One basic fact about 2-d surfaces is that they all belong to some ‘geometry’. Informally, a geometry is a space that is homogeneous (so every point is just like any other point), simply connected (meaning that it has no ‘holes’) and had as many symmetries as possible.

With this definition, there are three 2-dimensional geometries:

- Euclidean geometry (an infinite sheet of paper)
- Spherical geometry (the surface of a ball)
- Hyperbolic geometry (M.C. Escher’s Circle Limit)

These geometries all have a curvature. Euclidean geometry is not curved, spherical geometry is positively curved, and hyperbolic geometry is negatively curved, like a saddle or a pseudosphere.

The 3-dimensional geometries were described by Thurston, and Perelman proved that every ‘simple’ 3-manifold belongs to one of them.

Thurston reasoned this way. In 3-manifolds, you can be isotropic or non-isotropic; isotropic means looking the same in every direction.

If the geometry is isotropic, then it is curved the same way in every direction. So we get geometries of constant curvature. They are:

- Euclidean geometry (how we usually imagine space)
- Spherical geometry (the 3-sphere; this would be a finite universe)
- Hyperbolic geometry (look up dodecahedral tesselations)

The next group of geometries has one axis different from the other 2. The simplest examples are the product geometries:

- The Euclidean plane times a line. This is just Euclidean space, so we don’t count it again.
- The sphere times a line. This geometry is finite in two directions and infinite in another.
- The hyperbolic plane times a line. Comes up in knot theory.

Now things get weird. You can also have twisted products:

- The Euclidean plane twisted with a line. This is Nil geometry, where travelling along the ‘line’ direction makes the plane get more and more skewed. This geometry has been used for a century or more in Hamiltonian mechanics.
- A sphere twisted with the line. This gives us the 3-sphere again.
- The hyperbolic plane twisted with a line. This is the geometry of “the universal cover of SL2(R)” (they need a better name). Motion here occurs in screw-like patterns.

Finally, there is only one geometry where all 3 axes are different. This is:

- Sol geometry. In this geometry, travelling up the z axis contracts the x-axis and expands the y-axis. Going down does the opposite.

These are the eight geometries.

This post was originally posted on reddit.

Image by: TomRuen, from software by Jeff Weeks

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