One way to guarantee self-similarity is to build a shape by applying the same process over smaller and smaller scales. This idea can be realized with a process called initiators and generators.
| The initiator is the starting shape. |
| The generator is a collection of scaled copies of the initiator. |
| The rule is this: in the generator, replace each copy of the initiator with a scaled copy of the generator (specifying orientations where necessary). |
Examples
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Sierpinski gasket How can we turn "connect the midpoints and remove the middle triangle" into initiators and generators? |
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Koch curve Tents upon tents upon tents ... makes a shape we shall see is very strange, a curve enclosed in a small box and yet that is infinitely long. |
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Cantor set Cut all the tents out of the Koch curve and we are left with something that appears to be little more than holes. But we can be fooled by appearances. |
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Spinning gaskets Rotate the lower left third of the gasket and ... the lower left third of every piece rotates, each relative to the piece that contains it. |
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Fractal trees Simple binary branching generates some surprises. |
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Branching trees Here is Jeff Sorbo's software to animate binary branching trees |
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L-Systems A variation of initiators and generators originally developed to model the growth of plants, but also developed thousands of years ago in India as a method of decoration. |