Rule for Euclidean Tessellations by Regular Polygons
Theorem: A regular polygon in Euclidean (traditional) Geometry with p sides will tessellate with q copies of the polygon at each vertex if and only if .
Proof: Suppose a regular polygon with p sides tessellates with q copies at each vertex.
This is true if and only if .
This is true if and only if .
This is true if and only if .
This is true if and only if .
This is true if and only if . This is what we wanted to show. □
Questions:
What is the expression in second line of the proof? Have you seen that before?
Check that your (p,q) pairs from the previous slide satisfy the condition.
Why can we be sure that we've found all (p,q) pairs?