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# Ham sandwich theorem (easy to state, harder to motivate solution.  Possibly there's something better.

# Ham sandwich theorem (easy to state, harder to motivate solution.  Possibly there's something better.

A motivating problem: how can we prove that a sphere and a torus are different?

This is the sort of question that motivated the development of the field of topology for many years.  Recall that a [[torus]] is the shape of an inner tube, or the surface of a donut: it is a two-dimensional surface with one whole.  If a torus were made out of clay, one could stretch and squish it, but it is intuitively clear that no amount of stretching could possibly deform in into a sphere: it's simply impossible to get rid of the hole!  Similarly, a sphere made out of clay can't be deformed into a torus without ripping it.  The question of why the two shapes are not the same is fundamentally a topological one: we are interested in some intrinsic geometric properties of these objects, but not their specific rigid shapes.

 

It turns out that methods from [[algebra]] provide the easiest way to prove that it is impossible to deform a sphere into a donut.  Observe that a sphere has the following property: given any loop drawn on the sphere (say, a rubber band stretched around it somehow), it is possible to "contract" that loop to a point (i.e., to scrunch up the whole rubber band at one place).  On the other hand, if a rubber band is placed on a donut in such a way that it loops around the central whole, there is no way to contract it to a single point.  A crucial observation in the field of algebraic topology is that if a space can be obtained by squeezing and stretching a sphere, then it will still have the property that every loop is contractible.  Since a torus doesn't, we conclude that it must be fundamentally different from a sphere.  Arguments like this are the basis for the field of [[algebraic topology]], and this problem of contracting loops in a space is studied using the [[fundamental group]].