Anne Burns' Mathscapes
- In her Mathscape images, Anne M. Burns combines recursive algorithms for clouds, mountains, and various imaginary plant forms into one picture.
Computers make it possible for Burns to "see" the beauty of mathematics. The artworks in her Mathscapes gallery 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
PlantsMany of the plants in Burns' Mathscapes are made using recursive replaceme [...]
Many of the plants in Burns' Mathscapes are made using recursive replacement rules similar to the one used to create the Blue Fern. Unlike the Blue Fern, however, Burns' plants aren't straight-forward fractals that are exactly self-similar into infinity. Instead, some randomization is built in, so that the growth looks more natural.
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 continue to grow, die, or 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 possibility for variation built into the program, their shapes are not identical.
Burns also uses string re-writing to create images of plants. In string-rewriting, also called an L-system, 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
The clouds and the mountains in these images have essentially the same math behind them. The idea is to create a two-dimensional height field, which can either be assigned colors to look like clouds or projected as a three-dimensional image into two-dimensional space to create mountains.
The first step is to create a grid, and assign values to the four corner points. Next, the center point is assigned to be the average of the four corners plus a random number. In each stage, the grid is further subdivided, and the points at each location are assigned values based on nearby points, the same way the center point was. Over time, the random numbers added to the heights are scaled down. The rate at which these numbers scale down affects the shape of the height field, and manipulating it can produce jagged mountains and clouds or rounded mountains and soft clouds.
The first couple stages of assigning values to the height field are shown below:
To turn this height field into mountains, first it is translated into three dimensions, by letting the assigned values at each point be the z-coordinate of that point. Then, using trigonometry and calculus, the orientation of the field is changed, and the field is projected onto the two-dimensional canvas.
Why It's Interesting
Burns' images are interesting because they are excellent examples of the close connections between math, art, and nature. The landscapes she's created are beautiful and natural, and it can be surprising to learn that they were created through mathematics and computer programming.
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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.
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