What if the universe was hyperbolic?

A simply connected Euclidean or hyperbolic universe would indeed be infinite. But the universe might instead be “multiply connected,” like a torus, in which case there are many different such paths.

What would a hyperbolic universe look like?

A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood.

How does hyperbolic geometry differ from Euclidean geometry?

In hyperbolic geometry, two parallel lines are taken to converge in one direction and diverge in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles.

Is space a non Euclidean?

Non-Euclidean geometry is only applicable to space.

What kind of curvature geometry of space does the universe have if the universe is closed?

sphere
If the universe’s density is great enough for its gravity to overcome the force of expansion, then the universe will curl into a ball. This is known as the closed model, with positive curvature resembling a sphere. A mind-boggling property of this universe is that it is finite, yet it has no bounds.

Is space a non-Euclidean?

How do hyperbolic geometry and spherical elliptical geometry differ from Euclidean geometry?

The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line.

What is the difference between Cartesian space and Euclidean space?

A Euclidean space is geometric space satisfying Euclid’s axioms. A Cartesian space is the set of all ordered pairs of real numbers e.g. a Euclidean space with rectangular coordinates.

Is hyperbolic space real?

Hyperbolic space, developed independently by Nikolai Lobachevsky and János Bolyai, is a geometrical space analogous to Euclidean space, but such that Euclid’s parallel postulate is no longer assumed to hold.