fractals : general fractals : fractals by replacement (part 1)

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Fractals
	•	Introduction

General Fractals
	•	Fractals By Replacement • part 1 • part 2
	•	Fractals in Crystals
	•	Natural Fractals Activity

Bifurcations and Branching
	•	The Shortest Distance between 91 points
	•	Branching Networks

Hausdorf dimension
	•	Rocks as Fractals
	•	The Dimension of Shorelines

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Materials: • pencils (colored pencils make this fun), • straight edge, • paper.

Background: One way to make a fractal is to use "replacement". To do this, we make the fractal in a series of steps, and at each step we replace certain shapes by other shapes. For example, we could use the replacement scheme that looks like this:

 This replaces a right-side-up triangle with a grouping of 4 triangles, and leaves up-side-down triangles alone. Suppose at the first step we have a right-side-triangle. Then the next couple of steps look like this:

 

and eventually we'd get the Sierpinski triangle. (Notice, however, that if we started with an upside down triangle, using this replacement scheme would give us back the same triangle over and over again.)

Activity: There are 6 replacement excerises below. Work with a friend to iterate the process for one of these replacement schemes. You might want to try playing with color in addition to shape.

Start with a single white square, and use the replacement scheme below.

Start with a single black square, and use the replacement scheme below.

 

Continue to (part 2)