self-similar cross sections by class

What are the classes of cross sections through self-similar objects, according to my self-similar shape classification? Let's look at the 2D self-similar structures first:

2D

These are approximate, I will try to update these tables when I get more time and find mistakes. They are the 'almost surely' cross section structures. For instance a cross section through a 2D void-cluster almost surely will not hit any points in 2D so will be a completely empty void-void structure. Likewise, a cross-section through a cluster-cluster will almost surely cut through infinitely many of the blobs (over all cluster-clusters, not just a geometrically aligned one), making it also a cluster-cluster in 1D.

3D

In 3D things get more difficult so expect errors in this table. The main difficulty is that we get shapes that are combinations of more than two structural types. For instance a cross section through a sponge-sponge will definitely have separated objects (a cluster) but the objects will also have loops (sponge) when the cross section aligns with the loop, and hierarchical protrusions (tree) when we capture just the start of these loops in the cross-section. So it has the characteristics of all three. 

Typically this is written from the denser to the sparser type, in this case a sponge-tree-cluster. This is either interpreted as sponge-tree cluster: a cluster of sponge-trees (trees with holes in them in 2D); or as a sponge tree-cluster: a cluster of trees, but made of sponge (holes in it). These are the same thing, suggesting that A-B-C classes are associative. i.e. (A-B)-C = A-(B-C).

There is less obvious a pattern than in the 2D case, but it does have some consistency. The first word (the qualifier) is void, cluster, tree, sponge, sponge, sponge, solid from left to right and the last word (the noun) is void, cluster, cluster, cluster, tree, sponge, solid from top to bottom. 

These have both dealt with the two missing classes (shell & foam) at different ends. The first by repeating the second-last type (sponge) and the second by repeating the second type (cluster).

In order to accomodate the middle terms, there is a simpler way to work this out. Firstly, we look at the A-A diagonals. This has a pattern to it. 

Then all of the other cells of the form A-B are just the cross section result for A-A followed by the corss section result for B-B. There is some reduction going on. A tree-cluster-cluster will reduce to a tree-cluster and a tree-cluster-tree-cluster will reduce to a tree-cluster. Otherwise the cross section of the non-diagonal cells are just the piecewise combination of the cross section of the corresponding diagonal cells.

As for the A-A diagonals, they follow this pattern:

Numerically (with void-solid being 1-5) it is: 1-1, 2-2, 3-2, 4-3-2, 4-3, 4-4, 5-5. In 2D the pattern is 1-1, 2-2, 2-2, 2-2, 3-3. In general the n-dimensional cross section of the n+1 dimensional structure is:

with repeated terms removed in the triplet. Then X(A,B) is just X(A)-X(B) with appropriate reduction of repeated terms. '-' being a hyphen here.

The use of these tables is that you can guess the class of a higher dimensional structure just from its cross section. The most common case of a cross section is when the structure is n+1D so is a moving structure (a behaviour), and you have access to still images of it. So the above table can suggest the 2+1D structure from its 2D structure at any one time.

To guess a 3+1D structure we would need a 4D table.

My guess as to the A-A pattern would be: 1-1, 2-2, 3-2, 4-3-2, 5-4-3, 6-5-4, 6-5, 6-6, 7-7.

So the cross section of a shell-shell (number 5-5) would be: 5-4-3 which is a shell-sponge-tree. 

tables

Is turbulence a shell?

What shape is turbulence?

It sounds odd to give the chaotic motion of countless fluid molecules a shape. But we might be able to answer by being a bit broader with our use of the word shape.

We know that turbulence is statistically self-similar so it may fit into one of my structural classes of self-similar shapes. But turbulence is also moving, so we have to look at a 3+1D version of these structural classes. It turns out that mixing temporal and spatial dimensions doesn't have any significant effect on the set of classes, so the 3+1D 'behaviours' are the same classes as the 4D spatial structures. There is a 9x9 table of these.

