Geometric Analysis of Topology of 1-Manifolds of Constant Curvature

We explore and compare, via geometric methods, the topology of the hyperbolic-unit interval and the spherical-unit interval with respect to the Euclidean-unit interval. We do so using analysis of the density of rational and irrational numbers in the real plane by employing methods of refraction resulting from the geometric lens formed by the intersection of the circle, the aster, and the square rotated by $\pi$/4. Isometry of the square to its rotational rotational analogue allows for a mapping of hyperbolic and spherical subspaces to a Euclidean metric.

A sheet of foundational notation will be available at least a week before the talk at \url{http://jonathandavidevenboer.weebly.com/blog/foundational-notation-for-nums-15-talk}