Average Things

Well isn't that an inspiring title!

'average' is usually associated with uninteresting and featureless. An average image is uniform grey, an average car path is a straight line, an average surface is everywhere flat.

However, things get more interesting when the object in question is an equivalence class. For example, a triangle is the equivalence class of three coordinates under any rotation or translation operation. 

An average --- in particular a mean (μ) --- is defined in the general case as:

When the object x is a list of numbers, 

is just the sum of the squares of the differences, but for equivalence classes it is the minimum square distance between μ and any of the equivalent expressions of x.

For the example of the triangle, this means that you have to fit every randomly selected triangle onto the running average before adding it on. Do this for all randomly selected triangles (or at least a large enough number of them) and you get the average triangle. And it isn't a uniform equilateral triangle:

You can do the same for quadrilaterals, and five-sided polygons too:

Your average quadrilateral is not a square, and the average five-sided polygon isn't even convex, let alone a regular pentagon. More variants here.

We can do the same thing in 3D, e.g. with tetrahedra parameterised by their edge lengths:

Cuboidal structures are harder to parameterise, but if we restrict to just right-angled cuboids then the average shape is determined by the equivalence class of 90 degree rotations, giving:

What about general 3D blobs, what is the average blob shape? This is really hard to formalise, so I'll just try and get the rough characteristics of such a shape. My approach is to take the vertices of a regular polyhedron and parameterise their distances from the centre, to represent some general polyhedron that we can smooth to get our blob. In this case we choose a random selection of distance 'energies' (square distance), and the equivalence class is the discrete set of rotations that preserve the regular polyhedron:

24 for an octahedron:                                                                                  and 60 for an icosahedron:

Other than the icosahedron blob being fatter, the two attempts at an average blob are quite similar, they have bilateral symmetry, the two other axes are different elongations and are both asymmetric with an egg-like fatter and thinner end (info). No real blob has bilateral symmetry, it is because 'left handed' and 'right handed' blobs are equally likely, if you add reflection into the equivalence class then this symmetry is lost, but then you have two average blobs, itself and its mirror image.

What other equivalent classes are there? How about repeating signals. We 'forget' the starting point of a discrete signal, by defining every starting offset as an equivalent signal. This gives a different mean signal for each signal repeat length, here from 1 down to 10:

As you can see, these are nothing like a standard mean signal (which would be a flat y=0). In fact, rather than being featureless and uniform, they are close to the opposite. It seems that the mean signal (under the equivalent class) is something that is strongly differentiated horizontally, which all the random signals that make it up can anchor to horizontally.

I also find it interesting that the signals for each repeat length are all very different, and don't follow a simple pattern, which is unusual as it shows complex patterns emerging from a very simple premise: mean repeating signal.  (variants)

Here is a mean 4x4 repeating 2D signal, rendered as a greyscale image. For 2D we can include 90 degree rotations also in the equivalence class:

again, the signal has some structure to it, it isn't random shades, and it hasn't averaged to grey. This is an average 4x4 greyscale texture. (variants)

How about something a little less discrete, a mean set of angles in a circle:

In this case, the pattern is quite clear, the mean set includes one large angle gap (to anchor all the samples to) and decreasing gaps away from it (symmetrical on both sides). 

The equivalent in 3D is a mean set of points on a sphere, the equivalence class in now 3D rotations. Here for 5 points:

I haven't tried for a large enough number of points to see a pattern, but they aren't uniformly distributed, and the points are never close together. (info)

On that note, if we apply the previous 1D case iteratively so the points have a velocity, and the new mean velocity is calculated each step, then we can see this aversion to being close together as a form of repulsion of the particles:

time going bottom to top, we see the initially closely set particles separate, then the rightward acceleration of the left-most particle is due to proximity to the right-most particle (since we're in a circular universe here). 

So all of these examples are pretty simply defined average objects. It is just that I have included their equivalence class in the average, which is what ought to be done anyway. This shows that, far from averages being the least interesting example objects, they can often have unexpected complexity.

Why does any of this matter? Well it is growing complexity from almost nothing... There may be an extension of this to cosmology. Path integrals take the place of the average, particle paths represent each random sample, and the symmetries of the universe are the equivalence classes. The observed macroscopic universe is the integral of all possible universes but you rotate/translate/boost etc each candidate universe before adding into the integral. This could form an another way of getting complexity from 'nothing' than the unstable-vacuum approach.

Here is a visual summary of the previous structures, with blog links.