Regular Tessellation
A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons. [Remember: Regular means that the sides of the polygon are all the same length. Congruent means that the polygons that you put together are all the same size and shape.] Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled.
Here are examples
A tessellation of triangles
A tessellation of squares
A tesselation of hexagons
When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved.
You can work out the interior measure of the angles for each of these polygons:
Shape
Triangle
Square
Pentagon
Hexagon
More than six sidesAngle measure in degrees
60
90
108
120
more than 120 degreesSince the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.
RULE #1 : The tessellation must tile a floor (that goes on forever) with no overlapping or gaps.
RULE #2 : The tiles must be regular polygons - and all the same.
RULE #3 : Each vertex must look the same.
