Rule for Hyperbolic Tessellations by Regular Polygons
Theorem: A regular polygon in Hyperbolic Geometry with p sides will tessellate with q copies of the polygon at each vertex if and only if .
Proof: 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.
This is true if and only if and also if and only if .
Combining these yields .
Similar to the proof for Euclidean Tessellations by Regular Polygons, this can be re-written as . □
Questions:
Is there any concern that in manipulating that the inequality flipped? Why or why not?
Can you justify the second line further?
Based on this theorem, why is there not a (3,6) tessellation in Hyperbolic Geometry?
Is there a (6,3) tessellation in Hyperbolic Geometry?
Is there a (3,7) tessellation in Hyperbolic Geometry?
Is there a (5,4) tessellation in Hyperbolic Geometry?