Topology concerns geometric concepts such as distance, connectedness, and continuity, but in a fluid, flexible way. This can be contrasted with classical geometry which focuses on the geometry of rigid properties. The subject of topology can be developed in a beautifully abstract approach that has great simplicity, great generality, and yet great power. It has important applications to a host of fields, including cosmology (the shape of space), robotics, chaos, knot theory, and even internet searching.

In the popular consciousness, topology is linked with the illusive and sometimes surprising properties of flexible structures.  For example, topology tells us that the surface of a coffee cup can be deformed into a doughnut, but not into a soccerball.  The doughnut or torus and sphere are standard examples in topology, along with the Mobius band and the Klein bottle, illustrated below.  The Klein bottle, in particular, is famous as an example of an object that can only really exist in four (or more) dimensions, a fact that had tragic consequences for one unfortunate student.

Another surprising result from topology is illustrated below.  It states that the linked rings in the first figure can be unlinked to form the second, without cutting or tearing the material.  Can you figure out how to do it?  You are allowed to stretch, shrink, twist, and deform the material in any manner, as if it were infinitely flexible rubber or play dough.

For the solution, click here.