[b]Theorem[/b]: A regular polygon in Hyperbolic 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]: Consider a regular polygon in Hyperbolic Geometry with p sides that tessellates with q copies at each vertex. Let A be the angle at each vertex of the polygon.[br][br]This is true if and only if [math]A\cdot q=2\pi[/math] and also if and only if [math]\frac{\left(p-2\right)\pi}{p}\cdot q>A\cdot q[/math]. [br][br]Combining these yields [math]\frac{\left(p-2\right)\pi}{p}q>2\pi[/math].[br][br]Similar to the [url=https://www.geogebra.org/m/tn4qfh4f#material/qauajwes]proof for Euclidean Tessellations by Regular Polygons[/url], this can be re-written as [math]\left(p-2\right)\left(q-2\right)>4[/math]. □[br]

Questions:[br][br]Is there any concern that in manipulating [math]\frac{\left(p-2\right)\pi}{p}q>2\pi[/math] that the inequality flipped? Why or why not?[br][br]Can you justify the second line further?[br][br]Based on this theorem, why is there not a (3,6) tessellation in Hyperbolic Geometry?[br][br]Is there a (6,3) tessellation in Hyperbolic Geometry?[br][br]Is there a (3,7) tessellation in Hyperbolic Geometry?[br][br]Is there a (5,4) tessellation in Hyperbolic Geometry?