Harter-Heighway Dragon

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|ImageName=Harter-Heighway Dragon Curve (3D- twist)
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|ImageName=Harter-Heighway Dragon Curve
|Image=DragonCurve.jpg
|Image=DragonCurve.jpg
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|ImageIntro=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).
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|ImageIntro=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).
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[[Image:DragonCurve_Construction.png|600px|thumb|First 5 iterations of the Harter-Heighway Curve|left]]
 
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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 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.
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Then at each iteration,
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:*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).
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:*Each line is rotated alternatively to the left or to the right of the line it is replacing.
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[[Image:DragonCurve_Construction.png|900px|thumb|Base Segment and First 5 iterations of the Harter-Heighway Curve|center]]
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[[Image:DragonCurve_int15.gif|thumb|200px|right|15th iteration]]
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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 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.
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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 look closer and closer at the curve, the magnified parts of the curve continue 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]
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|Elementary=No
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|MiddleSchool=Yes
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==Properties==
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The Harter-Heighway Dragon curve has several different properties we can derive.
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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.
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===Perimeter===
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[[Image:DragonCurve_int15.gif|thumb|200px|right|15th iteration]]
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[[Image:DragonCurve_basic.png|thumb|right|1st iteration of the Harter-Heighway Dragon]]
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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.
 
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The perimeter of the Harter-Heighway curve increases by<math>\sqrt{2}</math> with each repetition of the curve.
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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 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.
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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.
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If the first iteration is split up into two triangles, the ratio of the perimeter of the first iteration over the base segment is:
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::<math>\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \frac{\sqrt{2}}{1}</math>
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===Number of Sides===
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[[Image:DragonCurve_Sides.png|900px|center]]
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The number of sides (<math>N_k</math>) 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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|ImageDesc====Perimeter===
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[[Image:DragonCurve_basic.jpg|thumb|left|250px|First iteration in detail]]
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For example, the third iteration of this curve should have a total number of sides <math>N_3 = 2^3 = 8\,</math>.
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The perimeter of the Harter-Heighway curve increases by a factor of <math>\sqrt{2}</math> for each iteration.
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===Fractal Dimension===
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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{2\sqrt{2}}{2s} = \sqrt{2}</math>
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The [[Fractal Dimension]] of the Harter-Heighway Curve can also be calculated using the equation: <math>\frac{logN}{loge}</math>.
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Let us use the second iteration of the curve as seen below to calculate the fractal dimension.
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[[Image:DragonCurveDimension1.png|center]]
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There are two new curves that arise during the iteration so that <math>N = 2\,</math>.
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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>.
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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>.
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==Changing the Angle==
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===Number of Sides===
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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 following fractals are the result of 13 iterations.
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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>.
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<gallery caption="" widths="305px" heights="205px" perrow="3">
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===Fractal Dimension===
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The [[Fractal Dimension]] of the Harter-Heighway Curve can be calculated to be: <math>\frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 </math>, so it is a space-filling curve.
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===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:
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<gallery caption="" widths="300px" heights="200px" perrow="3">
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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'''
Image:CurveAngle110.jpg|'''Curve with angle 110'''
Image:CurveAngle110.jpg|'''Curve with angle 110'''
</gallery>
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|other=Algebra
|other=Algebra
|AuthorName=SolKoll
|AuthorName=SolKoll
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|Field=Dynamic Systems
|Field=Dynamic Systems
|Field2=Fractals
|Field2=Fractals
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|FieldLinks=:To read about an alternate method of creating the Harter-Heighway Dragon http://sierra.nmsu.edu/morandi/coursematerials/JurassicParkFractal.html
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|References=
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Wikipedia, [http://en.wikipedia.org/wiki/Dragon_curve Wikipedia's Dragon Curve page]
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Cynthia Lanius, [http://math.rice.edu/~lanius/frac/jurra.html Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal]
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|ToDo=
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An animation of the fractal being drawn gradually through increasing iterations (a frame for each individual iteration)
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Also, an animation that draws the curve at the 13 or so iteration, but slowly to show that the curve never crosses itself.
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|HideMME = No
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Current revision


Harter-Heighway Dragon Curve
Fields: Dynamic Systems and Fractals
Image Created By: SolKoll
Website: Wikimedia Commons

Harter-Heighway Dragon Curve

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).


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.


Base Segment and First 5 iterations of the Harter-Heighway Curve
Base Segment and First 5 iterations of the Harter-Heighway Curve
15th iteration
15th iteration


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 look closer and closer at the curve, the magnified parts of the curve continue 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

The Harter-Heighway Dragon curve has several different properties we can derive.

Perimeter

1st iteration of the Harter-Heighway Dragon
1st iteration of the Harter-Heighway Dragon


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

If the first iteration is split up into two triangles, the ratio of the perimeter of the first iteration over the base segment is:

\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \frac{\sqrt{2}}{1}

Number of Sides

The number of sides (N_k) of the Harter-Heighway curve for any degree of iteration (k) is given by N_k = 2^k\,, where the "sides" of the curve refer to alternating slanted lines of the fractal.

For example, the third iteration of this curve should have a total number of sides N_3 = 2^3 = 8\,.


Fractal Dimension

The Fractal Dimension of the Harter-Heighway Curve can also be calculated using the equation: \frac{logN}{loge}.

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 N = 2\,.

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

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

Changing the Angle

The Harter-Heighway curve iterates with a 90 degree angle; however, if the angle is changed, new curves can be created. The following fractals are the result of 13 iterations.




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.




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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