# Mandelbrot Set

It's time to learn a "new" language and I'm learning *Go*. Actually, I need to learn *Go* out of necessity - for some work I'm doing and I might have to teach it as well.

The best way to learn is by doing. This morning I wrote a *Go* program to plot the Mandelbrot set. Back in the 90s when home computers had the power to compute these things effortlessly, many a night was spent implementing fractal graphics programs**. The set is the set of complex numbers *c* for which the function *f(z) = z ^{2} + c *repeated iterated remains bounded. A complex number can be represented by its real part and its imaginary part on a 2d plane. For each

*c*we plot a point of colour based on the number of iterations it takes for the iteration to cross a specified boundary.***

I found an old implementation I'd written in *Fortran**. Writing it in *Go* was more of an exercise of getting use to variable declarations, loops and file layout. For example, there are no *while* loops, only *for* loops and this takes a bit of getting used to.

For the graphical output, I found nice and easy to use library *gg** *that handles 2d graphics and outputs PNG files. The listing is below or here to download (with the Fortran too) for convenience. You will need to "go get" the *gg* library.

// Plot the Mandelbrot Set // // This is the Mandelbrot program based on one written in Fortran // in 1994. It still had the original date stamp on the files. //-rwxr-xr-x 1 cjep staff 2182 11 Mar 1994 mandel.for // // You will find better implementations and better explanations // package main // go get -d github.com/fogleman/gg import "github.com/fogleman/gg" func main() {`// Tune these to get different pictures`

`// We have provided the right values`

`// for the traditional "turtle"`

`//`

`const sleft float64 = -2.25`

`const sright float64 = 0.75`

`const stop float64 = 1.5`

`const sbottom float64 = -1.5`

`const srealp float64 = 0`

`const simagp float64 = 0`

`const outputFile string = "out.png"`

`// Maximum number of iterations and max magnitude`

`const maxIterations int = 2000`

`const magLimit float64 = 10000`

`// Vertical Size of the image in pixels`

`const size int = 1280`

`colors := [15]string {`

`"800000", // Maroon`

`"008000", // Green`

`"000080", // Navy`

`"c0c0c0", // Silver`

`"ff0000", // Red`

`"00ff00", // Lime`

`"0000ff", // Blue`

`"808080", // Grey`

`"800080", // Purple`

`"808000", // Olive`

`"008080", // Teal`

`"ffff00", // Yellow`

`"ff00ff", // Magenta`

`"00ffff", // Cyan`

`"ffffff", // White`

`}`

`// Pixel size converted to our X & Y axes`

`var deltaX float64 = (sright - sleft)/float64(size)`

`var deltaY float64 = (stop - sbottom)/float64(size)`

// Pixel indices and colours var pixelX int var pixelY int var color int // an index used to choose a colour XXX

`var conX float64 // Converted position from pixels to floats`

`var conY float64`

`var magnitude float64`

`var x0 float64`

`var x1 float64`

`var y0 float64`

`var y1 float64`

// Graphics - allowing for a wider image // dc := gg.NewContext(size+160, size) // We work through each pixel location regarded as a relative // location to our axes determined by sleft & sbottom above // // We are interested in the complex function f(z) = z^2 + c // where c is given by the pixel location. // Breaking this out into real & imaginary parts: // f(x, y) = x^2-y^2 + conX + i( 2xy + conY) // // where (conX, conY) is our "pixel" location. // For each point, we iterate through this function // starting at srealp, simagp. Here x0, y0 are set to srealp & simagp // x1, y1 take the "next value" then we past them to x0,y0 for the next // loop. // // We limit the number of iterations by maxIterations, // but also by the magnitude mod(z) = sqrt(x^2 + y^2) // - we haven't bothered to compute the square root here // // We want to plot how many iterations we go through at a given point // color is really counting the number of iterations we go through // we adjust it for 15 colours. Black (or "transparent") pixels // will occur in the middle if the iterations continue indefinitely // within the limit (we hit maxIterations) // f(z) = z^2 + c, c is our pixel location

`for pixelX = 0; pixelX <= size; pixelX++ {`

`for pixelY = 0; pixelY<=size; pixelY++ {`

`conX = sleft + float64(pixelX) * deltaX`

`conY = sbottom + float64(pixelY) * deltaY`

`color = 0 // Really counting iterations`

`x0 = srealp`

`y0 = simagp`

`magnitude = 0.0`

`for magnitude <= magLimit && color <= maxIterations {`

`x1 = x0*x0 - y0*y0 + conX`

`y1 = 2*x0*y0 + conY`

`color++`

`x0 = x1`

`y0 = y1`

`magnitude = x0*x0 + y0*y0`

`}`

`// Plot`

`if color < maxIterations {`

`dc.SetHexColor(colors[(color-1) % 15])`

`dc.SetPixel(pixelX, pixelY) // [1]`

`}`

`}`

`}`

`dc.SavePNG(outputFile)`

} // [1] in my Fortran code, I flipped the Y axis because the program // updated the screen interactively using a Fortran graphics library. // I haven't bothered here - we output a PNG once

** Yes, I'm old enough to have coded in Fortran. And yes, I'm old enough to have been taught it at University. We were forward thinking and learnt Fortran90.*

***We had fewer television channels in those days.*

****You will find both better explanations and better examples of code on other web sites :-)*