Harter-Heighway Dragon

From Math Images

(Difference between revisions)
Jump to: navigation, search
Line 66: Line 66:

Revision as of 15:18, 22 May 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
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: *Geometry


First iteration in detail
First iteration in detail

The per [...]


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{2\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

The Fractal Dimension of the Harter-Heighway Curve can be calculated to be: \frac{logN}{loge} = \frac{log2}{log\sqrt{2}} = 2 , so it is a space-filling curve.


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

Image:CurveAngle85.jpg Image:CurveAngle100.jpg Image:CurveAngle110.jpg

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.

Personal tools