### Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles.

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The behavior of special classes of isometric foldings of the Riemannian sphere ${S}^{2}$ under the action of angular conformal deformations is considered. It is shown that within these classes any isometric folding is continuously deformable into the spherical isometric folding ${f}_{s}$ defined by ${f}_{s}(x,y,z)=(x,y,|z\left|\right)$.

A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed....

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