From Polygons to Ultradiscrete Painlev\'e Equations
  The rays of tropical genus one curves are constrained in a way that defines a bounded polygon. When we relax this constraint, the resulting curves do not close, giving rise to a system of spiraling polygons. The piecewise linear transformations that preserve the forms of those rays form tropical rational presentations of groups of affine Weyl type. We present a selection of spiraling polygons with three to eleven sides whose groups of piecewise linear transformations coincide with the B\"acklund transformations and the evolution equations for the ultradiscrete Painlev\'e equations. 
Christopher Michael Ormerod and Yasuhiko Yamada
nlin.SI