Thursday, June 4, 2009

Topology Question

Ken's mention of Menger Sponges brought up the association with the Cantor set and the Sierpinski triangle. There must be a name for a topological property of objects from which a congruent copy/s of itself can be removed and the remaining portions of the object are equivalent to the removed piece. The interval, cube, and triangle are all such objects. It seems to me that there is now way that the circle is. Togologists help!


Anonymous said...

Do you mean self-similarity?

Ken Colangelo said...

If you subtract triangle B from a larger enclosing triangle A, the remaining portion C isn't a triangle. C can always be subdivided into other triangles with the same shape as A. Tesselated?
Is that what you mean?
If so, maybe you could call that self-similarity?

Unknown said...

I'm not sure self-similarity is exactly it. The post is poorly worded. I meant 'similar' instead of 'congruent'. I think the best way to state the property I'm thinking about is that an object A can be tiled by a scaled copy of itself, say A'. Is that self-similarity?