Wednesday, October 4, 2023

The complex and twisted geometry of the round trips

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Have you already I wondered what life would be like if the Earth wasn’t shaped like a sphere? We take for granted the smooth ride through the solar system and the seamless sunsets offered by the rotational symmetry of the planet. A Round Earth also makes it easy to determine the fastest way to get from a point A point B: You just have to go through the circle that passes through these two points and cut the sphere in half. We use these shortest paths, called geodesics, to plan airplane routes and satellite orbits.

But what if we instead lived on a cube? Our world would waver more, our horizons would be twisted and our shortest paths would be harder to find. You might not spend a lot of time imagining life on a cube, but mathematicians do: They study what travel looks like on all kinds of different shapes. And one recent discovery on back and forth on a dodecahedron has changed the way we see an object that we have been looking at for thousands of years.

Finding the shortest round trip on a given shape can seem as easy as choosing a direction and walking in a straight line. Eventually you’ll end up coming back to where you started, right? Well, it depends on what shape you’re walking on. If it’s a sphere, yes. (And, yes, we ignore the fact that the Earth is not a perfect sphere and its surface is not exactly smooth.) On a sphere, straight paths follow “great circles,” which are geodesics like the equator. If you walk around the equator, after about 25,000 miles you will make a full turn and end up where you started.

On a cubic world, the geodesics are less obvious. Finding a straight path on one side is easy because each side is flat. But if you were walking in a cubic world, how would you keep going “straight” when you reached an edge?

There is a fun old math problem that illustrates the answer to our question. Imagine an ant on a corner of a cube that wants to go to the opposite corner. What is the shortest path from the surface of the cube? A at B?

You can imagine many different paths for the ant.

Illustration: Samuel Velasco / Quanta Magazine

But which is the shortest? There is an ingenious technique to solve the problem. We flatten the cube!

If the cube was made out of paper, you could cut along the edges and flatten it to make a “net” like this.


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