Chapter 1
Flatland
Chapter 1 |
Flatland |
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Edwin Abbott's Flatland: A Romance of Many Dimensions weaves Victorian social satire with a gentle introduction to worlds of 2, 3, and 4 dimensions. This beautifully written book still costs less than $2. |
Chapter 2 |
Gluing |
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Play eight familiar games on a torus. The games let you scroll the board freely, to see that the torus is finite yet has no boundary. |
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Three-dimensional torus games should become available in a few years (once other projects are done). |
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The video The Shape of Space takes you on a computer-animated tour of the 3-torus and other spaces. |
Chapter 3 |
Vocabulary |
Chapter 4 |
Orientability |
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The eight familiar games may be played on a Klein bottle as well as a torus. Just choose Klein bottle from the games' topology menu. |
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When 3D torus games become available, there'll be an option for non-orientable spaces as well. |
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The video The Shape of Space constructs the Klein space and takes you on a computer-animated flight. |
Chapter 5 |
Connected Sums |
Chapter 6 |
Products |
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Product Spaces software lets you fly about a forest in S² × E. |
Chapter 7 |
Flat Manifolds |
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The Curved Spaces software includes flat spaces as well, in spite of its name. It has no games, but the graphics are good. Includes all ten flat 3-manifolds, as well as several geometrically different versions of the 3-torus, some of which are quite beautiful and surprising. You can see the manifolds in stereoscopic 3D if you have red-blue glasses. |
Chapter 8 |
Orientability vs. Two-Sidedness |
Chapter 9 |
The Sphere |
Chapter 10 |
The Hyperbolic Plane |
Chapter 11 |
Geometries on Surfaces |
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The Surface Explorer lets you create your own worlds, including mountains, lakes, houses, castles, fields, fences, etc., and walk about in them. The terrain topology can be any closed surface. (Thus the 3-dimensional space is a surface cross a line.) |
Chapter 12 |
The Gauss-Bonnet Formula and the Euler Number |
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The Surface Explorer includes a set of activites introducing the Euler number and the Gauss-Bonnet formula. |
Chapter 13 |
Four-Dimensional Space |
Chapter 14 |
The Hypersphere |
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Download the Curved Spaces software and open the sample file PoincaréDodecahedralSpace.gen. Press the left-arrow key a few times to open the walls, and you'll see a tiling of the hypersphere by 120 dodecahedra. |
Chapter 15 |
Hyperbolic Space |
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Download the Curved Spaces software and open the sample file HyperbolicSmall.gen. You'll see a tiling of the hyperbolic space by infinitely many copies of a small polyhedron. If you have a pair of red-blue glasses, turn off Fog, turn on Stereo, and start flying around. Even though hyperbolic space is much roomier than flat space, your stereoscopic vision creates the illusion that all of hyperbolic space sits inside a ball. |
Chapter 16 |
Geometries on Three-Manifolds I |
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The Curved Spaces
software includes this chapter's manifolds:
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Chapter 17 |
Bundles |
Chapter 18 |
Geometries on Three-Manifolds II |
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The Surface Explorer (see Chapter 11) lets you walk in any close surface cross a line, but you're stuck on the surface and can't fly. |
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Product Spaces software lets you fly freely about a forest in S² × E, E² × E, and H² × E. |
Chapter 19 |
The Universe |
Chapter 20 |
The History of Space |
Chapter 21 |
Cosmic Crystallography |
Chapter 22 |
Circles in the Sky |
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The latest information on the MAP satellite and efforts to detect the topology of space is available on the Cosmology News page. |
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