Cairo Solid Thwarted By Outlaw Polygons

As I looked at the assembled peels from Poly and speculated on a Cairo pattern wrap for the Snub Square Antiprism (j85), I noticed something odd about some of the pentagons around it's sides. Because every angle of these pentagons had it's roots in an equally divided equilateral triangle, they look very hexagon like. On further reflection, I began to wonder how they could be so odd, and still be equilateral pentagons? Attempts to construct them in 2 dimensions failed.The pentagons containing the "right" angles came in at 570 degrees of interior angles, while the troubling looking ones contained a whooping 600 degrees.   Further research revealed, the sum of the interior angles of a two dimensional pentagon is 540 degrees. Using the following formula (I found at mathsisfun.com) with n as number of sides,  (n-2) x180= the sum of polygon interior angles. Duh! 

The circular king of Flat Land would have had me bisected for even entertaining such a heresy as a pentagon with five 120 degree angles! Careful examination of the model on the right,  revealed that the pentagons on the Snub Cubeoctahedron (which presents the closest I've found so far to a spherical snub square network) were also impostors. They contained a total of 570 degrees of interior angles.these were only marks, representing the true dual.

Pentagonalicositetrahedron

 

Obviously something different happens when one strikes a line from the center of each polygon in a 2 dimensional snub square network. A careful examination of these pentagons reveals, three120 degree angles, and two 90 degree angles, in each.

 I still believe a visually similar and highly thought provoking result could be achieved on the snub square solid. While I'm pretty confident, I think a little more Photoshop and a working model will precede my fleece experiment. I found the true dual of the snub cube, but have yet to model it.