kolutastmr

2022-07-04

I was reading a paper on hyperbolic pascal triangle and the author stated that for Schlafli symbol $\left\{p,q\right\}$ , if $\left(p-2\right)\phantom{\rule{thickmathspace}{0ex}}\left(q-2\right)=4$ , it determines the Euclidean mosaic. For $\left(p-2\right)\phantom{\rule{thickmathspace}{0ex}}\left(q-2\right)<4$ a sphere is determined and for $\left(p-2\right)\phantom{\rule{thickmathspace}{0ex}}\left(q-2\right)>4$ a hyperbolic mosaic is defined.
On the nature of mosaic specified by Schlafli symbol $\left\{p,q\right\}$ ?

Step 1
The general idea is to compare the angles of a regular p-gon in $\left\{p,q\right\}$ 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 $\left(p-2\right)\pi /p$ . The angle sum in a p-gon in $\left\{p,q\right\}$ is $2\pi /q$ . The difference between these angle sums is
$\left(p-2\right)\pi /p-2\pi /q=\left(pq-2q-2p\right)\pi /pq=\left(\left(p-2\right)\left(q-2\right)-4\right)\pi /pq$
which has the same sign as $\left(p-2\right)\left(q-2\right)-4$ , hence verifying the statement you found in this paper.

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