Software and Resources
for
The Shape of Space
second edition

Chapter 1

Flatland

Flatland 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

Torus and Klein Bottle Games 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.
3D Games Three-dimensional torus games should become available in a few years (once other projects are done).
Shape of Space Video 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

Torus and Klein Bottle Games 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.
3D Games When 3D torus games become available, there’ll be an option for non-orientable spaces as well.
Shape of Space Video 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

Product Spaces Product Spaces software lets you fly about a forest in S² × E.
 

Chapter 7

Flat Manifolds

Curved Spaces 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

Surface Explorer 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

Surface Explorer 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

Curved Spaces 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

Curved Spaces 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

Curved Spaces The Curved Spaces software includes this chapter’s manifolds:
  • Seifert Weber Dodecahedral Space
  • Poincaré Dodecahedral Space
  • Tetrahedral Space
  • Quaternionic Space
  • Octahedral Space
 

Chapter 17

Bundles

Chapter 18

Geometries on Three-Manifolds II

Surface Explorer 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.
Product Spaces 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

Cosmology News 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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