The paper should be 1375 words 5-7 pages in length and be properly formatted and cited. The book that we are using is 2nd ed. of Henle, Modern Geometries. Objectives: Determine what properties a set of axioms possess (complete, consistent, independent, categorical, etc.) Use Euler’s Formula and construct a geometric interpretation of complex numbers Prove theorems and exhibit examples/models of finite geometry Compute hyper areas and hyper volumes Construct examples of fractals Compute the dimension of a self similar fractal Determine the geodesics on a general 2-Dimensional manifold Define and use Poincare’s Model Compare and Contrast with elliptic and parabolic geometries Determine the image of and graph the transformation of a given set of points under a Mobius Transformation Use the cross ratio, circle inversion, and homothetic transformations as proof tools Define, compute in, and solve equations in Quaternions Use Quaternions to model motion in 3-Dimensional Euclidean space Determine symmetries of patterns in 1, 2, and 3-Dimensional Euclidean space Use the axioms and theorems of classical Euclidean geometry as proof tools