Fractals in Mathcad
I first plotted the Mandelbrot set on a Spectrum 48k, sometime in my early teens. Writing the code probably took a few hours (hey…I never said I was fast at these things). I left my Spectrum running overnight, chugging away and number crunching. I woke up in the morning to find a highly-pixelated image of the fractal, plotted in wonderful black and white. I beckoned my sister to witness the mathematical glory I had produced, but she was too busy watching Going Live.
Since those halcyon days, one of the first things I do when I learn a new programming language or maths tool is to plot a fractal (it’s my warped version of “Hello World”). For example, here’s an earlier Maple worksheet and blog I wrote about the Mandebrot set.
In my time, I’ve plotted Mandelbrot Sets, Julia Sets, Quanternion fractals, Koch triangles and more. However, I always drift back to the Mandelbrot set. The deeper you dive into its complex canyons, the more beauty you discover.
I won’t discuss the mathematics or the algorithms in detail, but here’s part of a Mathcad worksheet that generates the Mandelbrot set (the download link is at the bottom of this post)
The first program generates a matrix giving the number of iterations before each point in the set tends to infinity (i.e is greater than a bailout value). The second program colours each point with custom RGB values; this is then plotted in a 2D plot.
A Mathcad routine to generate the Julia set follows the same process (again, the download link is at the bottom of this post).
Here’s a few colourful renderings of the Mandelbrot Set and the Julia Set I’ve generated using these Mathcad worksheets. The colouring algorithm is where much of the artistry comes into play. Just by tweaking a few numbers, you can produce remarkably different pictures.
These pictures were generated entirely inside Mathcad, with no other image editing.
So what are the downsides of using Mathcad for generating fractals? Mathcad’s an interpreted environment, so compiled C code will always be many orders of magnitude faster. However, a Mathcad worksheet is good for developing the initial algorithms – it’s a much more interactive, easy-to-use environment than a text editor and a compiler. Mathcad gives you much faster feedback on your algorithms than a traditional programming language.
The Next Steps
Ready To Buy?
What do our Customers say about us?
I have greatly appreciated your help, your restraint in response to some daft questions and your sheer professionalism.PL, Tunbridge Wells, UK
Your web page is simply brilliant. Congratulations!CF, Kirknewton, UK
“Origin has always been one of the most powerful packages for data analysis and graphing. With the new multi-sheet workbook’s full formatting features and its ability to embed both images and graphs, OriginLab is moving a big step forward. From raw data to final professional report creation, you can now keep all the stages of data processing in a single software.”Antoine Couturier, Ph.D. - Service Recherche, Institut National du Sport, de l'Expertise et de la Performance
“I am very satisfied with Origin. I think Origin is an excellent plotting tool. Its capabilities are vast, diverse, and beyond my needs. I appreciate the fact that almost anything on a plot can be adjusted according to a desired preference.”Matthew D. Sena - Sandia National Laboratories
For the time being we are unable to offer the following product ranges although we are currently working hard to increase the number of products we can offer in the future. Please contact us to talk about alternative products that we may be able to offer you.