hyperbolic tessellations image gallery

The hyperbolic plane can be tessellated by congruent regular
quadrilaterals (sides of equal length, all
interior angles being 2*pi/5 or 72 degrees).

Cf. e.g. Don Hatch's page
< http://www.hadron.org/~hatch/HyperbolicTesselations/ >

Row: {4,5} , Column: "Truncation = 0".
He writes:
"The dual of each tessellation or truncated tessellation is shown
in blue."

So for the tessellation by regular quadrilaterals, {4,5},
the quadrilaterals are in white and I think the dual in
blue is a pentagonal tessellation:
http://www.hadron.org/~hatch/HyperbolicTesselations/4_5_trunc0_512x512.gif

The vertices and edges of the white quadrilaterals form a graph.
From the vertex at the center of the circle, 5 vertices are
one edge away, and ~= 15 vertices are two edges away.
For three edges away, it's less easy to see how many
vertices there are. There might be an asymptotically valid
expression for the number of vertices n edges away, but
I don't know what it might look like.

David Bernier
.