Harter-Heighway Dragon
From Math Images
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An interesting property of this curve is that although the corners of the fractal seem to touch at various points, the curve never actually crosses over itself. Also, the curve exhibits ''self-similarity'' when iterated infinitely because as you look closer and closer at the curve, the curve continues to look like the larger curve. | An interesting property of this curve is that although the corners of the fractal seem to touch at various points, the curve never actually crosses over itself. Also, the curve exhibits ''self-similarity'' when iterated infinitely because as you look closer and closer at the curve, the curve continues to look like the larger curve. | ||
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| + | To learn another method to create the Harter-Heighway Dragon, click [http://sierra.nmsu.edu/morandi/coursematerials/JurassicParkFractal.html here] | ||
|Pre-K=No | |Pre-K=No | ||
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|ImageDesc= | |ImageDesc= | ||
==Properties== | ==Properties== | ||
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===Perimeter=== | ===Perimeter=== | ||
{{hide|1= | {{hide|1= | ||
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For example, if you look at the picture to the right, the straight red line shows the fractal as its base segment and the black crooked line shows the fractal at its first iteration. | For example, if you look at the picture to the right, the straight red line shows the fractal as its base segment and the black crooked line shows the fractal at its first iteration. | ||
| - | If the first iteration is split up into two | + | If the first iteration is split up into two triangles, the ratio of the perimeter of the first iteration over the base segment is: |
| + | ::<math>\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \frac{\sqrt{2}}{1}</math> | ||
}} | }} | ||
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Thus, the fractal dimension is <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, and it is a <balloon title="A space-filling curve in 2-dimensions is a curve with a fractal dimension of exactly 2. This means that the curve touches every point in the unit square."> space-filling curve</balloon>. | Thus, the fractal dimension is <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, and it is a <balloon title="A space-filling curve in 2-dimensions is a curve with a fractal dimension of exactly 2. This means that the curve touches every point in the unit square."> space-filling curve</balloon>. | ||
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|Field=Dynamic Systems | |Field=Dynamic Systems | ||
|Field2=Fractals | |Field2=Fractals | ||
| - | |FieldLinks= | + | |FieldLinks=:To read about an alternate method of creating the Harter-Heighway Dragon http://sierra.nmsu.edu/morandi/coursematerials/JurassicParkFractal.html |
|References= | |References= | ||
Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page] | Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page] | ||
Revision as of 13:21, 6 July 2009
| Harter-Heighway Dragon Curve |
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Contents |
Basic Description
This fractal is described by a curve that undergoes a repetitive process (called an iterated process). To begin the process, the curve has a basic segment of a straight line.Then at each iteration,
- Each line is replaced with two line segments at an angle of 90 degrees (other angles can be used to make fractals that look slightly different).
- Each line is rotated alternatively to the left or to the right of the line it is replacing.
The Harter-Heighway Dragon is created by iteration of the curve process described above, and is thus a type of fractal known as iterated function systems. This process can be repeated infinitely, and the perimeter or length of the dragon is in fact infinite. However, if you look to the image at the right, a 15th iteration of the Harter-Heighway Dragon is already enough to create an impressive fractal.
An interesting property of this curve is that although the corners of the fractal seem to touch at various points, the curve never actually crosses over itself. Also, the curve exhibits self-similarity when iterated infinitely because as you look closer and closer at the curve, the curve continues to look like the larger curve.
To learn another method to create the Harter-Heighway Dragon, click here
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Algebra
Properties
Perimeter
for each iteration.
For example, if you look at the picture to the right, the straight red line shows the fractal as its base segment and the black crooked line shows the fractal at its first iteration.
Number of Sides
) of the Harter-Heighway curve for any degree of iteration (k) is given by 
, where the "sides" of the curve refer to alternating slanted lines of the fractal.
.Fractal Dimension
.
.
, so that 
.
, and it is a
Changing the Angle
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
About the Creator of this Image
SolKoll is interested in fractals, and created this image using an iterated function system (IFS).
Related Links
Additional Resources
- To read about an alternate method of creating the Harter-Heighway Dragon http://sierra.nmsu.edu/morandi/coursematerials/JurassicParkFractal.html
References
Wikipedia, Wikipedia's Dragon Curve page Cynthia Lanius, Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal
Future Directions for this Page
An animation of the fractal being drawn gradually through increasing iterations (a frame for each individual iteration)
Also, an animation that draws the curve at the 13 or so iteration, but slowly to show that the curve never crosses itself.
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
