Solution to Isoperimetric problem and still tiles the plane? And is smooth everywhere upon rotation?

Question

We all know that the solution to the isoperimetric problem in the Euclidean plane is the circle. Unfortunately, the circle does not tile the plane without adding at least 2 bowed in triangles. Because of the shape formed by adding the triangles I looked at prototiles formed by the function: $f(j):=\begin{cases} Sin[x-\frac{1}{4}(3+(-1)^{1+j})\pi]^j+\frac{1}{2}(3+(-1)^{1+j}) & \\ Sin^j[x] \end{cases},\frac{\pi}{2}\leq \theta\leq \frac{(4+(-1)^{1+j})\pi}{2}, j \in \mathbb{N}$

I would show the first 12 examples but can't embed pictures yet.

I wrote an essay for fun on why they are nice and have a small perimeter to area ratio for a given string of fixed length and indicated that the best answer was likely for j=1,2 without proof. But they are not smooth all the way around since they have two cusp points on the left and right ends. I was wondering if there is any previous research generalizing the isoperimetric theorem to tilings. Is my function for j=1,2 the best? Does a tiling that is smooth all the way around the centroid exist?

Answer 1

A bounded domain with smooth boundary cannot tile a plane. Indeed, suppose $\Omega$ is such a domain. It has finitely many neighbors $\Omega_1, \dots,\Omega_n$ in the tiling. Let $\Gamma_j = \partial\Omega_j\cap \partial \Omega $ be the common part of the boundary of $\Omega_j$ and $\Omega$. These sets are closed, and together they cover $\partial\Omega$. Since $\partial\Omega$ is connected, there is a point $z$ that belongs to the boundaries of $3$ or more domains. It is impossible for all of these domains to have smooth boundaries at $z$, because each domain smooth boundary takes up a sector of angular size $\pi-\epsilon$ in a small neighborhood of $z$. There is not enough room within the total angle of $2\pi$.

any previous research generalizing the isoperimetric theorem to tilings

Theorem 4 of The Honeycomb Conjecture by Thomas C. Hales is an inequality of isoperimetric type for which the optimal domain is a regular hexagon. The Honeycomb conjecture itself concerns a tiling with "smallest total boundary length" (properly interpreted), also an problem of isoperimetric type.