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.
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<div id=planttypes>[[Image:Planttypes.gif|thumb|center|380px|[[Anne_Burns'_Mathscapes#planttypes|Image 2]]. Several common inflorescences]]</div>
  
 
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.
 
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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{{{!}}border=0 cellpadding=10 cellspacing=10
 
{{{!}}border=0 cellpadding=10 cellspacing=10
 
{{!}}
 
{{!}}
<div id=planttypes>[[Image:Planttypes.gif|right|thumb|380px|[[Anne_Burns'_Mathscapes#planttypes|Image 2]]. Several common inflorescences]]</div>
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[Image:trees.gif|thumb|380px|[[Anne_Burns'_Mathscapes#trees|Image 3]]. Recursively created trees.]]</div>
{{!}}{{!}}
 
<div id=trees>[[Image:trees.gif|thumb|380px|[[Anne_Burns'_Mathscapes#trees|Image 3]]. Recursively created trees.]]</div>
 
 
{{!}}{{!}}
 
{{!}}{{!}}
<div id=purplebush>[[Image:Stringrewritingbush.gif|thumb|250px|[[Anne_Burns'_Mathscapes#purplebush|Image 4]]. Bush created using string-rewriting.]]</div>
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<div id=purplebush>[[Image:Stringrewritingbush.gif|thumb|right|250px|[[Anne_Burns'_Mathscapes#purplebush|Image 4]]. Bush created using string-rewriting.]]</div>
 
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<div id="stringplantsplusone">[[Image:Rewrite flowers .gif|center|frame|[[#stringplantsplusone|Image 6]]. The plants from [[#simplestringplants|Image 5]] after the replacement rule ''L'' &rarr; ''I(L)F'' has been applied once.]]</div>
 
<div id="stringplantsplusone">[[Image:Rewrite flowers .gif|center|frame|[[#stringplantsplusone|Image 6]]. The plants from [[#simplestringplants|Image 5]] after the replacement rule ''L'' &rarr; ''I(L)F'' has been applied once.]]</div>
  
Notice that the second image didn't change, because it's string didn't contain an ''L''. Also, it is important to note that this particular replacement rule could be carried out infinitely, since ''I(L)F'', the replacement string, contains ''L'', the original string. Several replacement rules can also be used in combination.
+
Notice that the second plant didn't change, because it's string didn't contain an ''L''. Also, it is important to note that this particular replacement rule could be carried out infinitely, since ''I(L)F'', the replacement string, contains ''L'', the original string. Several replacement rules can also be used in combination.
  
 
==Clouds & Mountains==
 
==Clouds & Mountains==

Revision as of 10:35, 14 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.

Image 2. Several common inflorescences

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:trees.gif


Why It's Interesting

Burns' images reveal the fractal geometry behind many natural phenomena. 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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