Let's Tessellate

Think about regular polygons for just a minute. A regular polygon is a polygon with sides of equal length and angles of equal measure. There is an infinite number of regular polygons, and the more sides they have, the more they approach looking like a circle. (On a side note, this reminds me a lot of what we were talking about in class about how calculus allows the denominator to be zero because it can be so small that it is almost zero, but not quite zero. Similarly, a polygon can have so many sides that we could almost consider it to be a circle, but it will never truly be a circle. I love the idea of infinity!) So even though the number of regular polygons is infinite, there are only three that tessellate. The fraction of the number of regular polygons that tessellate out of all regular polygons is 3/infinity...so the fraction is infinitely small. That's a weird thought but it's veering off this post's main idea: why do only triangles, squares and hexagons tessellate?

(from mathisfun.com)
The answer isn't too complicated. The equation for finding the measure of an interior angle of a regular n-gon is 360/n. So the measure of an interior angle of a triangle is 60, and that of a square is 90, and that of a hexagon is 120. In a tessellation, the corners of the shape being tessellated must meet. And the golden reason for a regular tessellation is having those corners all add up to 360. When triangles tessellate, six corners meet. 6 corners x 60 degrees= 360 degrees. With a square, it's 4 x 90...which is also 360. And with a hexagon, it's 3 x 120 (yes- 360!). I knew that triangles tessellated (and it was pretty easy to figure out that squares tessellate too). If you remember that six regular triangles make up a hexagon (as you can see in the tessellating triangles picture) it's easy to realize that a hexagon would also tessellate. So to figure out if a regular n-gon would tessellate, you would use the equation:
360/n x number of corners meeting in tessellation
If this again equals 360, then the regular n-gon tessellates. But let me remind you right now--unless its a triangle, square or hexagon, that equation won't work