# Perimeter of $(p,q)$ tiling of the hyperbolic plane

Consider a $(p,q)$ regular tiling of the hyperbolic plane projected on the Poincare disc (that is, a tiling of q p-gons joining at each vertex).

Obviously the area of all tilings converge to $\pi$, but what about the total perimeter? that is, the sum of all the lengths of the tiling edges?

Is the total perimeter infinite? and if it is finite, is there a known formula for the total length?

• "Obviously the area of all tilings converge to $\pi$ " Why do you think that? – Willemien Nov 29 '14 at 10:07
• because the tilings cover the Poincare disc completely, which has area $\pi$ – lurscher Dec 2 '14 at 18:56
• I thought you would say something like that , but it is wrong , the (hyperbolic) area of the Poincare disk is infinite,(the length contracts towards the edge), The maximum area of a triangle is $\pi$ (the triangle in an $( 3, \infty)$ tiling ) – Willemien Dec 2 '14 at 20:12
• Which is why I made the comment of $\pi$ as the total area, to make it clear that I was referring to the lengths of the projected (euclidean) perimeters on the Poincare disc coordinates, not on the underlying hyperbolic plane – lurscher Dec 3 '14 at 5:22

• Great answer! This leaves the interesting question of whether the outer perimeter (of, say, all tiles distance $\leq n$ from a center tile) converges or not. – Steven Stadnicki Feb 21 '15 at 4:08