## Rendering the Mandelbrot Set in Clojure

Clojure is a functional language that brings dynamic types and lisp syntax to the JVM. It has a number of features that make it interesting which will be the subject of another post but for now, let’s dive right into some code.

## The Mandelbrot Set

The Mandelbrot set is an example of a fractal which is basically something that can be described using a simple formula and yet exhibit fascinating levels of detail and beauty. I thought it would be a good application to get me started learning Clojure.

The Mandelbrot formula is:

**Z = Z ^{2} + C**

Where **Z** is a complex number and **C** is a constant. To find out whether or not **C** is in the set, we apply the formula iteratively until the magnitude of **Z** becomes larger than 2, or we reach a certain number of maximum iterations.

Most 2D fractals are described mathematically using complex numbers which are just like regular (real) numbers but with an added 2nd dimension. They can be represented by two regular numbers so another way to describe the formula above might be:

**[x, y] = [x, y] ^{2} + [a, b]**

Where **[a, b]** is equivalent to **C** and represents the coordinates on the complex plane we’re interested in. By applying the formula we can find out whether or not **C** is in the Mandelbrot set. If it is, we draw it with a black pixel, otherwise with some other colour depending the number of iterations taken by the formula.

## Defining a Functional Mandelbrot Set

Let’s try to write this in Clojure code. First of all, how do you square a complex number? First let’s represent **Z** as a vector of regular numbers **[x, y]**. To square them we just apply the formula:

**[x, y] ^{2} = [x^{2} – y^{2}, 2xy]**

** **In Clojure, it looks like this:

(defn mandel [[x y]] [(- (* x x) (* y y)) (* 2 x y)])

But this represents just half of the Mandelbrot formula, we need to add that constant** C** and also somehow call our `mandel`

function iteratively. Normally, in an imperative language, this would be easy. We just assign C to our starting value, then calculate Z in a for loop. In Clojure, it’s a bit different:

(defn mandelformula [xcoord ycoord] (iterate #(vec (map + (mandel %) [xcoord ycoord])) [xcoord ycoord]))

This function uses the `iterate`

function to create successive calls to the `mandel`

function passing our initial starting value **[xcoord, ycoord]** which represents **C** in the original formula. The `mandel`

function is actually wrapped in an anonymous function which adds the return value to the initial value hence giving us the **Z**^{2}** + C**. The reason we have to use `map`

is that we cannot directly add the two vectors together. The individual **x** and **y** values must be added separately and then constructed back into a vector with the `vec`

function.

That’s quite a dense piece of code but I think represents the mathematical version of the formula fairly well. It’s certainly not the fastest implementation but the goal of functional programming is to favour correctness and readability rather than performance. The great thing about this implementation is that that `iterate`

function returns a lazy infinite sequence. This is very much like the mathematical definition of the formula given at the start; it says nothing about how many times we’re actually going to keep applying the formula.

The next part of the definition of the set says that we need to apply the formula to a given point until either the magnitude of **Z** becomes greater than 2, or we reach a given number of iterations.

(defn mag [[x y]] (+ (* x x) (* y y))) (take 50 (take-while #(<= (mag %) 4) (mandelformula xcoord ycoord)))

Firstly, we define the function `mag`

which returns the magnitude (actually, the magnitude squared) of the given vector pair. Next we use this function with the `take-while`

function so we’ll only evaluate the terms of the mandelbrot formula while the value of each term is less than or equal to 4 (we’ve squared 2 to avoid having to perform an expensive square root in the `mag`

function). `take-while`

returns another lazy sequence (which might also be infinite) so we use `take`

to restrict the number of terms evaluated to 50.

Now we’re pretty much done with all the maths stuff. All we need to do is count the number of terms calculated and from that number, calculate a colour.

(defn coord-colour [[xcoord ycoord] max-iters] (let [num-iters (count (take max-iters (take-while #(<= (mag %) 4) (mandelformula xcoord ycoord))))] (if (= max-iters num-iters) *set-colour* (iter-colour num-iters max-iters))))

This function takes a coordinate, a defined number of maximum iterations and returns the colour that the coordinate should be rendered with by applying the Mandelbrot formula. You can see that we’re assigning **num-iters** to the count of the terms returned by our sequence calculated earlier. We then compare **num-iters** with **max-iters**, if they are equal, then the given coordinate is likely to be in the Mandelbrot set so we return ***set-colour***. If **num-iters** is less than **max-iters** then the coordinate cannot be within the set so we pass the values to another function `iter-colour`

which applies a gradient to the colours.

## Calculating a Colour Gradient

First of all we define three vectors representing the colours we want to draw with as RGB values.

;; Colours used to draw the set (def *grad-colour-a* [255, 255, 0]) ;yellow (def *grad-colour-b* [0, 0, 255]) ;blue (def *set-colour* [0, 0, 0]) ;black

We want to calculate a gradient colour such that if **num-iters** is 0, we return yellow, if it is equal to **max-iters** we return blue and if it is somewhere in between, we return a mix between the two colours.

(defn grad-colour "Returns the colour that is the given fraction of the way between the first and second colours given. Returns as a vector of three integers between 0 and 255." [colA colB frac] (vec (map #(+ (* frac (- %2 %1)) %1) colA colB))) (defn iter-colour "Returns the colour needed to paint a point with the given number of iterations" [num-iters max-iters] (grad-colour *grad-colour-a* *grad-colour-b* (/ (double num-iters) max-iters)))

The `iter-colour`

function calculates the number of iterations as a fraction of the maximum number of iterations to give a value between 0 and 1 and passes this along with the two colours to the `grad-colour`

function. This function simply calculates the difference between each of the R, G and B values of the two colours and weights it by multiplying by **frac**. Again, similar to how we calculate the sum of a vector in the `mandelformula`

function, we use `map`

to apply this calculation to each of the three values in the two vectors.

