Mapping of regular polygons: octagon

 

√2+1 lines, one being the side of 
the regular octagon.

√ 2

√ 2

 a2 + b2 = c2.
(√ 2+1)2 +(1)2 = c2

Lines of one

 a2 + b2 = c2.
(√2/2)2 + (√2/2+1)2 =c2

Lines of 2