1.E. Iterated Function Systems

Invariance of the Gasket

Note that applying all three of these transformations to the gasket gives the gasket again.

Click the gasket picture to see the effect of applying these transformations to the gasket.

That is, the gasket is invariant under the simultaneous application of these three transformations.

In fact, the gasket is the only (compact) shape left invariant.

What happens if we apply these transformations to some shape other than the gasket?

What happens if we apply these transformations to the resulting shape?

What happens if we iterate this process? Click the drawing for an animation, or try your own drawing in the deterministic IFS software.

On the left is an instance of this idea applied to a sketch of my cat, on the right to a sketch of my brother. Click each picture to see the process applied. Click the last picture in the sequence to return to the first.

We observe a sequence of pictures that converges to the gasket, independently of the starting shape. With a scanner and imagination, some mischief can be achieved.

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