  We discover two distinct topological pathways through which the pentagonal
Cairo tiling (P5), a structural model for single-layer $AB_2$ pyrite materials,
respectively transforms into a crystalline rhombus-hexagon (C46) tiling and
random rhombus-pentagon-hexagon (R456) tilings, by continuously introducing the
Stone-Wales (SW) topological defects. We find these topological transformations
are controlled by the orientation correlations among neighboring $B$-$B$ bonds,
and exhibit a phenomenological analogy of the (anti)ferromagnetic to
paramagnetic transition in two-state Ising systems. Unlike the SW defects in
hexagonal 2D materials such as graphene, which cause distortions, the defects
in pentagonal 2D materials preserve the shape and symmetry of the fundamental
cell of P5 tiling and are associated with a minimal energy cost, making the
intermediate R456 tilings realizable metastable states at room temperature.
Moreover, the intermediate structures along the two pathways are neither
crystals nor quasicrystals, and yet these random tilings preserve
hyperuniformity of the P5 or C46 crystal (i.e., the infinite-wavelength
normalized density fluctuations are completely suppressed), and can be viewed
as 2D analogs of disordered Barlow packings in three dimensions. The resulting
2D materials possess metal-like electronic properties, making them promising
candidates for forming Schottky barriers with the semiconducting P5 material.
