Template
For a hexagonal tile with lattice spacing $a_\text{lat}=1$, we use the following function to generate the landmass:
$$
h(x,z) = \exp\left[-\left(\frac{x^2+z^2}{R^2(x,z)}\right)^{16}~\right]
$$
where $R(x,z)$ is the radial function composed of the hexagonal and the noise part:
$$
R(x,z) = r\times\left(\text{hex}(x,z)+\lambda~\text{noise}(x,z)\right)
$$
the coefficient $r$ is the ideal radius of the tile (default value is $0.64$ to allow some overlap) and $\lambda$ is the noise strength (default value is $0.03$). The hexagonal part is defined by
$$
\text{hex}(x,z) &= \frac{\sqrt{3}}{\sin(t(x,z))+\sqrt{3}\cos(t(x,z))};\\
t(x,z)&=t_0(x,z)+\frac{\pi}{6}-\frac{\pi}{3}\left\lfloor\frac{3}{\pi}t_0(x,z)+\frac{1}{2}\right\rfloor,\\
t_0(x,z)&=\arctan(z,x)\sim\tan^{-1}\left(\frac{x}{z}\right).
$$
This function involves rotating a line section 6 times into a hexagon. The noise part is defined by
$$
\text{noise}(x,z) = 3+\cos(12f\cdot(y+t_0(x,z))) + 2\sin(7f\cdot(y+t_0(x,z)))
$$
where $f$ is the noise frequency (must be integer for periodicity; default is $1$). The parameter $y$ determine the noise shift which, by default, depends on the tile center
$$
y = 1.23 + x_\text{tile} + z_\text{tile}^2.
$$