Decomposing the Four-Dimensional Hypersphere
Professor Davide P. Cervone
February 21, 2005
Tea and pastries will be served
Have you ever wondered what an object in four dimensions might be like? What shapes in four space correspond to the concept of spheres and cubes in three dimension? A sphere in three dimensions can be broken down into two hemispheres that are glued together along their boundaries, so can we do something similar to make a "hypersphere" in four dimensions? It turns out that we can, but there are some even more interesting ways to break apart a four-dimensional hypersphere into two congruent pieces. In this talk, we will explore these ideas more carefully by investigating first the three-dimensional cube and its four-dimensional analog, the "hypercube", and will develop an unusual decomposition of the hypersphere that has connections to a wide range of mathematical areas, from dynamical systems to complex analysis. No previous knowledge of four-dimensional geometry is necessary, just an open mind and a willingness to move beyond your three-dimensional preconceptions. Computer-generated animations will play a crucial role in visualizing these objects, so if nothing else, at least come and enjoy the pictures!
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