## Surfing the Complex Plane

Now we have all we need to calculate the colour of a given coordinate on the complex plane, we’re close to being able to render a fractal in Clojure! But how do we go from the pixels you find on your screen to the mathematical coordinates of the complex plane? Clearly, we need another function.

(defn get-coord "Returns the coordinates of the given pixel in the complex plane" [x y xstart ystart xsize ysize width height] [(+ xstart (* (/ x width) xsize)) (+ ystart (* (/ y height) ysize))])

This function takes the x and y `pixel`

coordinates and returns the corresponding `complex`

coordinates. The other parameters are the complex coordinate of the top-left of our screen (**xstart** and **ystart**), the complex dimensions of the area we’re drawing (**xsize** and **ysize**) and the dimensions of the area we are drawing in pixels (**width** and **height**).

Now all we need is a way to get all the pixels in our drawing area as vectors to pass on to the this function.

(defn get-pixels [width height] (for [y (range height) x (range width)] [x y]))

This function returns a lazy sequence of pixel coordinates for the given width and height. We just need to map the `get-coord`

function onto this sequence and we’ll have a lazy sequence of *all* the complex coordinates in our drawing area. Then all that’s left is to get the colour of each of those complex coordinates (according the mandelbrot formula) and assign the colour to the corresponding pixel.

(defn render [xstart ystart xsize ysize width height max-iters wr] (dorun (pmap (fn [pixel] (let [[x y] pixel] (.setPixel wr x y (int-array (coord-colour (get-coord (double x) (double y) xstart ystart xsize ysize width height) max-iters))))) (get-pixels width height))))

Here we are in the heart of the fractal renderer. Note that we use `pmap`

to process the sequence of pixels which works just like `map`

but runs across multiple threads which allows us to make the most of multi-core system and render each pixel in parallel. Also, we must use `dorun`

because `get-pixels`

returns a lazy sequence, so `pmap`

is forced to actually process each pixel (rather than return a lazy sequence as it would do normally). Finally, the colour returned by each call to `coord-colour`

is converted into a Java int[] array so that it can be passed to the `.setPixel`

function of **wr** which is an instance of a `WritableRaster`

.

Note also how we `coerce`

the pixel values of **x** and **y** into doubles. This is because by default, Clojure will use arbitrary precision using Java’s BigDecimal class which would incur a huge performance hit when it came to applying the formula. Coercing to double ensures we use either double primitives or boxed Double objects.

## Setting up the Swing Stuff

All the remaining code is to initialise the Java Swing components needed to display the image generated.

(defn get-img [width height] (BufferedImage. width height (BufferedImage/TYPE_INT_RGB))) (defn get-panel [width height img] (proxy [JPanel] [] (paint [g] (.drawImage g img 0 0 (Color/red) nil)))) (defn construct-frame [width height panel] "Creates and displays a JFrame of the given dimensions with the panel added to it" (let [frame (JFrame.)] (.setPreferredSize panel (Dimension. width height)) (doto frame (.add panel) .pack (.setLocationRelativeTo nil) .show))) (defn mandelbrot [xstart ystart xsize ysize width height max-iters] "Returns a function to render the mandelbrot set with the given parameters on a frame" (let [img (get-img width height) panel (get-panel width height img) wr (.getRaster img)] (construct-frame width height panel) (fn [] (do (render xstart ystart xsize ysize width height max-iters wr) (.repaint panel)))))

We have defined a number of functions for creating the Swing components and also a function for putting them all together called `mandelbrot`

. Notice, rather than call `render`

immediately, the `mandelbrot`

creates and returns an anonymous function instead. This function is actually a `closure`

which means we can call it repeatedly without having to keep passing in the parameters for the window size and number of iterations. When running a program in a REPL, being able to call individual parts of your code can be very handy.

Finally, we assign the function to a var, and then call it!

(def my-mandelbrot (mandelbrot -2 -1.25 3 2.5 1200 1000 50)) (future (my-mandelbrot))

We wrap the call to `my-mandelbrot`

in a `future`

which returns immediately and then executes the function passed to it in a separate thread. The reason for this is that rendering the fractal can take a significant amount of time. Using a `future`

allows you to continue to use the REPL while the rendering happens in the background which can be helpful for debugging or general development. After about 30 seconds or so, you’ll see something like this:

## Conclusion

We’ve seen how to create a working fractal rendering program by building on mathematical formulas and making use of Clojure features such as lazy, infinite sequences, parallel processing and closures to do it. These features allow us to keep the code close to the problem domain and hopefully, improve its readability and correctness. In the next post we’ll see how to profile the renderer and *drastically* improve its performance.

The full source code can be found here on github.

Stay tuned for more Clojure goodness in the future!

Thanks for this post, it’s awesome. Me and a friend of mine are trying to implement a graphical renderer that you could use with Clojure to draw and display 3D Object. We would like to use it as the bottom layer for a fractal 3D renderer and manipulator, and I think your post will be useful to us.

Bye!

Alfredo

Alfredo Di NapoliOctober 20, 2010 at 17:31

That’s great that the post has inspired you to explore the world of fractals and Clojure. 3D fractals are another level complexity and fascination. You might find this page interesting if you haven’t already seen it. There’s tons of useful information and amazing renderings: http://www.skytopia.com/project/fractal/mandelbulb.html

I would suggest as a start you could try extending the Clojure code I wrote so that it can render any part of the fractal and allow you to zoom into any depth. You will probably find there is still plenty of room for optimizations as well as lots of other interesting challenges in order to do this. If you do start work on any changes it would be great if you did them as a fork of the code I have on github. 🙂

alexOctober 20, 2010 at 21:14