I am trying to subdivide a finite portion of the plane into a set of at least 20 or so random interlocking shapes, with the shape boundaries having a fractal dimension around that of borders between countries that aren’t simply straight lines (which I crudely estimate to be $ 1.15\pm0.1$ , based upon visual inspection of Wikipedia’s list of fractals by Hausdorff dimension). Preferably, the plane would have toroidal boundary conditions, although generating another single country-like shape to serve as the overall boundary would also be acceptable.

Running Kruskal’s algorithm for maze generation on a torus (and keeping only the boundary) gives about the right fractal dimension, but that only creates one shape that tiles the plane by translation. Running several instances of the DFS algorithm at once (again, keeping only the boundaries between the various sub-mazes) generates highly unrealistic narrow portions. (Additionally, in both cases, the result is made of tiny discrete units, which may or may not be a problem). Using a Delauney triangulation is an excellent method to make a graph representing a system of countries, but its corresponding Voronoi mesh makes for perfectly straight borders.

While it is likely implied by solving the main question, do also let me know if there is a simpler procedure for generating a single country-like shape. (I’ve tried a polar plot of fractional Brownian motion, but the anomalies look decidedly non-random to the eye, as they all point directly towards or away from the pole. Using a 3D terrain generator (and slicing it at a certain z-value) works, but one has to generate the entire terrain, not just the boundary in question; this method also produces many excess islands (which might actually be beneficial, but that could interfere with the border-making process).)