Between a tetrahedron and a sphere

What's in between a tetrahedron and a sphere? You could smooth the tetrahedron partially but this no longer has any flat sides, or you could go to higher solids like an octahedron then an icosahedron, but these still have sharp edges, unlike a sphere.

A polyhedron has C0 continuity and a sphere has Cinfinity continuity. It makes sense that an intermediate could have C1 continuity, which is what this shape has:

We start with a tetrahedron:

then slice off the corners and the edges:

and repeat:

the limit as the iterations goes up is:

I am just showing the edges, but this is a convex solid with flat surfaces in all the circles. In this one the corners are cut off so the slices touch, and the edges cut off one third of the way along the sliced edges. The result has octahedral symmetry, with 8 largest circle faces. 

It is C1 continuous because its face angles (the first derivative) follow a devil's staircase function across an edge, this is a continuous function. 

Here's the solid object:

and here's the same thing but starting with a cube:

Here is the Octahedral rendered with transparency and refraction:

Mean Lumps

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:

vertices: x: 1,-1, 0, ϕ, ϕ, 0,-ϕ, 0, 1, 0,-ϕ,-1
               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.

Mean particles

The results of the previous post on mean points shows that average sets of points tend to not be next to each other. If we introduce time into the system, then this can be seen as a repulsion of the points. So let's try this out. 

Looking at just points in a 1+1D world, where the spatial dimension is in a circle. We can evolve a set of points as follows:

1. Pick any set of starting points, these are also the current mean point set.
   
2. Create a large population of sets, each set is equal to this starting set. 
3. For each time step, add a random offset to each point set, and rotate it around the circle to be as close as possible to the mean set. Accumulate the deltas (relative angle of these rotated points to the mean points).
4. add the total delta onto the mean point set. (i.e. generate a new mean set)
5. go to 3.

 The results are shown with time going from bottom to top, with starting positions 1,1.1,7,8  (angles out of 10 ticks around the circle):

Notice that points appear to repel each other, and the two points closest to each other repel the quickest.

The size of the random offset applied at each time frame affects how fast the particles communicate and 'see each other', with no repelling until the random dispersion of each point starts to interfere with its neighbour.

Since we use a large population of random sets, it basically represents a uniform dispersion of the sets over time, with a chosen dispersion rate. The dispersion rate does not have any effect other than changing the time scale on the resulting trajectories. So the results are effectively parameter-free. They are simply the mean trajectories of a set of n points, given the circle's rotational symmetry, and a maximum communication speed. As such, the trajectories are otherwise entirely defined by the initial position of the points.

The behaviour isn't particularly interesting though, but we can make it more so by going from first order to second order dynamics. This is done by adding the change in the mean points on to all the candidates at each time step, reflecting the nature of particles to maintain their velocity. The result has a little more oscillation:

This is basically degeneracy pressure in action, just without complex numbers. However, there is an additional behaviour in this system: unlike degeneracy pressure in absolute coordinates, which converges (if it loses energy over time) to equidistant points, with circular symmetry included the points stabilise to the non-uniform dispersal seen in the previous post. At least it would if the communication distance was wide enough.

This makes it a system that has repulsion and a sort of attraction (as the converged set of points cluster on one side of the circle). This is not visible on the above graph due to the communication distance being lower, but it would be interesting to investigate this behaviour further.  

I expect the 2D version (on a sphere) to show even more unusual behaviour, it could well display orbiting behaviour around the lowest energy point locations (seen in the previous post).

Also note that the set of particles conserves momentum. At least, it appears to.