Harter-Heighway Dragon
From Math Images
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- | {{hide|1= | + | ==Properties== |
- | [[Image:DragonCurve_basic.png|thumb| | + | {{Hide|1= |
- | + | '''Perimeter''' {{hide|1= | |
- | + | [[Image:DragonCurve_basic.png|thumb|right|First iteration in detail]] | |
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The perimeter of the Harter-Heighway curve increases by a factor of <math>\sqrt{2}</math> for each iteration. | The perimeter of the Harter-Heighway curve increases by a factor of <math>\sqrt{2}</math> for each iteration. | ||
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For example, if the first iteration is split up into two isosceles triangles, the ratio between the base segment and first iteration is: <math>\frac{2s\sqrt{2}}{2s} = \sqrt{2}</math> | For example, if the first iteration is split up into two isosceles triangles, the ratio between the base segment and first iteration is: <math>\frac{2s\sqrt{2}}{2s} = \sqrt{2}</math> | ||
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- | + | '''Number of Sides''' {{hide|1= | |
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The number of sides of the Harter-Heighway curve for any degree of iteration (''k'') is given by <math>N_k = 2^k\,</math>, where the "sides" of the curve refer to alternating slanted lines of the fractal. | The number of sides of the Harter-Heighway curve for any degree of iteration (''k'') is given by <math>N_k = 2^k\,</math>, where the "sides" of the curve refer to alternating slanted lines of the fractal. | ||
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- | + | '''Fractal Dimension''' {{hide|1= | |
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[[Image:DragonCurveDimension.png|right|thumb|225px|2nd iteration of the Harter-Heighway Dragon]] | [[Image:DragonCurveDimension.png|right|thumb|225px|2nd iteration of the Harter-Heighway Dragon]] | ||
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- | == | + | ==Changing the Angle== |
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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: |
Revision as of 11:56, 25 June 2009
- This image is an example of a Harter-Heighway Curve (also called Dragon Curve). This fractal is an iterated function system and 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 (3D- twist) |
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Contents |
Basic Description
This fractal is described by a curve that undergoes an iterated process. To begin the process, the curve starts out as a line as the base segment. Each iteration replaces each line with two line segments at an angle of 90 degrees (other angles can be used to make various looking fractals), with each line being rotated alternatively to the left or to the right of the line it is replacing. To learn more about iterated functions, click here.
The Harter-Heighway Dragon is created by iteration of the curve process described above. This process can be repeated infinitely, and the perimeter 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.
The perimeter of the Harter-Heighway curve increases by with each repetition of the curve.
An interesting property of this curve is that the curve never crosses itself. Also, the curve exhibits self-similarity 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).
Related Links
Additional Resources
- Reference used - Wikipedia, Wikipedia's Dragon Curve page
- Reference used - 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.