So what does a mean shape look like overall? I'm talking about any sort of blob-like shape. A rock, a piece of dough, a pebble, a chunk of ice in saturn's ring.
This parameter space is pretty hard to pin down, and each of the above have different processes that form them. But one common thing is that they are fairly smooth, and they have a continuous, simple surface. We aren't talking about sea urchins or sponges or tree shapes.
Regardless of the exact parameter space, the average shape is constrained by the equivalence class, and it is due to this that we can get a rough shape, without knowing the actual probability density of the parameter space.
In this post I make an attempt at an average shape using an octahedron with randomly selected radial distances of the six vertices, using a random distribution of 'energies' (the square of the distance) as used in previous posts. In fact we already have the result, it is from the dice numbers in the mean textures post. The dice values for the sides 1,2,3,4,5,6 are: 3.35, 2.57, 2.86, 2.86, 4.58, 5.98, which translating to radial distances gives this shape:
The left side is the octahedron from three angles, and the right is a smoothed version to give an idea of the blob shape. As you can see, it has bilateral symmetry, but is elongated and somewhat egg-shaped in the two other axes.
We can do the same thing with icosahedra. The result is:
y: 0, 0, ϕ, 1,-1,-ϕ, 1, ϕ, 0,-ϕ,-1, 0
z: ϕ, ϕ, 1, 0, 0, 1, 0,-1,-ϕ,-1, 0,-ϕ (ϕ is the golden ratio)
lengths: 1.03,0.91,0.91,0.92,0.92,0.90,0.93,1.65,0.93,0.93,0.93,1.04
The result is a bit different, but the main features are similar to the octahedron case: It has bilateral symmetry on one axis, and an egg-shaped elongation on the two other axes, the proportions of these two elongations are similar with the octahedral case, the main difference being that the icosahedral blob is fatter (bottom row).
Nevertheless, these two blobs are quite similar looking, and remind me of an almond a bit.
