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?