The Crooked, Complex Geometry of Round Trips

Have you ever ever questioned what life can be like if Earth weren’t formed like a sphere? We take as a right the sleek journey via the photo voltaic system and the seamless sunsets afforded by the planet’s rotational symmetry. A spherical Earth additionally makes it simple to determine the quickest method to get from level A to level B: Simply journey alongside the circle that goes via these two factors and cuts the sphere in half. We use these shortest paths, known as geodesics, to plan airplane routes and satellite tv for pc orbits.

However what if we lived on a dice as a substitute? Our world would wobble extra, our horizons can be crooked, and our shortest paths can be more durable to seek out. You won’t spend a lot time imagining life on a dice, however mathematicians do: They research what journey seems to be like on every kind of various shapes. And a recent discovery about spherical journeys on a dodecahedron has modified the way in which we view an object we’ve been taking a look at for 1000’s of years.

Discovering the shortest spherical journey on a given form may appear so simple as selecting a course and strolling in a straight line. Finally you’ll find yourself again the place you began, proper? Properly, it is dependent upon the form you’re strolling on. If it’s a sphere, sure. (And, sure, we’re ignoring the truth that the Earth isn’t an ideal sphere, and its floor isn’t precisely clean.) On a sphere, straight paths observe “nice circles,” that are geodesics just like the equator. When you stroll across the equator, after about 25,000 miles you’ll come full circle and find yourself proper again the place you began.

On a cubic world, geodesics are much less apparent. Discovering a straight path on a single face is straightforward, since every face is flat. However should you have been strolling round a cubic world, how would you proceed to go “straight” while you reached an edge?

There’s a enjoyable previous math drawback that illustrates the reply to our query. Think about an ant on one nook of a dice who needs to get to the other nook. What’s the shortest path on the floor of the dice to get from A to B?

You can think about plenty of totally different paths for the ant to take.

Illustration: Samuel Velasco/Quanta Journal

However which is the shortest? There’s an ingenious approach for fixing the issue. We flatten out the dice!

If the dice have been made from paper, you may reduce alongside the sides and flatten it out to get a “web” like this.

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