# Topology

## Famous Topologists

Topological ideas can be traced back to Gottfried Wilhelm Leibniz (1646-1716), but three of the most famous figures in the development of topology are Möbius, Riemann, and Klein.

Augustus Ferdinand Möbius (1790-1868) is best known for his invention of the **Möbius strip**, which is a simple strip of **paper** that is twisted and connected so that it has only one side. Normally, cutting a strip of paper into a long, narrow **rectangle** and connecting the ends will result in a belt-like loop with two sides. A person cannot draw a single line with a pencil on both sides of the belt-like loop without crossing an edge. In constructing the Möbius strip, however, the strip of paper is twisted as it is looped, and the result is a one-sided object.

At first, this one-sided construction seems impossible, but if a person draws a straight, continuous line on the Möbius strip, the line will cover the entire length of both sides of the strip without ever crossing an edge, and it will return to its starting point in one long stroke.

Georg Friedrich Bernhard Riemann (1826-1866) developed some of the most important topological ideas about the stretching, bending, and twisting of surfaces, but he died prematurely at the age of 39 before he could expand significantly upon his ideas.

Felix Klein (1849-1925) is best known for the paradoxical figure which was named after him: the Klein bottle.

The Klein bottle is a one-sided object that has no edge. It is a tapered tube whose neck is bent around to enter the side of the bottle. The neck continues into the base of the bottle where it flares out and rounds off to form the outer surface of the bottle. Like the Möbius strip, any two points on the bottle can be joined by a continuous line without crossing an edge, which gives the impression that the inside and outside of the Klein bottle are continuous.

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