line(0, 0, 0, -len);
  translate(0, -len);

  if (len > 2) {

    let n = int(random(1,4));

    for (let i = 0; i < n; i++) {

      let theta = random(-PI/2, PI/2);

      push();
      rotate(theta);
      branch(h);
      pop();
    }
  }

8.6 L-systems

In 1968, Hungarian botanist Aristid Lindenmayer developed a grammar-based system to model the growth patterns of plants. L-systems (short for Lindenmayer systems) can be used to generate all of the recursive fractal patterns we’ve seen so far in this chapter. You don’t need L-systems to do the kind of work I'm doing here; however, they are incredibly useful because they provide a mechanism for keeping track of fractal structures that require complex and multi-faceted production rules.

In order to create an example that implements L-systems in p5.js, you are going to have to be comfortable with working with (a) recursion, (b) transformation matrices, and (c) strings. So far you've worked with recursion and transformations, but strings are new here. I will assume the basics, but if that is not comfortable for you, I would suggest taking a look at the p5.js tutorial Strings and Drawing Text.

An L-system involves three main components:

• Alphabet. An L-system’s alphabet is comprised of the valid characters that can be included. For example, I could say the alphabet is “ABC,” meaning that any valid “sentence” (a string of characters) in an L-system can only include these three characters.

• Axiom. The axiom is a sentence (made up with characters from the alphabet) that describes the initial state of the system. For example, with the alphabet “ABC,” some example axioms are “AAA” or “B” or “ACBAB.”

• Rules. The rules of an L-system are applied to the axiom and then applied recursively, generating new sentences over and over again. An L-system rule includes two sentences, a “predecessor” and a “successor.” For example, with the Rule “A —> AB”, whenever an “A” is found in a string, it is replaced with “AB.”

Let’s begin with a very simple L-system. (This is, in fact, Lindenmayer’s original L-system for modeling the growth of algae.)

As with our recursive fractal shapes, we can consider each successive application of the L-system rules to be a generation. Generation 0 is, by definition, the axiom.

Let’s look at how we might create these generations with code. We’ll start by using a String object to store the axiom.

let current = "A";

And once again, just as we did with the Game of Life and the Koch curve array examples, we will need an entirely separate string to keep track of the “next” generation.

let next = "";

Now it’s time to apply the rules to the current generation and place the results in the next.

for (let i = 0; i < current.length; i++) {
  let c = current.charAt(i);

  if (c === 'A') {

    next += "AB";

  } else if (c === 'B') {

    next += "A";
  }
}

And when it's done, current can become next.

current = next;

To be sure this is working, I will package it into a function and and call it every time the mouse is pressed.

Example 8.9: Simple L-system sentence generation

{p5 sketch link}

let current = "A";

let count = 0;

function setup() {
  print("Generation " + count + ": " + current);
}

function draw() {
}

function mousePressed() {
  let next = "";

  for (let i = 0; i < current.length; i++) {
    let c = current.charAt(i);
    if (c === 'A') {
      next += "AB";
    } else if (c === 'B') {
      next += "A";
    }
  }

  current = next;
  count++;
  print("Generation " + count + ": " + current);
}

You may find yourself wondering right about now: what exactly is the point of all this? After all, isn’t this a chapter about drawing fractal patterns? Yes, the recursive nature of the L-system sentence structure seems relevant to the discussion, but how exactly does this model plant growth in a visual way?

What I've left unsaid until now is that embedded into these L-system sentences are instructions for drawing. Let’s see how this works with another example.

To read a sentence, I'll translate it in the following way:

Let’s look at the sentence of each generation and its visual output.

Look familiar? This is the Cantor set generated with an L-system.

The following alphabet is often used with L-systems: “FG+-[]”, meaning:

This type of drawing framework is often referred to as “Turtle graphics” (from the old days of LOGO programming). Imagine a turtle sitting on your computer screen to which you could issue a small set of commands: turn left, turn right, draw a line, etc. p5.js isn’t set up to operate this way by default, but by using translate(), rotate(), and line(), we can emulate a Turtle graphics engine fairly easily.

Here’s how I would translate the above L-system alphabet into p5.js code.

Assuming I have a sentence generated from the L-system, I can walk through the sentence character by character and call the appropriate function as outlined above.

for (let i = 0; i < sentence.length; i++) {

  let c = sentence.charAt(i);

  if (c == 'F') {
    line(0, 0, len, 0);
    translate(len, 0);
  } else if (c == 'G') {
    translate(len, 0);
  } else if (c == '+') {
    rotate(theta);
  } else if (c == '-') {
    rotate(-theta);
  } else if (c == '[') {
    push();
  } else if (c == ']') {
    pop();
  }

}

The next example will draw a more elaborate structure with the following L-system.

The example available for download on the book’s website takes all of the L-system code provided in this section and organizes it into three classes:

• Rule: A class that stores the predecessor and successor strings for an L-system rule.
• LSystem: A class to iterate a new L-system generation (as demonstrated with the StringBuffer technique).
• Turtle: A class to manage reading the L-system sentence and following its instructions to draw on the screen.

We won’t write out these classes here since they simply duplicate the code we’ve already worked out in this chapter. However, let’s see how they are put together in the main tab.

Example 8.10: LSystem

https://editor.p5js.org/embed/By4ED6mOg

let lsys;
let turtle;

function setup() {
  createCanvas(600,600);

  let ruleset = [];
  ruleset[0] = new Rule('F',"FF+[+F-F-F]-[-F+F+F]");

  lsys = new LSystem("F",ruleset);

  turtle = new Turtle(lsys.getSentence(), width/4, radians(25));
}

function draw() {
  background(255);

  translate(width/2, height);
  turtle.render();

}

function mousePressed() {

  lsys.generate();

  turtle.setToDo(lsys.getSentence());

  turtle.changeLen(0.5);

}