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What is topology?
Meet the topology
Topology is branch of geometry that deals with the properties of surfaces and general shapes but is unconcerned with the measurements of angles and lengths. High on the agenda are qualities that 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 the people that can not 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 place of the handle.
Coffee cup to donut
The most basic shapes studied by topologists are polyhedra ‘poly means many and hedra means faces). An example of a poly hedron is a cube, with 6 square faces ,8vertices (points at the junction of the faces) and 12edges (the lines joining the vertices). The cube is a regular polyhedron because:
- all the faces are the same regular shape,
- all the angles between edges meeting at a vertex are equal.
Topology is a relatively new subject, but it can still be traced back to the Greeks, and indeed the culminating result of Euclid’s Elements is to show that there are exactly five regular polyhedra. These are the Platonic solids:
- tetrahedron (with 4 triangular faces),
- cube (with 6 square faces),
- octahedron (with 8 triangular faces),
- dodecahedron (with 12 pentagonal faces),
- icosahedron (with 20 triangular faces).
Platonic solids and their properties
Euler’s formula is that the number of vertices V, edges E and faces F, of a polyhedron are connected by the formula: V – E + F = 2
For example, for a cube, V=8, E=12 and F=6 so V – E + F = 8 – 12 + 6 = 2 and, for buckminsterfullerene, V – E + F = 60 – 90 + 32 = 2. This theorem actually challenges the very notion of a polyhedron.
Classification of surfaces
A topologist might regard the donut and the coffee cup as identical but what sort of surface is different from the donut? A candidate here is the rubber ball. There is no way of transforming the donut into a ball since the donut has a hole but the ball does not. This is a fundamental difference between the two surfaces. So a way of classifying surfaces is by the number of holes they contain. Let’s take a surface with r holes and divide it into regions bounded by edges joining vertices planted on the surface. Once this is done, we can count the number of vertices, edges, and faces. For any division, the Euler expression V – E + F always has the same value, called the Euler characteristic of the surface:
V – E + F = 2 – 2r.
If the surface has no holes (r=0) as was the case with ordinary polyhedra, the formula reduces to Euler’s V–E+F=2. In the case of one hole (r=1), as was the case with the cube with a tunnel, V–E+F=0.
This is how Mobius strip looks like
Ordinarily a surface will have two sides. The outside of a ball is different from the inside and the only way to cross from one side to the other is to drill a hole in the ball– a cutting operation which is not allowed in topology (you can stretch but you can not cut). A piece of paper is another example of a surface with two sides. The only place where one side meets the other side is along the bounding curve formed by the edges of the paper.
The idea of a one-sided surface seems far-fetched. Nevertheless, a famous one was discovered by the German mathematician and astronomer August Möbius in the 19th century. The way to construct such a surface is to take a strip of paper, give it one twist and then stick the ends together. The result is a ‘Möbius strip’, a one-sided surface with a boundary curve. You can take your pencil and start drawing a line along its middle. Before long you are back where you started! It is even possible to have a one-sided surface that does not have a boundary curve. This is the ‘Klein bottle’ named after the German mathematician Felix Klein. What’s particularly impressive about this bottle is that it does not intersect itself. However, it is not possible to make a model of the Klein bottle in three-dimensional space without a physical intersection, for it properly lives in four dimensions where it would have no intersections. Both these surfaces are examples of what topologists call ‘manifolds’– geometrical surfaces that look like pieces of two-dimensional paper when small portions are viewed by themselves. Since the Klein bottle has no boundary it is called a closed 2-manifold.
Topology by Numberphile
The Poincare conjecture
For more than a century, an outstanding problem in topology was the celebrated Poincaré conjecture, named after Henri Poincaré. The conjecture centres on the connection between algebra and topology. The part of the conjecture that remained unsolved until recently applied to closed 3-manifolds. These can be complicated– imagine a Klein bottle with an extra dimension .Poincaré conjectured that certain closed3-manifolds which had all the algebraic hallmarks of being three-dimensional spheres actually had to be spheres. It was as if you walked around a giant ball and all the clues you received indicated it was a sphere but because you could not see the big picture you wondered if it really was a sphere. No one could prove the Poincaré conjecture for 3-manifolds. Was it true or was it false? It had been proven for all other dimensions but the 3-manifold case was obstinate.There were many false proofs, until in 2002 when it was recognized that Grigori Perelman of the Steklov Institute in St Petersburg had finally proved it. Like the solution to other great problems in mathematics, the solution techniques for the Poincaré conjecture lay out side its immediate area, in a technique related to heat diffusion.