reperiendi

Regular tilings of three-dimensional spaces

Posted in Astronomy, Fun links, General physics, Math by Mike Stay on 2011 January 31

If you start at the north pole and make an equilateral triangle 6000 miles on a side, the bottom will lie on the equator, each of the angles will be 90 degrees, and only four of them will fit around the pole.

In a similar way, large enough tetrahedra would tile the surface of a hypersphere. This paper identifies the eleven regular tilings of three-dimensional spaces and whether they’re spherical, Euclidean, or hyperbolic tilings, and then looks at the geometry of spacetime to see how it might be tiled.

The “cubic” tilings (where eight polyhedra meet around a vertex like cubes do in Euclidean space) are amenable to taking cross-sections; this tiling of hyperbolic space with dodecahedra

has a cross section with a tiling of the hyperbolic plane with pentagons:

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