If the universe is a bottle then it is a bottle with no inside or outside. In the advanced geometry of topology, it is called a Klein bottle. This unique bottle can be described by hyperbolic non-Euclidean geometry of two negative curvatures and one minimum distance between curvatures. On the one hand, a sphere which has no edges but has two sides, the inside and the outside. Although it is not practically possible to construct a Klein bottle using a square sheet of flexible material in 3D, the result is nonetheless valid in mathematics.


Consider, in the plane, the square of all points with Cartesian coordinates (x,y) such that -1 ≤ x ≤ 1 and -1 ≤ y ≤ 1. The mathematical operation of identifying the point (x,1) with the point (x,-1), for all x, can be thought of as forming a cylinder in 3D with 2 sides, the outside and the inside, and 2 edges. The operation of identifying the point (1,y) and (-1,-y) is like forming a cylinder with a twist resulting into a Möbius topology. The Klein bottle is constructed by doing these 2 operations simultaneously.