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 ) &lt; 4 a sphere is determined and for ( p − 2 ) ( q − 2 ) &gt; 4 a hyperbolic mosaic is defined.On the nature of mosaic specified by Schlafli symbol { p , q } ?

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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  )  &amp;lt;  4 a sphere is determined and for   (  p  −  2  )    (  q  −  2  )  &amp;gt;  4 a hyperbolic mosaic is defined.On the nature of mosaic specified by Schlafli symbol   {  p  ,  q  } ?

persstemc1 · Accepted Answer

Step 1The 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&#039;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  qwhich has the same sign as   (  p  −  2  )  (  q  −  2  )  −  4 , hence verifying the statement you found in this paper.

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

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