These Thurston maps are NET maps for every choice of translation term. They have degree 16. Not only are they are Euclidean, they are even flexible Lattes maps. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0,lambda1,lambda2,lambda1+lambda2} This pure modular group Hurwitz class contains infinitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 3. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 1/16, 2/8, 4/4, 8/2, 16/1 FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(2,7)(3,14)(4,12)(5,13)(6,11)(10,15)", "b=(1,13)(2,3)(4,16)(5,9)(6,7)(8,12)(10,11)(14,15)", "c=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)", "d=(1,14)(2,5)(3,16)(4,7)(6,9)(8,11)(10,13)(12,15)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,5)(2,7)(3,14)(4,16)(6,11)(8,12)(9,13)(10,15)", "b=(2,3)(5,13)(6,7)(8,16)(10,11)(14,15)", "c=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)", "d=(1,14)(2,5)(3,16)(4,7)(6,9)(8,11)(10,13)(12,15)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,a*b,1,a*b,1,1,1,1,1,1,1,1,c*d,1,c*d,1>(1,6)(2,13)(3,8)(4,15)(5,10)(7,12)(9,14)(11,16)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)", "c=(1,13)(2,3)(4,16)(5,9)(6,7)(8,12)(10,11)(14,15)", "d=(1,9)(2,15)(3,6)(7,10)(8,16)(11,14)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=<1,a*b,1,a*b,1,1,1,1,1,1,1,1,c*d,1,c*d,1>(1,6)(2,13)(3,8)(4,15)(5,10)(7,12)(9,14)(11,16)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)", "c=(2,3)(5,13)(6,7)(8,16)(10,11)(14,15)", "d=(1,13)(2,15)(3,6)(4,8)(5,9)(7,10)(11,14)(12,16)", "a*b*c*d");