Math 17: Mathematics Beyond Calculus Winter 2007 The Geometry of the Fourth Dimension This course will explore various aspects of the fourth dimension through a variety of perspectives. Just as an anthropologist studies other cultures and returns with new insights about our own society, we hope to gain a deeper understanding of our 3D world after "going outside of it" to see other alternatives. Several topics to be explored are: * Geometric: For example, what comes next in the sequence: segment, polygon, polyhedron, ...? How can we rigorously study and appreciate the mathematical properties of 4D geometric objects? * Computational: How do we represent 4D structures in a computer model, to analyze, manipulate or visualize them? * Literature and popular culture: How has the fourth dimension appeared in literature, e.g., Edwin Abbott's Flatland, and in the popular imagination? * Formal mathematics: What are the rigorous axiomatic (Euclidean definition-theorem-proof) and coordinate (x, y, z, w) methods of exploring patterns and relationships in 4D geometry? * Artistic: How have artists found inspiration and insight from ideas of hyperspace? * Visualization: How can lower-dimensional shadows and cross-sections of higher-dimensional objects help us to develop intuition about structures we can not directly apprehend? * Historical: How has study of the fourth dimension developed over time? * Physics: Given that geometrical insight is essential to many aspects of science and physics, how can the geometry of the fourth dimension help us to understand the relationship between space and time, and lead to Einstein's theory of special relativity?