Harter-Heighway Dragon

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|HighSchool=Yes
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|ImageDesc====Perimeter===
|ImageDesc====Perimeter===
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[[Image:DragonCurve_basic.png|thumb|left|250px|First iteration in detail]]
[[Image:DragonCurve_basic.png|thumb|left|250px|First iteration in detail]]
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===Number of Sides===
===Number of Sides===
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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>.
The number of sides of the Harter-Heighway curve for any degree of iteration (''k'') is given by <math>N_k = 2^k\,</math>.
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===Fractal Dimension===
===Fractal Dimension===
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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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===Angle===
===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:
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Image:CurveAngle110.jpg|'''Curve with angle 110'''
Image:CurveAngle110.jpg|'''Curve with angle 110'''
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|other=Algebra
|other=Algebra

Revision as of 12:57, 3 June 2009


Harter-Heighway Dragon Curve (3D- twist)
The image above is an example of a Harter-Heighway Curve (also called Dragon Curve). This fractal was first described in 1967 by American Martin Gardner 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)
Fields: Dynamic Systems and Fractals
Created By: SolKoll

Contents

Basic Description

First 5 iterations of the Harter-Heighway Curve
First 5 iterations of the Harter-Heighway Curve







The curve itself is fairly simple, with 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.

15th iteration
15th iteration


The Harter-Heighway Dragon is created by iteration of the curve described above. The curve can be repeated infinitely, so that 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\sqrt{2} 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

Perimeter

<span class="_togglegroup _toggle_initshow _toggle _toggler toggle-visible" style="di [...]

Perimeter

First iteration in detail
First iteration in detail


The perimeter of the Harter-Heighway curve increases by a factor of \sqrt{2} for each iteration.

For example, if the first iteration is split up into two isosceles triangles, the ratio between the base segment and first iteration is: \frac{2s\sqrt{2}}{2s} = \sqrt{2}


Number of Sides

The number of sides of the Harter-Heighway curve for any degree of iteration (k) is given by N_k = 2^k\,.


Fractal Dimension

2nd iteration of the Harter-Heighway Dragon
2nd iteration of the Harter-Heighway Dragon

The Fractal Dimension of the Harter-Heighway Curve can also be calculated.


As seen from the image of the second iteration of the curve, there are two new curves that arise during the iteration and N = 2.

Also, the ratio of the lengths of each new curve to the old curve is: \frac{2s\sqrt{2}}{2s} = \sqrt{2} , so e = \sqrt{2}.


Thus, the fractal dimension is \frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 , and it is a space-filling curve.


Angle

The Harter-Heighway curve iterates with a 90 degree angle. However, if the angle is changed, new curves can be created:

Curve with angle 85

Curve with angle 100

Curve with angle 110




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).







If you are able, please consider adding to or editing this page!

Have questions about the image or the explanations on this page?
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.






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