Park City Mathematics Institute
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Summer 2015
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Henry Segerman [@henryseg]

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Title: How to Make Sculptures of 4-Dimensional Things
Abstract: Our brains have evolved in a 3-dimensional environment, and so we are very good at visualising 2- and 3-dimensional objects. But what about 4-dimensional objects? The best we can really do is to look at 3-dimensional "shadows". Just as a shadow of a 3-dimensional object squishes it into the 2-dimensional plane, we can squish a 4-dimensional shape into 3-dimensional space, where we can then make a sculpture of it. If the 4-dimensional object isn't too complicated and we choose a good way to squish it, then we can get a very good sense of what it is like. We will explore the sphere in 4-dimensional space, the 4-dimensional polytopes (which are the 4-dimensional versions of the 3-dimensional polyhedra), and various sculptures and puzzles that have come from thinking about these things.

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Related Links
Henry's Math(s) Website
Shapeways: Mathematical Art by Henry Segerman


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