Topology lends itself to some interesting animations, which I found on the web. Take for example a look at the animation below.
A mug and a doughnut have the same topology, since it's only dependent
on the number of holes.
If you would like to role up two dimensions, this could be accomplished in many ways. From a piece of paper one may produce a sphere, a donut or a lemniscate with three holes. These are all two-dimensional surfaces with a different topology: they all posses a different number of holes, namely zero, one or three.
One of the most famous objects in topology is Möbius Ring.
It is a surface with one edge and one side!
There is a magnificent story of the brilliant physicist Sir Richard Philip
Feynman who conquered the charms of the most attractive girl on campus
by disproving the teacher of philosophy that everything in the world
has two sides to it.
So it happened that this girl later became his first wife!
Klein's bottle is an object without any edges and just one side, as opposed to a sphere which, although it has no edges either, possesses two sides (inside and outside, of course).
Klein's bottle may be obtained by glueing the open edges of the Möbius strip together. Maybe a bit less obvious, this cannot be accomplished in three dimensions. In three dimensions, Klein's bottle has to intersect itself. This intersection is resolved in four dimensions, very similar to a string that crosses itself may be "untangled" by lifting it a bit from its plane at the intersection.
Can you turn a sphere inside out without cutting and pasting, and without making (infinitely) sharp edges? For the answer, watch the animation below. It is entertaining and educating at the same time, and represents a piece of true animation art as well.
