Harter-Heighway Dragon

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Revision as of 11:59, 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).

Basic Description

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

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

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Perimeter

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

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:

About the Creator of this Image

SolKoll is interested in fractals, and created this image using an iterated function system (IFS).