23 Topology
Topology is the branch of geometry that deals with the properties of surfaces and
general shapes but is unconcerned with the measurement of lengths or angles. High on
the agenda are qualities which do not change when shapes are transformed into other
shapes. We are allowed to push and pull the shape in any direction and for this reason
topology is sometimes described as ‘rubber sheet geometry’. Topologists are people
who cannot tell the difference between a donut and a coffee cup!
A donut is a surface with a single hole in it. A coffee cup is the same, where
the hole takes the form of the handle. Here’s how a donut can be transformed
into a coffee cup.
Classifying polyhedra
The most basic shapes studied by topologists are polyhedra (‘poly’ means
‘many’ and ‘hedra’ means ‘faces’). An example of a polyhedron is a cube, with 6
square faces, 8 vertices (points at the junction of the faces) and 12 edges (the
lines joining the vertices). The cube is a regular polyhedron because: