I was reading a paper on hyperbolic pascal triangle and the author stated that for Schlafli symbol

kolutastmr 2022-07-04 Answered
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.
On the nature of mosaic specified by Schlafli symbol { p , q } ?
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Answers (1)

persstemc1
Answered 2022-07-05 Author has 18 answers
Step 1
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 ) π / p . The angle sum in a p-gon in { p , q } is 2 π / q . The difference between these angle sums is
( p 2 ) π / p 2 π / q = ( p q 2 q 2 p ) π / p q = ( ( p 2 ) ( q 2 ) 4 ) π / p q
which has the same sign as ( p 2 ) ( q 2 ) 4 , hence verifying the statement you found in this paper.
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