The Hat Family
The Hat is one member of a continuous family of aperiodic monotiles parameterized by edge length ratio a.

a = 0: Chevron shape
a = sqrt(3): The Hat
a = 3: Comet shape

At exactly a = 0 and a = sqrt(3)*2, the tile can tile periodically (liminal cases).

Hat Family Continuum

Morph between tile shapes. The parameter controls edge length ratios.