1. Tessellations

2. What is a Tessellation?
A tessellation is a tiling, kind of like the floor, except it goes on forever.
There must be no overlapping or no gaps.

3. Regular Tessellations
It must tile a floor (flat surface) with no gaps or overlapping.
The tiles must be regular polygons. (Remember that regular means all the sides and angles are congruent.)
Each vertex must look the same.
This is a vertex.

4. Things We Can Tessellate with These Rules.
Hexagons
120°+120 °+120 °=360 °
Squares
Notice that at the vertex the angles add up to 360°.
90°+90°+90°+90°=360°
Triangles
The same thing happens here.
60°+60°+60°+60°+60°+60°=360°

5. Guess what! Any regular polygon with more than six sides will overlap.
What Won’t Tessellate
Pentagons
It makes a gap!
108 °+108 °+108 °=324 °
Octagons
Now there is an overlap.
135 °+135 °+135 °=405 °
Guess what! Any regular polygon with more than six sides will overlap.
So the only polygons that will work are the triangle, square, and hexagon.

6. Semi-Regular Tessellations
Semi-regular tessellations are almost the same as regular tessellations, except you can use two or more regular polygons.
3, 6, 3, 6
3, 3, 3, 3, 6
Not a tessellation