Anne Burns' Mathscapes

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Field: Fractals
Image Created By: Anne M. Burns
Website: [1]


In her Mathscape images, Anne M. Burns combines recursive algorithms for clouds, mountains and various imaginary plant forms into one picture.

Basic Description

Image 1. Fractal Scene I.

Computers make it possible for Burns to "see" the beauty of mathematics. The artworks in the gallery of "Mathscapes" were created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector.

A More Mathematical Explanation


Many of the plants in Burns' Mathscapes are made using recursive replacement rules similar [...]


Many of the plants in Burns' Mathscapes are made using recursive replacement rules similar to the one used to create the Koch Snowflake.

Image 2 shows some basic plant formations similar to those that Burns' used in her replacements. Each offshoot in this figures can be replaced by a smaller version of the original image, and so on and so on until the pictures are quite complicated. The pictures can be made to look more natural by using random numbers to vary the length of the branches or the angle at which they connect.

Another way to vary an image is to use the deReffye method. In this method, at each stage a branch can either 1) continue to grow, 2) die, or 3) wait until the next stage. Each option has a certain probability, and all the probabilities add up to one. You can also make the probabilities change over time, so that at later stages, the branches are more likely to die (and thus the plant eventually stops growing). The trees in Image 3 were created from the same program using the deReffye method. Because they're from the same program, they look similar, but because of the three options at every stage and the changing probabilities, their shapes are not identical.

Image 2. Several common inflorescences
Image 3. Recursively created trees.
Image 4. Bush created using string-rewriting.

Burns also uses string re-writing to create images of plants. In string-rewriting, a simple plant image composed of sticks, leaves, and flowers is represented by a string of letters and punctuation marks. At each stage, a set of re-writing rules replaces each character with a string of other characters. The end result string is then translated back into a plant image. A bush created using this method can be seen in Image 4.

Clouds & Mountains

Why It's Interesting

Image 5. Mountains in Spring.

Teaching Materials

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About the Creator of this Image

Anne M. Burns is a professor at Long Island University's C.W. Post campus. She received her Ph.D. in Mathematics from SUNY Stony Brook in 1976. Her research interests include discrete dynamical systems, scientific visualization, and using mathematics and computer graphics to describe nature.


Burns, A. Recursion in nature, mathematics and art.]

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