Smaller on the inside

This week, I’ve been reading about the hyperbolic plane. I became curious about this peculiar concept in mathematics after reading a short parody of Alice’s Adventures in Wonderland, in which the Mad Hatter’s tea party took place at a hyperbolic table.

The basic idea is that the distances in a hyperbolic plane are longer when they are closer to the boundary. When we illustrate this in the normal Euclidian geometry that everyone learns in school, a regular shape such as a regular pentagon looks distorted and irregular, since the side nearest the boundary will appear much smaller to make up for the fact that distances near the edge are actually much larger.

Timothy Gowers illustrates this in his Mathematics: A Very Short Introduction with a map of the Earth. Although the Earth is spherical, the maps we use are flat. This means that places near the poles of the Earth, such as Greenland, appear a lot larger than they really are. There is still a difference between perceived and actual distance, but this time the difference is in the opposite direction: the actual distance is smaller than it looks near the boundary.
The Earth analogy also helps us to see why a pentagon drawn on a hyperbolic disc would appear to have curved sides. When an aeroplane travels from the UK to Canada, it flies over the Arctic because this is the shortest distance between the two places that is above the Earth’s surface. However, when we draw the path of this aeroplane onto our flat map, it appears to curve upwards. This is because on the flat map, distances are shorter than they appear the closer we are to the poles, so although the path upwards appears to be longer than the line across the ocean between the UK and Canada, this is compensated for by the fact that further up the curve represents a shorter distance than it appears to.
We define a straight line as the shortest distance between two points. On the Earth’s surface, the shortest distance between two points is a curved line; this also applies to the hyperbolic disk. A pentagon’s straight sides will therefore appear to curve inwards, just like the plane appeared to fly in an outwards curve over the Arctic. Distances on a hyperbolic disk appear larger than they really are towards the centre, so the shortest path between the corners of a pentagon appears to bend towards the centre of the circle, because the distances are smaller on the inside.

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