Harter-Heighway Dragon

(Difference between revisions)
 Revision as of 13:27, 29 June 2009 (edit)← Previous diff Revision as of 13:41, 1 July 2009 (edit) (undo)Next diff → Line 23: Line 23: 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 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$\sqrt{2}$ 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. 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. Line 35: Line 34: {{Hide|1= {{Hide|1= '''Perimeter''' {{hide|1= '''Perimeter''' {{hide|1= - [[Image:DragonCurve_basic.png|thumb|right|First iteration in detail]] + [[Image:DragonCurve_basic.png|left]] + + The perimeter of the Harter-Heighway curve increases by a factor of $\sqrt{2}$ for each iteration. 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: $\frac{2s\sqrt{2}}{2s} = \sqrt{2}$ + For example, if you look at the picture to the left, 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 isosceles triangles, the ratio of the first iteration over the base segment is: $\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \sqrt{2}$ }} }} '''Number of Sides''' {{hide|1= '''Number of Sides''' {{hide|1= + [[Image:DragonCurve_Sides.png|left]] 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. 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 $N_3 = 2^3 = 8\,$, which is consist with the number of sides appearing in third iteration in the image to the left. }} }}

Revision as of 13:41, 1 July 2009

Harter-Heighway Dragon Curve (3D- twist)
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).

Basic Description

First 5 iterations of the Harter-Heighway Curve

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.

15th iteration

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.

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

Perimeter

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 left, 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 isosceles triangles, the ratio of the first iteration over the base segment is: $\frac{s\sqrt{2} + s\sqrt{2}}{s + s} = \frac{2s\sqrt{2}}{2s} = \sqrt{2}$

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 $N_3 = 2^3 = 8\,$, which is consist with the number of sides appearing in third iteration in the image to the left.

Fractal Dimension

2nd iteration of the Harter-Heighway Dragon

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

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.

Changing the Angle

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

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

• An animation for the showing the fractal being drawn gradually through increasing iterations