Friday, November 29, 2024

Mean Voronoi blocks

Returning back to the article that implied that the most common 3D Voronoi cell would be cubic. From memory the paper found that the most likely 3D cell would have the topology of a cube. But that doesn't mean it has the geometry of a cube. 

When taking the average geometry we need to rotate the objects first before averaging them. 

Here I do this by creating a random 3D delauney triangulation and picking out only those vertices with 6 edges (representing a 6-faced polyhedron, i.e. a hexahedron).

To average these hexahedra pairwise, I compare their face plane equations (meaning the vectors from the centre point to the face surface and orthogonal to the face. I currently compare ever permutation of these faces to find the closest. Before averaging these vectors. 

There is a chance that these permutations could pick an illegal permutation, but I consider that to be quite unlikely. Nevertheless, that does mean the result still has a little bit of doubt remaining.

The result is not quite as I expected, in particular it seems to have one edge that is very close to being 0 in length:


The plane equation coordinates are (shown as face dots above): 
-0.194 -0.799542  0.177
-0.194 0.799542 0.177
-0.0325489  0.0  -0.919887
 0.32221 0.0   0.86651
 0.853741 0.0  -0.206742
-1.80479 0.0  -0.206742

this has corners at:
 -1.72998 -0.656383 -0.859826
   -2.02784 -0.00848857     1.74037
 -1.72998  0.656383 -0.859826
  -2.02784 0.00848857    1.74037
 0.674983  -1.25876 -0.944922
 1.04826 -1.00809 0.596529
 0.674983   1.25876 -0.944922
 1.04826  1.00809 0.596529

The shape has bilateral symmetry as expected.

While the topology doesn't look to be the same as a cube, any tiny deviation in the plane equations returns it to being cubic geometry. 

So maybe (I might even say likely) the average 3D Voronoi cell looks like this. It being the modal topology and mean morphology.

Here is the version with mirror symmetry included, from side and top views:

As expected, it lacks the bilateral symmetry of the previous polyhedron.   
Plane vectors:
0.901108  -0.245847 -0.0036255
0.284537 0.768137 0.307837
-0.0607338  -0.702541   -0.54646
-0.238406  0.624325 -0.635057
  -1.78769  -0.245847 -0.0036255
-0.149459 -0.245847  0.713359
corners: 
  -1.7839 -0.259842  -0.92409
-1.67419 -1.07282 0.108903
-2.01822  1.41839 0.813744
-2.01419  1.38803 0.885761
 0.978475 0.0619286  -1.64478
0.557456 -1.51177 0.425191
  1.21246  0.908574 -0.900279
 1.02075 0.176343  1.10404


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