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# About The Shape of Space Video

The Geometry Center

The Shape of Space is part of a continuing series of Geometry Center productions which use computer graphics to communicate concepts on the cutting edge of mathematics.

The series has focused on topology, an area of mathematics not traditionally encountered before advanced undergraduate mathematics classes. However, when presented visually, many topological ideas are accessible even to elementary school students. Nontechnical language and exciting graphics are used to present a comprehensible, step-by-step exploration which retains mathematical depth. By communicating the vibrancy and beauty of contemporary research mathematics, the video series combats the common misperception that math is boring and discoveries in mathematics ended with the ancient Greeks.

The Shape of Space is more focused than previous Geometry Center videos, in terms of both intended audience and mathematical ground covered. The pilot for a series of smaller scale videos, The Shape of Space is explicitly intended for distribution to high schools as part of a module including curriculum materials and supplementary microcomputer software. It examines a few of the ideas appearing in the Geometry Center's 1991 film Not Knot in a more elementary and detailed fashion.

The video examines the question, what does it mean for space to have a shape? Gluing together the edges of a square, we construct two-dimensional spaces which differ from the infinite plane. Having seen what this means for the Flatlanders who live in those spaces, we then explore corresponding 3-D spaces which are finite but have no boundary. Our own physical universe may have just such a structure: what seems to be a star in a distant galaxy could be our own sun.

The Shape of Space is the result of collaboration between mathematicians, programmers, and designers. Custom software written in Perl, as well as Softimage's commercial animation package and the Geometry Center's own Geomview, were used in production of the video.

### Still frames from the video

``Life in 2 dimensions has its problems. When two Flatlanders want to pass one another, they can't go around each other or to the side as we would, since they can't leave the plane. Our athletic Flatlanders must jump over one another to continue on their way.''

``The Flatlanders can still travel about their universe as before. When looking at a fundamental domain, we must imagine that its edges are glued together - anything leaving one glued edge returns at the other. The way we glue them determines the shape of the space.''

``Let's ride the spaceship inside the 3-torus. Even though the 3-torus is finite, we have the illusion of flying in an infinite space. There are only two stars in this universe, but we see each one over and over.''

The tiled picture view of the the two dimensional manifold based on a Möbius strip: ``They see infinitely many images of their universe - half reversed and half not.''

``We fly this way, and see ships in neighboring rows flying in opposite directions. The mirrored images turn, as we do, to fly along paths that seem to cross ours, but they can never hit us - that's impossible in this space.''

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