# Curvature (InstanceTopic, 3)

### From Hi.gher. Space

**Curvature** is a geometric property of a space. It can generally be expressed as a real number.

If this number is zero, the space is said to be *flat*. In this case, the sum of the angles in any triangle adds up to 180 degrees, and two distinct parallel lines will never touch. If a flat space A is embedded within another space B, the shortest route between any two points in space A does not leave A. Flat space observes Euclidean geometry.

If the number is positive, the space is *curved*. In this case, the sum of the angles in a triangle may add up to more than 180 degrees. We cannot call two lines "parallel" any more in such a space, but if one constructs three coplanar lines E, L1, L2 such that E is perpendicular to both L1 and L2, then the lines L1 and L2 may intersect. On a sphere, they will intersect at two points and will actually appear as circles. On a paraboloid, they may intersect at zero points (as circles perpendicular to the central line of symmetry), exactly one point (as parabolas in a plane of symmetry), or (possibly? depending on how continity works at the pole - this may produce zero points instead) at infinitely many distinct points (in any other case).

If the number is negative, the space is *hyperbolic*. In this case, the sum of the angles in a triangle may add up to less than 180 degrees. This allows an infinite number of distinct regular infinite hyperbolic tilings which are not possible in non-hyperbolic space.