Continue exploring Einstein's Special Relativity in this advanced mathematics lecture. Delve into a unique explanation using a one-dimensional Newtonian world inhabited by bats that use echolocation to determine event positions and times. Develop fundamental Lorentz transformations for transitioning between different bat coordinate systems with constant relative motion. Examine the rational parametrization of hyperbolas, a crucial component in understanding the theory. Cover topics including space-time coordinates, matrix formulation of Lorentz transformations, canonical affine normalization, and the complexities that arise in this framework. Gain deeper insights into the mathematical foundations of Special Relativity through this innovative approach.