A new simple family of non-periodic tilings with square tiles

We define a new family of non-periodic tilings with square tiles that is mutually locally derivable with some family of tilings with isosceles right triangles.

Both families are defined by simple local rules, and the proof of their non-periodicity is as simple as that of the non-periodicity of Robinson's tilings.

We also relate the construction to the classical chair tiling: forgetting the decoration, our tilings map almost one-to-one onto the chair, our substitution is essentially a square root of the chair substitution, and our local rules are perfect, defining exactly the family of substitution tilings.