Tessellates

The first question I had was "what exactly is a tessellate?" After scouring the internet, I found my answer. According to Wikipedia, "a tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps."  Tessellations can also be generalized to higher dimensions. 

There are four different types of regular polygons tessellations: regular, semi-regular, nonuniform periodic, and nonuniform non-periodic. "A regular tessellation is periodic and uniform and consists of congruent regular polygons." There are only three regular tessellations because only triangles, squares, and hexagons have interior angles that divide 360 degrees evenly. This is important because the tessellate has to be uniform, meaning "the same type of polygon(s) are at each vertex." 

Here is a really cool chart I found on the Theory of Regular Polygon Tessellations webpage that explains and shows most of the different combinations a regular tessellate must be (they all have to add up to 360 for the combination to be a regular tessellate): 

Combination of Polygons	Combination of Angles
6 triangles	6X 60 = 360
4 squares	4 X 90 = 360
3 hexagons	3 X 120 = 360
3 triangles and 2 squares	(3 X 60) + (2 X 90) = 360
3 triangles and 2 squares - another formation	(3 X 60) + (2 X 90) = 360
2 hexagons and 2 triangles	(2 X 120) + (2 X 60) = 360
2 hexagons and 2 triangles - another formation	(2 X 120) + (2 X 60) = 360
1 hexagon, 2 squares, and 1 triangle	(1 X 120) + (2 X 90) + (1 X 60) = 360
1 hexagon, 2 squares, and 1 triangle - another formation	(1 X 120) + (2 X 90) + (1 X 60) = 360
1 hexagon and 4 triangles	(1 x 120) + (4 X 60) = 360
2 octagons and 1 square	(2 X 135) + (1 X 90) = 360
1 dodecagon, 1 square, and 1 hexagon	(1 X 150) + (1 X 90) + (1 X 120) = 360
2 dodecagons and 1 triangle	(2 X 150) + (1 X 60) = 360
1 dodecagon, 2 triangles, and 1 square	(1 X 150) + (2 X 60) + (1 x 90) = 360

Some common tessellates are in beehives and soccer balls, as well to form:  

<--- scales         tiles --->

   <--- bark on trees 

Sources: 
Wikipedia: 
http://en.wikipedia.org/wiki/Tessellation

Theory of Regular Polygon Tessellations: http://www.beva.org/math323/asgn5/tess/regpoly.htm