As everyone knows, only the triangle, square and hexagon tile the plane by themselves. However for other regular polygons it seems we can obtain tilings of the plane composed of pairs of tiles, one of which is a regular polygon and the other an equilateral polygon where tilings are edge to edge with all edges in both tiles are the same size. * A regular polygon is a polygon that has all sides equal and all interior angles equal. * An 'equilateral' polygon is a polygon that has all sides equal but not all interior angles equal. (by this definition an equilateral triangle is not an 'equilateral' polygon but a regular polygon) Definitions from; www.mathopenref.com/polygonregular.html www.mathopenref.com/polygonirregular.html 1/ What is the smallest m-sided equilateral polygon for a given n-sided regular polygon in a tiling of the plane? Tyler from http://www.superliminal.com/geometry/tyler/Tyler.htm by Melinda Green and Don Hatch is ideal for experimenting with these kinds of tiles. Using Tyler I obtained these values for n and m. Polygon pair tessellations n-sided m-sided regular equilateral polygon polygon n m graphics of tilings -- --- ------------------------------------------- 3 0 www.squaring.net/gfx/3n.png 4 0 www.squaring.net/gfx/4n.png 5 4 www.squaring.net/gfx/5n-4m.png 6 0 www.squaring.net/gfx/6n.png 7 8 www.squaring.net/gfx/7n-8m.png 8 6* www.squaring.net/gfx/8n-4m.png, www.squaring.net/gfx/8n-6m.png, 9 12 www.squaring.net/gfx/9n-12m.png 10 6 www.squaring.net/gfx/10n-6m.png 11 16 www.squaring.net/gfx/11n-16m.png 12 8* www.squaring.net/gfx/12n-3m.png, www.squaring.net/gfx/12n-8m.png 13 20 www.squaring.net/gfx/13n-20m.png 14 10 www.squaring.net/gfx/14n-10m.png 15 24 www.squaring.net/gfx/15n-24m.png 16 12 www.squaring.net/gfx/16n-12m.png 17 28 www.squaring.net/gfx/17n-28m.png 18 6 www.squaring.net/gfx/18n-6m.png 19 32 www.squaring.net/gfx/19n-32.png 20 16 www.squaring.net/gfx/20n-16.png 21 36 www.squaring.net/gfx/21n-36.png 22 18 www.squaring.net/gfx/22n-18.png 23 40 www.squaring.net/gfx/23n-40.png 24 9 www.squaring.net/gfx/24n-9.png 25 44 www.squaring.net/gfx/25n-44.png The pairs satisfy these formulae except for 8 6* and 12 8* where the formula gives 8 4 and 12 3. 4 and 3 are regular polygons, not equilateral polygons as defined above. if n = 0(mod 6), then m = n/2-3 else if n = 0(mod 2), then m = n-4 else if n = 1(mod 2), then m = 2n-6 2/ Is it always possible to tile the plane with these 2 prototiles; An n-sided regular polygon and an m-sided equilateral polygon pair for all n? It appears that in the 3 cases above the Conway criterion can always be applied to do so. 3/ Is the minimum m-sided equilateral polygon shape unique for given n? Stuart