## A maths puzzle: Equivalent constructions of a fractal

Take an equilateral triangle and skip to the beginning of the next paragraph unless you need reminding how to make in into a Sierpinski Gasket… Split it into 4 equal triangles and remove the middle one. Repeat the process ad infinitum.

Now, you can get the same shape by taking the boundary of that triangle and adding the boundary of the middle triangle to it and then repeating that process ad infinitum.

How can one show that these two processes lead to the same shape? These two different ways of constructing a shape obviously lead to the same shape, and both could be given explicit Iterated Function Scheme expressions. One is a union of closed curves (lines) the other is an intersection of closed discs (triangles) How could one show that the two IFSs had the same limit? I haven’t really thought about this much. Maybe it is obvious if you write out the IFSs explicitly. But in the general case it might not be obvious. How would one check?

Incidentally, conservapedia is the best thing ever. If you were trying to make a “mock conservative” wikipedia clone, it wouldn’t be half as funny as the real thing… Also, GodTube

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