Difference between revisions of "Anne Burns' Mathscapes"

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[[Anne_Burns'_Mathscapes#planttypes|Image 2]] shows some basic plant formations similar to those that Burns used in her replacements. Each offshoot in this figure is replaced by a smaller version of the original image, which is again replaced by a smaller version, a process that repeats until the picture is 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.
 
[[Anne_Burns'_Mathscapes#planttypes|Image 2]] shows some basic plant formations similar to those that Burns used in her replacements. Each offshoot in this figure is replaced by a smaller version of the original image, which is again replaced by a smaller version, a process that repeats until the picture is 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 [[Anne_Burns'_Mathscapes#trees|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.
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The plant images can also be made to look more natural by using the deReffye method. In this method, at each iteration of the growth algorithm a branch can either continue to grow, die, or do nothing until the next iteration. Each option is assigned a certain probability, and the probabilities change over time. At later stages the branches are more likely to die, which means that plant eventually stops growing. The trees in [[Anne_Burns'_Mathscapes#trees|Image 3]] were created from the same program using the deReffye method. Because they're generated by the same algorithm, they look similar, but because of the possibility for variation built into the program, their shapes are not identical.
  
 
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Revision as of 15:29, 6 July 2011

Inprogress.png
Mathscape
Mathscape.gif
Field: Fractals
Image Created By: Anne M. Burns
Website: [1]

Mathscape

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.



Anne Burns' Mathscape images are natural looking landscapes whose components are created using math. The plants are generated through recursive algorithms similar to those used to make the plant-like fractal Blue Fern, and through string rewriting. The mountains and clouds are created using height fields (see below) and trigonometry.


A More Mathematical Explanation




Plants

Some of the algorithms Burns uses to create plants are similar to the fract [...]




Plants

Some of the algorithms Burns uses to create plants are similar to the fractal algorithms used to generate the Blue Fern. Unlike the Blue Fern, however, Burns' plants aren't perfect fractals that are exactly self-similar into infinity. Instead, some aspects of growth are randomized, so that the final image looks more natural.

Image 2 shows some basic plant formations similar to those that Burns used in her replacements. Each offshoot in this figure is replaced by a smaller version of the original image, which is again replaced by a smaller version, a process that repeats until the picture is 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.

The plant images can also be made to look more natural by using the deReffye method. In this method, at each iteration of the growth algorithm a branch can either continue to grow, die, or do nothing until the next iteration. Each option is assigned a certain probability, and the probabilities change over time. At later stages the branches are more likely to die, which means that plant eventually stops growing. The trees in Image 3 were created from the same program using the deReffye method. Because they're generated by the same algorithm, they look similar, but because of the possibility for variation built into the program, 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, 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. For example, Burns describes a system where the letter I represents a stem, the letter L represents a leaf, and the letter F represents a flower. Branches to the left are enclosed with by brackets, and branches to the right are enclosed by parentheses. Some simple plant images and the strings that describe them are shown in Image 5:

Image 5. Simple plants and their string translations.

At each stage, a set of substitution rules replaces each character with a string of other characters. The end result string is then translated back into a more complicated 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:

Heightfield.gif

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.

Image 8. Mountains in Spring.



Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

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.



References

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





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