Three Fathoms
Observatory


In cosmology, there are the possibilities of a flat space, a positively curved space and a negatively curved space. While empirical data strongly suggests a flat Universe conforming to Euclidean geometry we learnt in school, the other forms of spatial geometries are quite counterintuitive to our human spatial sense. A positively curved space can be visualized as the surface of a 2sphere (e.g. the crust of Earth) embedded in a 3dimensional space. Viewed from any point on the surface, a distant object subtends larger than normal (Euclidean) angles while the sum of the three angles of a triangle >180 degree. On a stereographic projection plate, objects appear larger and larger as the distance increases. Space at increasing distance from any observer changes according to a sin function from 0 to pi, first increases gradually and then decreases gradually. The geometry of a hyperbolic space is the negative curvature analogue of the 2sphere. Viewed from any point in a hyperbolic space, distant objects subtend smaller than normal (Euclidean) angles while the sum of the three angles of a triangle <180 degree. Objects appear smaller and smaller as the distance increases. Space at increasing distance from the observer expands exponentially according to a hyperbolic sin function (sinh) from 0 to infinity. As space at increasing distance exponentially expands, any distant objects rapidly diminish in angular size (but of same actual size) until they become infinitely small, while the expanded space accommodates more and more of these objects. This view of space is shared by any other observers within the homogeneous universe. Interestingly, this hyperbolic space geometry can be visualized by a pattern transformation composed by a mathematical artist M. C. Escher: <iframe width="640" height="360" src="http://www.youtube.com/embed/7nTW1EVXDBY?feature=player_detailpage" frameborder="0" allowfullscreen></iframe>
