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.
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(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.

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(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.

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(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:

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

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(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.

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(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.

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## 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.

Table of Contents
Lisa Eckstein