Does there exist a $C^1$ curve $\alpha:\mathbb{S}^1 \to\mathbb{R}^2$ such that the plane can be tessellated by a union of congruent shapes, each has boundary identical to $\text{Image}(\alpha)$ up to translation and rotation?
Where can I find a classification of such creatures?
I might be a bit imprecise in my formulation, but I hope that the essential question is clear:
We all know that there are regular polygonal tessellations of the plane. The question is whether one can use a "building-block shape" whose boundary is $C^1$, without "corners"?
Note that I do want all the building-blocks to be congruent, so they should be the same up to an isometric transformation of $\mathbb{R}^2$. And I want the intersection between two such shapes in the tiling be only at the boundaries.
Intuitively, the thing I am looking for is something like that:
Picture Source:
(By Dmharvey - w:en:Image:Wallpaper_group-p3-1.jpg, Public Domain, https://commons.wikimedia.org/w/index.php?curid=856136).