Regular Tessellation Write-up

From EscherMath

In the exploration Angles of Polygons and Regular Tessellations Exploration we see that there are only three regular tessellations. The exploration is really an outline for a proof.

After doing the exploration, your assignment is to provide the justifications and details of the arguments. Below are 5 steps that make up the general argument. Write down a short paragraph for each step showing why this is true.

• Step 1 – In every n-gon there are n-2 triangles (this is the best case scenario)
• Step 2 – The sum of the angles in an n-gon is (n-2)*180
• Step 3 – If our n-gon is regular, then the angle measure is exactly (n-2)*180 / n
• Step 4 – Show that the angles associated with the 3-, 4- and 6-gon are the only ones that divide 360.
• Step 5 – Show that this means that only 3-, 4- and 6 gons give regular tessellations