Once a specific problem is solved, we start to relax the conditions. Of course, that’s never enough for mathematicians. And with that, the regular, monohedral, edge-to-edge tilings of the plane are completely understood. ![]() Similarly for squares: Four squares around a single point at 90 degrees each gives us 4 × 90 = 360.Ī similar argument will show that after the hexagon - whose 120-degree angles neatly fill 360 degrees - no other regular polygon will work: The angles at each vertex simply won’t add up to 360 as required. This works out perfectly: The measure of each internal angle of an equilateral triangle is 60 degrees, and 6 × 60 = 360, which is exactly what we need around a single point. This chart raises all sorts of interesting mathematical questions, but for now we just want to know what happens when we try to put a bunch of the same n-gons together at a point.įor the equilateral-triangle tiling, we see six triangles coming together at each vertex. Here they are up to n = 8, the regular octagon. We can make a chart for the measure of an interior angle in regular n-gons. ![]() ![]() What do these two facts have to do with the tiling of regular polygons? By definition, the interior angles of a regular polygon are all equal, and since we know the number of angles ( n) and their sum (180( n − 2)), we can just divide to compute the measure of each individual angle.
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