Torus and Klein bottle tessellations with a single tile of pied de poule (Houndstooth)

We design a 3D surface made by continuous deformation applied to a single tile. The contour edges are aligned according to the network topology of a Pied-de-poule tessellation. In a classical tessellation, each edge is aligned with a matching edge of a neighbouring tile, but here the single tile acts as a neighbouring tile too. The continuous deformation mapping the Pied-de-poule tile to the 3D surface preserves the staircase nature of the contour edges of the tile. It is a diffeomorphism. The 3D surface thus appears as a torus with gaps where the sides of the tile meet. Next we present another surface, also a single Pied-de-poule tile, but with different tessellation type, a Klein bottle. Both surfaces are 3D printed as innovative art works, connecting topological manifolds and the famous fashion pattern.