A solid body of fluid still doesn't have a shape or structure in the conventional sense of a subset of 3+1D. But there are various features in turbulent fluid that we can use to define a shape which is quite consistent over time. The main ones are present in 2D also:

Left is a hyperbolic field, defined by its centre point, magnitude and its two 'invariant' axes. In 3D it is defined by the central line (or curve) and the two intersecting surfaces (a type of 'coherent turbulent structure'). 

Right is a vortex, defined by its centre point and magnitude, in 3D it is defined by the central vortex line (or curve) and a magnitude.

In 2D all fluid flow is composed of these two features, for example we define the vortex in red and the hyperbolic structure in green here:

The infinitesimal transformation at the vortex is defined by an anti-symmetric matrix (a rotation) and at the hyperbolic point by a symmetric matrix (a nonuniform scale), both are unitary (determinant=1) for incompressible fluids. Halfway between these two matrices are the lower or upper triangular matrices, which represent a shearing action rather than rotation or 'squashing'. This can be our boundary between points inside the 3+1D set and points outside.

In other words, the shape that we are classifying is the set of points in 3+1D whose infinitesimal transform according to the vector field is more anti-symmetric than symmetric. It is the set of points that are in the vortex and not those between vortices.

In 3D vortices look like long limbs, which tend to stretch out over time. Here is a nice image where the vortex curves in a turbulent flow are separated by approximate scale:

This shows the scale symmetry, with many vortices at the small (blue) scale and a few at the large (red) scale.

Even though they look to be distinct curve segments, this is just a threshold on the vorticity, each vortex should extend until it meets another vortex or drops to exactly zero vorticity. So I *think* that most of these curves would connect either to themselves or to the boundary. 

That means that the central vortex lines would make something like a void-sponge structure in 3D and therefore either a void-shell or void-foam structure in 3+1D. If we are talking about the solid shape I described above with the thicker vortex limbs, then it would be a sponge in 3D and either a shell or foam in 3+1D.

To further investigate its 3+1D behaviour we need to look at how it behaves. Much of the time these vortices move around but do not interact. They seem to somewhat repel each other. However, there are interactions that occur, such as this:

In 3+1D this is just a single surface in a saddle-like shape. Likewise, it is possible for a ring vortex to appear from nothing, e.g. from a pushed limb of water such as out of a pipe. Again, in 3+1D this looks like a (paraboidal) surface.

So is the structure in 3+1D just a collection of wobbly surfaces? Well, I think that it also has branching, just not very commonly. Firstly, I think there may be branching at the points where the vorticity goes to 0, such as when 4 vortex lines meet in a plane, or where 8 meet as though from the vertices of a cube to the centre. Secondly, we can see examples of a thick vortex limb flattening out and splitting into two thinner limbs:

This is real data, and even shows a second splitting to even smaller vortex limbs:

So the structure can be branching limbs in 3D and therefore branching surfaces in 3+1D. This makes it a shell or a foam.

In this paper we also see branching happening in the last images, and at a smaller scale. For anti-parallel vortex streams:

and for orthogonal vortex streams:

The difference with a 3+1D foam is that these loops always reconnect back into thicker limbs. With a shell they may branch again and again until they disappear, in terms of turbulence that is descending the energy cascade until the movement becomes thermal energy.

So, all of this considered, I think that there is a good chance that turbulence is a 3+1D shell (specifically a shell-shell), if we are describing the vortex regions. If we describe just the non-vortex regions we will get the set complement of the shell-shell in 3+1D, which is in fact also a shell-shell. 

And none of this is affected by how thick we make the vortices, so we needn't necessarily stop at the 'shear surface'. What we can't do is use the 'vortex surface' used in many visualisations because this is based on a fixed vorticity value, so it disconnected and therefore looks more like blobs than a connected sponge-like structure in 3D. 

Incidentally, there is also a lot to be said about knots in turbulent vortices, but it doesn't matter for the 3+1D structural class because a knotted string in 3D is not a knotted surface in 3+1D. 

