*The Shape of Space*Jeff Weeks explores the geometry and topology of two and three dimensional spaces, attempting above all to convey an intuition for their structure. He eschews rigorous mathematics, but informally introduces the idea of a manifold and a range of associated concepts. There are exercises, but they appear inline in the text and are not intimidating; this is more challenging than a "popular" work, but a far cry from a textbook.

Part I begins with surfaces and introduces some key contrasts: geometry versus topology (whether smooth deformations matter or not), intrinsic versus extrinsic (what properties can someone on a surface or in a space ascertain without leaving it), local versus global, homogeneous versus nonhomogeneous (the same everywhere?), and closed versus open. Other concepts include orientability and connected sums and products and bundles. Weeks also looks here at the simple (flat) three-manifolds.

Part II considers geometry on the sphere and the hyperbolic plane,
and general geometries on surfaces: "Except for the sphere and the
projective plane, which have elliptic geometry, and the torus and the
Klein bottle, which have Euclidean (flat) geometry, *all* surfaces can
be given hyperbolic geometry." This leads to an explanation of the
Gauss-Bonnet formula and the Euler number.

Part III proceeds to geometry on three-manifolds, looking at four-dimensional space, hyperspheres, hyperbolic space, fibre bundles, and the full range of geometries on three-manifolds. Attempts are made to prime our intuitions for these, subject to the limitations of the printed page:

"The geometry of H² x E is also homogeneous but not isotropic. Like S² x E, it has different sectional curvatures in different directions. Vertical slices have zero curvature, while horizontal slices have negative curvature. Figure 18.2 provides a rough illustration of H² x E geometry."

The mathematical conclusion of *The Shape of Space* comes here, with a
statement of Thurston's Geometrization Conjecture (which was not yet
proven in 2002), classifying three-manifolds into eight kinds. This
second edition has a new Part IV, "The Universe", discussing applications
to cosmology and the possibility of determining from observation something
of the universe's topology or geometry, but that's really a digression.

*The Shape of Space* is moderately involved, but nothing in it should
be beyond engaged high school students. On the other hand, it could
profitably be read by undergraduate mathematics students, since it treats
material poorly covered in most university curricula, and even research
students who have studied differential geometry might find something
new in it, perhaps some novel intuitions for visualising spaces.

The cosmology is now a bit dated, but the mathematics is not, the proof of the Geometrization Conjecture notwithstanding. There is also a web site with supporting videos and games and visualisation software, some of it pitched at younger children (tic-tac-toe and mazes on toruses) and some at undergraduates and researchers:

"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."

October 2017

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- books about mathematics

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