What Is Topology?

By Kanaga Rajan

You may think our exhibition, Taping Shape, is all fun, no science. Well, it certainly IS fun, but it’s an excellent study of topology. Topology is a field in math that studies the geometric and spatial properties of shapes that are maintained after deformations such as stretching or twisting.

What does that mean? Imagine an elastic sphere. What do you think happens if you stretched the sphere? You’d create an ellipsoid. What about if you pinch eight equidistant points on the same sphere and stretch? You’d create a cube. The amazing fact is that though you’ve transformed the sphere into two different shapes, you haven’t destroyed the surface of the original sphere! The surface was manipulated into entirely new shapes, but the continuous outer surface of the ellipsoid and cube are still the same as that of the sphere. Believe it or not, these objects are topologically the same!

When we think of geometry, we usually think of the math classes we took in school. Those classes focused on Euclidean geometry, which studies the angles and lengths of objects. Topology focuses on how an object is connected. In the example of the sphere turning into a cube, the new shapes are both topologically equivalent, or homeomorphic.

Another shape, called a torus, is a surface that’s made from two circles: one fixed circle and another that stands perpendicularly to the first, and moves through space along the path of the fixed circle to generate the torus. It creates a shape that is basically a donut. Try to imagine an elastic torus and try to deform it back into a sphere without breaking the surface … It’s not possible! A torus is not homeomorphic to the sphere. For objects to be topologically related, they can only be deformed by non-destructive transformations.

Though topology seems like an abstract type of geometry, it's widely applicable in different sciences outside of strict mathematics. In biology, a branch of topology called knot theory is used to estimate properties of the enzymes that unknot DNA. Topology is also used to understand and describe the shape of the universe. Additionally, topology is applicable to physics and computer science.

If you’re interested in seeing topology in action, check out Taping Shape, an interactive exhibition open through September 11, 2016, at the Fleet!