According to the wikipedia article on pentagonal tilings, the 14th type of pentagonal tiling is

completely determined, with no degrees of freedom

However, I was wondering why an affine stretch of the plane would (thus preserving the tiling) would not introduce a degree of freedom for the angles. If this affine stretch yields a subcase of a different type of pentagon, then this 14th case itself would be a subcase. Thus my question is:

Why is the 14th convex pentagon tiling uniquely determined, and not able to be stretched with an affine transformation to give it a degree of freedom?


Since the tiles aren't all oriented the same way, a transformation like stretching or skewing would affect different tiles differently, so you wouldn't have a tiling with only one kind of tile any more. Of course, you can apply translations, rotations and scalings (stretching equally in all directions), as they are trivially degrees of freedom of any tiling.


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