Let’s begin with an n-dimensional ball. A homeomorphism \(f\) of the ball is a continuous map \[f:D^n \rightarrow D^n \] with a continuous inverse mapping. What a homeomorphism does is preserve topological structure, which means the image of a homeomorphism will have roughly the same relationship between points as the domain did. For example, in 2 dimensional space a square is homeomorphic to a disk, while a line is not homeomorphic to the unit circle. Let’s say we have 2 distinct homeomorphisms \(f\) and \(g\) of the n-dimensional ball, and we know that \(f|_{S^{n-1}}=g|_{S^{n-1}}\), or more plainly that the homeomorphisms agree with each other on the boundary of the ball. Then there is a result called Alexander’s Trick which states that there is a homotopy between these homeomorphisms.

A homotopy is best thought of as a function that creates an animation between two topological spaces. If a homotopy is a continuous deformation from a space \(X\) to a space \(Y\), then is defined as such:

\[ H: X \times [0,1] \rightarrow Y \]

The animation comes from viewing \(t\) as a time parameter that ranges from 0 to 1. At \(t=0\) the still of the video would display \(X\) and at \(t=1\) the still would show \(Y\).

Given the result of Alexander’s Trick, we now have a way of visualizing the continuous deformation from one homeomorphism of the disk to another. To make the situation tractable, lets consider a situation where the homeomorphism \(g\) is the identity on the disk.

For Alexander’s Trick to apply to this situation we know that the restriction of the homeomorphism \(f\) to the boundary must be the identity. In the 2 dimensional disk we can apply rotation by an angle \[\theta=\theta’(1-x^2-y^2)\] where \(\theta’\) is a fixed angle. Close to the origin this rotates points approximately \(\theta’\) degrees, but ranging over the points approaching the boundary we see rotation is steadily reduced until we reach the boundary where the set \[x^2+y^2=1\] is fixed.

Now we are left to apply the homotopy that Alexander’s Trick suggests, a function defining a homotopy between a homeomorphism of the disk whose boundary is the identity, and the identity on the disk. The function we apply is as follows:

\[ H(x,t)= \begin{cases} tf(x/t). & 0\leq ||x|| < t \\ x, & t \leq ||x|| \leq 1\\ \end{cases} \]

Now this is where the real fun comes in, we feed this function into a Mathematica notebook, define a grid on the disk so we can see what’s happening to the points, and the result is this:

As can be seen from the illustration, the result of applying the Alexander’s Trick homotopy is that starting at the boundary the image of the homeomorphism is scaled down by a factor of \(t\), squeezing down to a single point, leaving the image of the identity in the wake of this action.

The animation illustrates the resolution of homeomorphism to identity very nicely, but since we’re dealing with continuous mappings, and since we’re in 2 dimensional space, with a third dimension being introduced as a time parameter, there is another way of visualizing this where rather than \(t\) being a time parameter, it is instead a physical axis. Then we can create a sculpture, known as a kymograph, which can better illustrate the scaling down of the homeomorphism to a single point. Observe:

This tornado-esque sculpture gives a physical and tangible illustration of a functional topological result. Topology has a rich visual and physical intuition, and with technologies such as rendering software, Wolfram Mathematica, 3D printing, and laser cutting, we have the power to project abstract mathematical structures into the physical world, displaying their beauty, complexity, and simplicity.