# Harter-Heighway Dragon

(Difference between revisions)
 Revision as of 11:56, 25 June 2009 (edit)← Previous diff Revision as of 14:49, 25 June 2009 (edit) (undo)Next diff → Line 42: Line 42: '''Number of Sides''' {{hide|1= '''Number of Sides''' {{hide|1= - The number of sides 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. }} }} Line 49: Line 49: [[Image:DragonCurveDimension.png|right|thumb|225px|2nd iteration of the Harter-Heighway Dragon]] [[Image:DragonCurveDimension.png|right|thumb|225px|2nd iteration of the Harter-Heighway Dragon]] - The [[Fractal Dimension]] of the Harter-Heighway Curve can also be calculated. + The [[Fractal Dimension]] of the Harter-Heighway Curve can also be calculated using the equation: $\frac{logN}{loge}$

## Revision as of 14:49, 25 June 2009

Harter-Heighway Dragon Curve (3D- twist)
This image is an example of a Harter-Heighway Curve (also called Dragon Curve). This fractal 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.

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

## Properties

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 ($N_k$) 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.

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).

Reference used - Wikipedia, Wikipedia's Dragon Curve page
Reference used - Cynthia Lanius, Cynthia Lanius' Fractals Unit: A Jurassic Park Fractal

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