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Continuing discussion from chrisboyle/sgtpuzzles#479

A very interesting grid type to add would be an Altair tiling with squares, pentagons, hexagons, heptagons, and octagons (though not all are regular).

One concern is finding integer ratios for both sqrt(2) and sqrt(3) (possibly more?) without getting exceedingly large. Another possibility is finding ratios for like sqrt(2)/2 or sqrt(3)/2, if these play nicer with integers.

Another compromise that must be made is determining which shapes are regular (or as close to regular as integers will allow) and which are going to be skewed a bit. In the example image from krazydad.com, the heptagons are the most obviously irregular, but the hexagons or pentagons must be too, since the internal angles of 5-, 6-, and 8-gons do not add up to 360° (108+120+135=363°).