In order to generalize Danzer’s 7-fold tiling to n-fold symmetry,
where n>5 is odd, L. Danzer and D. Frettlöh introduced trapezoidal tiles,
each one the union of two triangles with edge lengths of the form
It needs some further effort, including the introduction of three additional prototiles (two pentagons, one non-trapezoidal quadrangle),
but one obtains an infinite series of substitution rules based on n-fold symmetry (n odd).
Unfortunately, none of these tilings show perfect n-fold symmetry, as Danzer’s 7-fold does,
thus loosing aesthetic appeal.