Shapes of Space: Notes and References

Notes

(1) Throughout the text, phenomena in three-dimensional space will be explained with analogies to two-dimensional Flatland, which is easier to visualize. It is necessary to imagine Flatland as existing in a surface, not on it. A Flatlander perceives any local space as a flat plane, and cannot detect curves, bumps, or corners in the surface without a great deal of clever experimentation.

(2) The problem with imagining a spherical Flatland is that it is easy to start thinking about Flatland as a planet like Earth. But Flatland is really Flatuniverse, with circular planets and stars in the surface of a plane of aether.

(3) A great circle is a circle on a sphere which is as big as possible. On an idealized, perfectly spherical Earth, the equator and all the lines of longitude are great circles. To an inhabitant of a spherical Flatland, great circles appear to be straight lines.

(4) A flat torus can be a lot of fun. Many games are played on a square board, and imagining that the edges are glued allows greater freedom of movement and changes the strategy of the game.Weeks explains how to play torus tic-tac-toe and torus chess. He suggests the following starting position for torus chess:

(5) A Klein bottle can also be constructed by attaching the edges of two Möbius bands.

(6) The projective plane is involved in a non-Euclidean geometry called elliptic geometry. For a good explanation of this geometry and the reasoning behind the construction of the projective plane, see Banchoff, pages 187-188.

(7) Weeks explains how to make "hyperbolic paper" by taping equilateral triangles together so that seven meet at each vertex.

This paper approximates a hyperbolic plane. Chapter 10 of The Shape of Space offers an excellent description of hyperbolic geometry.

References

Banchoff, Thomas. Beyond the Third Dimension. New York: Scientific American Library, 1990.

Rucker, Rudy. The Fourth Dimension: Toward a Geometry of Higher Reality. Boston: Houghton-Mifflin, 1984.

Weeks, Jeffrey R. The Shape of Space. New York: Marcel Dekker, 1985.

And, as always, thanks to Edwin Abbott Abbott for giving us Flatland.