The Fundamental Group of the Menger Sponge and the Towers of Hanoi

Hanspeter Fischer
Ball State University
Professor of Mathematics

“The classical fractal known as the “Menger sponge” is what remains of a solid cube after drilling   infinitely many (ever thinner) holes in each of the x, y, and z-direction, using a dense pattern. While it is topologically only 1-dimensional, its local structure is so complex that it contains a (topological) copy of every (separable metric) space of (topological) dimension 1.
Due to this universal property, it is desirable to have some detailed understanding of the algebraic characteristics of the Menger sponge’s geometry, as captured by Poincaré’s fundamental group (defined via the concatenation of loops). However, the group in question contains uncountably many elements and standard methods fail to model it.
In this talk, we present an explicit and systematic description of the fundamental group of the Menger sponge (along with a generalized Cayley graph) in terms of word sequences. Our word calculus requires only two letters and can be mechanically represented using a variation on the popular “Towers of Hanoi” puzzle. This is joint research with Andreas Zastrow (University of Gdańsk, Poland)”.

Light refreshments will be served @ 1:30pm EMS E424A.