Half of a Klein bottle with Möbius strip

Figure-8 Klein Bottle

Klein Bottle with two Möbius Bands

The two Möbius bands of a Klein bottle are connected by an ordinary two-sided
loop whose back and front sides are colored white and blue respectively

Möbius Band

The Möbius band is one-sided

Mirror image and orientability

If you move this face around the Möbius band, it returns as its mirror image (and upside-down). This means that the Möbius band is non-orientable.

Half of a Klein bottle with Möbius strip

Walking along the Möbius band allows to reach both sides of the Klein bottle.

Building a Klein Bottle

A Klein bottle is formed by joining two sides of a sheet to form a cylinder, then looping the ends of a cylinder back through itself in such a way that the inside (green) and outside (white) of the cylinder are joined

The Riemann Hypothesis is probably the hardest unsolved problem in all of mathematics, and one of the most important. It has to do with prime numbers - the building blocks of arithmetic. Nick Mee, together with Sir Arthur C. Clarke, tells us about the patterns hiding inside numbers.

One million dollars is waiting to be won by anyone who can solve one of the grand mathematical challenges of the 21st century. But be warned...these problems are hard. In the first of two articles, Chris Budd explains how to hit the bigtime.