In early eighteenth-century Königsberg, Prussia, a river ran through the city. There were two small islands in the river, connected to each other and to the banks by seven bridges.
Someone posed a question: how could a pedestrian cross all seven bridges exactly once, without repeating or missing any, and finally return to their starting point? Later, the mathematician Euler transformed it into a one-stroke drawing problem, like the five-pointed stars many children often drew when they were young.
Euler pioneered and created graph theory, an immensely important and far-reaching field of mathematical study. Further research and additions in many areas eventually gave rise to topology.
In the nineteenth century, two German mathematicians discovered the Möbius strip, a typical topological shape. Take a long strip of paper, twist one section 180 degrees, then join the two ends together. Draw a line along it with a pen, and you would find that the line could cover the entire surface.
A Möbius strip looked three-dimensional, but it was actually one infinite surface, so it was two-dimensional. Creating one required twisting space. Human left and right hands were extremely alike, yet fundamentally different. It was impossible to fit a left glove perfectly over a right hand, or a right glove perfectly over a left hand, no matter how one twisted them. But if the gloves were moved onto a Möbius strip, the problem became easy to solve.
Because it was a non-orientable surface, left and right had no meaning on it. To distinguish left from right, one first had to establish orientation. Assuming oneself as the center, when facing forward, left was left and right was right. While Phoebe could still hear the people outside speaking, Filius told her to raise her left hand.
Telling left from right was second nature to some