Mean polyhedrons

The previous post was a method to find the average polygons of a certain number of sides, assuming that one defines a polygon by its interior angles and edge lengths. 

If we just define shapes by their edge length then we can obtain average triangles and also average tetrahedrons. Without loss of generality, we pick randomly distributed edge lengths which sum to 1.

For triangle edge lengths a,b,c, we get the means (shown over 4 independent trials): 

a,b,c: 0.4, 0.2, 0.4

An isosceles triangle.

when we include reflections, we get an obtuse triangle:

a,b,c: 0.444, 0.361, 0.194

For tetrahedrons we pick 6 edge lengths, and throw out any invalid tetrahedrons. Those being any with faces that disobey the triangle inequality (sum of shortest sides must be greater than longest side) or disobey the tetrahedron inequality (sum of smallest face areas must be greater than the largest face area). 

With these remaining tetrahedra, we find the closest match (dot product of the vector of edge lengths) for all relative orientations. For tetrahedra that is 12 orientations. We convert the mean edge lengths into coordinates like so:

Under this system, the mean tetrahedron is fairly unstable, but seems to look a bit like this:

a,b,c,d,e,f: 0.157258, 0.184638, 0.178683, 0.143683, 0.16418, 0.171558
r,s,t,u,v:     0.115537, 0.144023, 0.113931, 0.035557, 0.123238

coords: (0,0,0), (0.179,0,0), (0.116, 0.144,0), (0.114, 0.0356, 0.123)

If we allow reflection when matching shapes, then the result look more like this:

a,b,c,d,e,f: 0.195577, 0.184195, 0.126073, 0.159834, 0.193624, 0.140698
r,s,t,u,v: 0.0458932, 0.178386, 0.0402287, 0.0351516, 0.130161
coords: (0,0,0), (0.126,0,0), (0.0459, 0.178,0), (0.0402, 0.0352, 0.130)     

Here it is top down: 

Cuboids

For normal right-angled cuboids, we know from the mean signals post that the mean edge lengths must be proportional to 2, 5 and 11:

This is because the cuboid has no further constraints on its edge length, and its symmetries are the same as a length 3 sequence with reflection allowed.

 

In order to do the same thing as the last post for more complicated polyhedrons, we need some invariants. For n-sided polygons we knew that the sum of angles adds up to pi*(n-2). For polyhedrons it isn't quite as simple. 

A very general invariant is called the Gram-Euler theorem , this equates the n-dimensional angles of each 'cell' together in one formula. Where the cells of a polyhedron are the volume, the faces, the edges and the vertices. 

This post might slightly help explain it.

A simpler theorem (which may be what you end up with by applying above to 3D polyhedra) is called Descarte's theorem, it says that the sum of the deficit angles equals 4 pi. Deficit angles are 2pi minus the sum of the corner angles for the faces that touch that corner.

For the cuboid example, the 8 deficit angles together with 12 edge lengths gives 20 degrees of freedom. But the above invariant, and the scale invariance of the shape reduce that to 18 degrees of freedom.

How many does an irregular cuboid have anyway? Well there's 3x8 = 24 vertices, but there are 3 translation symmetries, 3 rotation symmetries and one scale symmetry, reducing that down to: 17 degrees of freedom.

So the deficit angles + edge lengths is a fairly canonical description of an irregular cuboid, we just need to throw away that extra degree of freedom of constructions that don't make a closed cuboid. 

Summary: 

So far, the following seem to be approximately true, and they may well be exactly true:

• The mean triangle over all edge lengths is isoceles with lengths l,2l,2l
• The mean cuboid over all edge lengths has dimensions 2l,5l,11l,
  
• The mean tetrahedron (incl reflections) over all edge lengths is scalene (all edge lengths different).
• The mean triangle over all edge lengths and interior angles is also isoceles.
  
• The mean quadrilateral over all edge lengths and interior angles is a kite
• The mean convex pentagon over all edge lengths and interior angles has bilateral symmetry
• The mean quadrilateral (incl reflections) over all edge lengths and interior angles is an isoceles triangle, approx lengths l,2l,2l