[b]Theorem[/b]: 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 [math]\left(p-2\right)\left(q-2\right)=4[/math].[br][br][b]Proof[/b]: Suppose a regular polygon with p sides tessellates with q copies at each vertex. [br][br]This is true if and only if [math]\frac{\left(p-2\right)\pi}{p}\cdot q=2\pi[/math] .[br][br]This is true if and only if [math]\left(p-2\right)\cdot q=2p[/math].[br][br]This is true if and only if [math]pq-2q-2p=0[/math].[br][br]This is true if and only if [math]pq-2q-2p+4=4[/math].[br][br]This is true if and only if [math]\left(p-2\right)\left(q-2\right)=4[/math]. This is what we wanted to show. □

Questions:[br][br]What is the expression [math]\frac{\left(p-2\right)\pi}{p}[/math] in second line of the proof? Have you seen that before?[br][br]Check that your (p,q) pairs from the previous slide satisfy the condition. [br][br]Why can we be sure that we've found all (p,q) pairs?