# Harter-Heighway Dragon

### From Math Images

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Also, the ratio of the lengths of each new curve to the old curve is: <math>\frac{4(s\sqrt{2})}{4(s)} = \sqrt{2} </math>, so that <math>e = \sqrt{2}</math>. | Also, the ratio of the lengths of each new curve to the old curve is: <math>\frac{4(s\sqrt{2})}{4(s)} = \sqrt{2} </math>, so that <math>e = \sqrt{2}</math>. | ||

- | Thus, the fractal dimension is <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, and it is a space-filling curve. | + | 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</ballono>. |

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## Revision as of 14:25, 1 July 2009

{{Image Description |ImageName=Harter-Heighway Dragon Curve |Image=DragonCurve.jpg |ImageIntro=This image is an artistic rendering of the Harter-Heighway Curve (also called the Dragon Curve), which is a fractal. This curve 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). |ImageDescElem=

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. 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 the curve never crosses itself. 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.

|Pre-K=No |Elementary=No |MiddleSchool=Yes |HighSchool=Yes |ImageDesc=

## Contents |

## Properties

{{Hide|1=

### Perimeter

### Number of Sides

### Fractal Dimension

{{hide|1= The Fractal Dimension of the Harter-Heighway Curve can also be calculated using the equation: .

Let us use the second iteration of the curve as seen below to calculate the fractal dimension.

There are two new curves that arise during the iteration so that .

Also, the ratio of the lengths of each new curve to the old curve is: , so that .

Thus, the fractal dimension is , and it is a space-filling curve

}}

|other=Algebra |AuthorName=SolKoll |AuthorDesc=SolKoll is interested in fractals, and created this image using an iterated function system (IFS). |SiteName=Wikimedia Commons |SiteURL=http://commons.wikimedia.org/wiki/File:Harter-Heighways_dragon_curve_(3D_twist).jpg |Field=Dynamic Systems |Field2=Fractals |FieldLinks= |References= Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page] Cynthia Lanius, [http://math.rice.edu/~lanius/frac/jurra.html Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal] |ToDo=*An animation for the showing the fractal being drawn gradually through increasing iterations |HideMME = No }}