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Friday, October 21, 2005

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 The inner beauty of math

Math is beautiful:
The sculpture, designed by Adrian Ocneanu, professor of mathematics at Penn State, presents a three-dimensional "shadow" of a four-dimensional solid object. Ocneanu's research involves mathematical models for quantum field theory based on symmetry. One aspect of his work is modeling regular solids, both mathematically and physically.

In the three-dimensional world, there are five regular solids -- tetrahedron, cube, octahedron, dodecahedron, and icosahedron -- whose faces are composed of triangles, squares or pentagons. In four dimensions, there are six regular solids, which can be built based on the symmetries of the three-dimensional solids. Unfortunately, humans cannot process information in four dimensions directly because we don't see the universe that way. Although mathematicians can work with a fourth dimension abstractly by adding a fourth coordinate to the three that we use to describe a point in space, a fourth spatial dimension is difficult to visualize. For that, models are needed.

"Four-dimensional models are useful for thinking about and finding new relationships and phenomena," said Ocneanu. "The process is actually quite simple -- think in one dimension less." To explain this concept, he points to a map. While the Earth is a three-dimensional object, its surface can be represented on a flat two-dimensional map.

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