Close up photo of the map's cartouche, displaying the formula for the Mandelbrot set

The Mandelbrot set

The Mandelbrot set is a fractal shape that reveals fascinating details upon magnification. It gets its name from Benoit Mandelbrot, who is often referred to as the father of fractal geometry. Fractals are a new class of geometry that are known for their roughness, their self-similarity over many different scales, and their resemblance of naturally occurring forms — all of which can easily be observed in the Mandelbrot set. Perhaps the most stunning feature of the set is that all of it's infinite complexity emerges spontaneously and reliably from one very simple equation, through the process of iteration. Anyone with a computer can run the equation and it will produce the exact same shape every time.

More formally, the Mandelbrot set is the set of all numbers in the complex plane that remain bounded under repeated iterations of the equation. These points form the interior of the shape. Points that lay outside the set escape to infinity when the equation is iterated, and the boundary between the two regions is where all of the beautiful chaotic details are found.

mock up of the 54x36 inch poster, photo by hobvias sudoneighm

The Poster

Although the Mandelbrot set is very complex, there are definite patterns to the chaos. The Mandelmap poster helps to make sense of it by mapping out many of the most common features. The text introduces and explains many of the core concepts, providing an overview of the subject in a style that is accessible to non-experts. If you are using one of the many programs available for exploring the Mandelbrot set, this map will help you find your way around visually as you zoom in to this amazing fractal.

Every element in the poster was created from scratch — even the antique look was made by scanning in coffee-stained paper. The various mappings of the fractal were rendered at high resolution with custom software, some of which was coded just for this poster, allowing for an exacting level of control. No corners were cut and nothing was rushed in the making of this poster; everything was created to the highest level of finish possible.

The posters are available in two sizes: a standard 36x24 inches and a giant 54x36 inches. Both sizes are printed with offset lithography at high resolution for professional-quality results, and are treated with a moisture- and scuff-resistant coating. The larger size is printed on a heavier and stiffer paper to help extend its life and prevent sagging when it is on the wall. Whichever size you choose, the crisp details and vintage style will look great in any home, office, or classroom. Check out the gallery page to see customer images of the Mandelmap, or click here to buy yours now! (ships worldwide)

The printing of the Mandelmap poster was made possible through the amazing support given by the backers of the Kickstarter campaign. Huge thank you!!

The Designer

Bill Tavis is the artist and researcher who created the Mandelmap poster. Driven by his love of fractals and math, he spent more than a year carefully crafting this map to be both accurate and aesthetically appealing. He initially started making the poster because he wanted a printed map of the Mandelbrot set for himself, to use for his own explorations. Tavis often incorporates patterns from the Mandelbrot set and other fractals into his artwork, which you can see on his Tumblr blog and Instagram page.

Tavis rendered the various views of the Mandelbrot set with software he coded himself, and used shapes from the Mandelbrot set to create various custom elements of the map like the compass rose and cartouche. He grew up fascinated by the maps from the Age of Exploration, and quickly saw that as an ideal style for the Mandelmap. The sea monsters and other animals that were often depicted on those maps inspired Tavis to commission artist Helena Martin to draw pen-and-ink illustrations representing several of the most common locations, such as Elephant Valley.

Tavis currently resides in Austin, TX. Always happy to hear from you, if you have anything to say please contact him at or use the contact form.