How to Create a Sierpinsky Triangle

What is the Sierpinsky Triangle?

The Sierpinski triangle is a fractal with the general shape of an equilateral triangle that has been subdivided recursively into smaller equilateral triangles. There are infinitely many different stages of the Sierpinski triangle, depending on how often you would like to subdivide the original triangle.

Task

Create the first four stages of the Sierpinsky triangle.

Instructions

1.		Activate the Regular Polygon tool and create an equilateral triangle.
2.		Use the Midpoint or Center tool in order to create the midpoints of the triangle's sides.
3.		Use the Polygon tool in order to create a small triangle inside the equilateral triangle by using point A and the two midpoints next to it as the vertices of the new triangle.
4.		In the same way, create two more triangles for vertices B and C respectively.
		Note: You have now created stage 1 of the Sierpinsky triangle.

Instructions (continued)

Note: We have created a custom tool called

Sierpinsky Triangle tool for you in order to simplify constructing the next stages for you. This tool recreates instruction steps 2 to 4 for you, i.e. constructing the midpoints as well as the additional three triangles in one instance.

5.		Activate the Sierpinsky Triangle tool.
6.		With the Sierpinsky Triangle tool activated, successively select each of the smaller triangles you created in steps 3 and 4.
		Note: You have now created stage 2 of the Sierpinski triangle.
7.		Create stage 3 of the Sierpinski triangle.
		Hint: You might want to use the Zoom In tool in order to reach the small triangles more easily.
8.		Create stage 4 of the Sierpinski triangle.

What is a Custom Tool?

GeoGebra allows you to create custom tools that are not part of the default GeoGebra Graphics Toolbar. By providing a custom tool, certain steps of a constructions may become less tedious to recreate or new types of objects can be created on the fly (e.g. a rectangle).