Thread by @QuantaMagazine

Most of us imagine the universe as extending forever in all directions. Does it have to be so? (Thread)
quantamagazine.org/what-is-the-ge…

After all, because of the planet’s very subtle curvature, everyone once thought that the Earth was flat. Now, of course, we know it’s shaped like a sphere.

Is our contemporary mental model of a vast, infinitely expansive universe similarly flawed?

In our new explainer, @EricaKlarreich and @LucyIkkanda visually explored three alternative geometries of the universe: flat, spherical and hyperbolic. Let’s learn more about each shape.

In school, most of us learned flat Euclidian geometry, in which the angles of a triangle add up to 180 degrees. A “flat” universe could be shaped like the surface of a doughnut (roll a flat sheet into a cylinder and bend it into a torus)

Life inside a two-dimensional torus would be like living in an infinite two-dimensional array of identical rectangular rooms. You would theoretically be able to see yourself far in the distance.

In a three-dimensional torus, life would be like living in an infinite three-dimensional array of identical cubic rooms.

What if the universe is spherical? Imagine you and a friend are two-dimensional creatures living on the surface of a sphere. As your friend strolls away from you, they’ll shrink relative to the size of your visual field.

But once your friend passes the equator, something strange happens. They’ll start looking bigger and bigger the farther away they walk from you. That’s because they’re taking up a larger percentage of your visual circle.

And if you’re a 3-D creature alone in a 3-D spherical universe, you’ll see yourself upside down, filling the entire backdrop of the sky.

There’s also the possibility of a hyperbolic universe. In a hyperbolic shape like a floppy hat or saddle, the sides open outward. Here, for example, is a distorted view of the hyperbolic plane known as the Poincaré disk:

In hyperbolic geometry, the circumference of a circle grows exponentially compared to the radius. We can see that exponential pileup in the masses of triangles near the boundary of the hyperbolic disk.

Mathematicians like to say that it’s easy to get lost in hyperbolic space. If your friend walks away from you, they will soon recede into an exponentially small speck.

Most cosmological measurements seem to favor the theory that we live in a “flat” universe, though we can’t say that definitively.
quantamagazine.org/what-shape-is-…

For more explainers like these, and for ongoing reporting on new developments in physics, mathematics, computer science, biology, and more, visit quantamagazine.org.
(/thread)