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Sierpinski Gaskets


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Fractals are characterised by their property of self-similarity, they are both similar to themselves in all regions and similar to themselves on different length scales, so that an enlargement of a fractal looks the same as the original.

The Polish mathematician Waclaw Sierpinski discovered a famous fractal structure now known as the Sierpinski gasket. Sierpinski's method of construction was to start with an equilateral triangle, then remove the central triangle to leave three smaller triangles.

Each of these triangles is a smaller version of the original triangle. The central triangle is then removed from each of the smaller triangles to leave even smaller equilateral triangles. This process can be continued indefinitely. The end result is the Sierpinski Gasket, which is composed of a dust of disconnected points.

A similar procedure may be applied to a regular tetrahedron. Removing an octahedron from the centre of the tetrahedron leaves four smaller tetrahedra. Octahedra may then be removed from each of these smaller tetrahedra. Continuing indefinitely produces a tetrahedral Sierpinski gasket.

The above animation is derived from Pascal's Triangle, as shown on the website: PascGalois Triangles and Hexagons and other Group-related Cellular Automata
Web Link: http://faculty.ssu.edu/~kmshanno/pascal/

Text and Animated Julia Set and Tetrahedral Gaskets by Nick Mee from 'Art and Mathematics' CD-ROM Published by Virtual Image