I was reading a paper on hyperbolic pascal triangle and the author stated that for Schlafli symbol $\{p,q\}$, if $(p-2)\;(q-2)=4$, it determines the Euclidean mosaic. For $(p-2)\;(q-2) <4$ a sphere is determined and for $(p-2)\;(q-2) > 4$ a hyperbolic mosaic is defined.
I cannot seem to understand how to draw these inferences. As a machine learning scientist, I did not have a formal training in advanced geometry and so would appreciate any references or assistance on this.
On
One walk through of the math could be seen in my blog Hyperbolic Polygon-and-Tessellation
Some of the tessalations generated from it are showed here https://animadversio.github.io/portfolio/hyperbolic-art
generated with this repo
The general idea is to compare the angles of a regular $p$-gon in $\{p,q\}$ to a Euclidean $p$-gon to see whether there is angular defect or excess. If the Euclidean $p$-gon has a greater angle sum, the tiling is hyperbolic; if the Euclidean $p$-gon has a lesser angle sum, the tiling is spherical.
So let's compare! The angle sum in a Euclidean $p$-gon is $(p-2)\pi/p$. The angle sum in a $p$-gon in $\{p,q\}$ is $2\pi/q$. The difference between these angle sums is
$$(p-2)\pi/p - 2\pi/q = (pq-2q-2p)\pi/pq = ((p-2)(q-2)-4)\pi/pq$$
which has the same sign as $(p-2)(q-2) - 4$, hence verifying the statement you found in this paper.