Harter-Heighway Dragon
From Math Images
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- | This fractal is described by a <balloon title="A crooked line, like the the one used in this fractal, can be considered a | + | This fractal is described by a <balloon title="A crooked line, like the the one used in this fractal, can be considered a curve."> curve</balloon> that undergoes a repetitive process (called an [[Iterated Functions|iterated process]]). To begin the process, the curve has a basic segment of a straight line. |
- | Then at each iteration | + | 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 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 | + | :*Each line is rotated alternatively to the left or to the right of the line it is replacing. |
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- | 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 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 | + | 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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The Harter-Heighway curve iterates with a 90 degree angle. However, if the angle is changed, new curves can be created: | The Harter-Heighway curve iterates with a 90 degree angle. However, if the angle is changed, new curves can be created: | ||
- | <gallery caption="" widths=" | + | <gallery caption="" widths="305px" heights="205px" perrow="3"> |
Image:CurveAngle85.jpg|'''Curve with angle 85''' | Image:CurveAngle85.jpg|'''Curve with angle 85''' | ||
Image:CurveAngle100.jpg|'''Curve with angle 100''' | Image:CurveAngle100.jpg|'''Curve with angle 100''' |
Revision as of 11:57, 6 July 2009
- This image is an artistic rendering of the Harter-Heighway Curve (also called the Dragon Curve), which is a fractal. It is often referred to as the Jurassic Park Curve because it garnered popularity after being drawn and alluded to in the novel Jurassic Park by Michael Crichton (1990).
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.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Algebra
Properties
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).
References
Wikipedia, Wikipedia's Dragon Curve page Cynthia Lanius, Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal
Future Directions for this Page
- An animation for the showing the fractal being drawn gradually through increasing iterations
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.