Inversive cluster-shells

Here is a cluster-shell:

to tell that is is made from a cluster (of balls) we can take a cross section through it:

here it is from a more top-down angle:

This one is a bit of a complicated shape to engineer, it has five steps. Step 1, make an airtight strip of spheres:

Then add a strip at half that scale below it, and half again below that etc, making the resulting strip twice as deep:

Now do the same stacking but on a 2D grid of these strips instead, with the big spheres at the grid vertices:

It is now already a cluster-shell, but it has smooth areas and it is unbounded, so we can do better.

Next stereographically project this onto a sphere:

Lastly replace each sphere in the above with this shape, as shown in the first image. Here's a last pic from a more side on angle:

The inversive tree-foam is just the set complement of this. Here I have applied a sphere inversion around the largest inner ball to make it bounded in a sphere, and I'm rendering it backwards (with transparent and solid swapped) so you can see the inner structure:

------------ OLD --------------------

The below almost works but doesn't quite work, but I'm leaving in here for reference:

Cluster-shells are another structural class that are hard to generate with inversive geometry. They therefore are one of the five unspecified classes in this description of all 49 classes of the 7x7 structural classification of self-similar shapes in 3D.

One way that I *think* counts as a cluster-shell is shown here:

It is built in a similar way to the cluster-tree. It starts with a Ford sphere packing, using the triangular lattice version, here the spheres are shown at half size:

At full size they all are packed in so no sphere can move:

Between each touching pair of spheres we can make a watertight concave triangle bridging them. This is true for the underneath spheres too:

This is a cluster-foam as the lower 'triangular mesh' of spheres butt up against the big spheres. We can now stereographically project this infinite mesh of spheres onto the (Riemann) sphere:

Much like the sphere tree (cluster-tree) structure, we can then recursively replace each sphere with this above shape (ignore the thin gaps):

The problem with this is that it remains a cluster-foam as the recursed sphere is aligned with the small underneath sphere mesh, so it has millions of air pockets.

Instead we can turn the sphere shape upside down every time we recurse each sphere. Giving the resulting shape, the (probable) cluster-shell:

A slice-through shows the solid spheres from which is it made:

Unforunately this isn't a true cluster shell as some of the intermediate spheres don't match together properly.

Inversive void-shells

The void-shell is one of the trickiest structural classes to generate using sphere inversion, being one of only five that are missing in the 7x7 classification of self-similar geometry here.

A void-shell is the boundary surface of a space-filling tree, so is a fractal with no volume. There are two simple ways to make this using linear transformations, the Menger-shell:

and the Viscek-shell:

(both my own naming convention as I don't know of any better names).

My first successful attempt at this class with sphere inversions is based around the fairly well-known square-symmetric Apollonian gasket:

We use this pattern on each each principle axis. The large white circles are mapped to the full disk such that the two inner contact points become vertical or horizontal opposites. This creates a sort of cocoon with the triangular hole seen in the centre of the below images. 

The four white circles remain as 3-plane shapes after being transformed to the full disk. The remaining asymmetrical concave quadrilateral black areas are Mobius transformed into the inner concave quadrilateral, and this is then treated as a single-plane structure like the one above.

A tree-solid is simply the set inversion of the above. However, inkeeping with having the structure within a sphere, we can do this by:

1. orthographically transform the unit sphere to an infinite plane

2. repeatedly scale down any points larger than the above 'tri-disk's radius.

3. using the single-plane structure, but one-sided at the start, so features are only on the inside of the sphere

The result is an inversive tree-solid:

Rules for Inversive Classes

This classification of recursive shapes can be applied to inversive geometry in 3D, which is geometry that is symmetric to inversions around various 'generating spheres'. It is a 7x7 table that looks like this:

How we go from generating spheres to the class above is what I describe in my new paper preprint. It is a connection through about 4 layers of abstraction. 

Firstly, we have the generating spheres. The inversive limit sets are generated by repeatedly inverting points within these spheres such that the sphere surface remains unchanged. 

Secondly we have the connectivity graph, which represents many sets of generating spheres that all have the same connectivity. The nodes are the spheres and the edge numbers represent the 'branch order' n of the two intersecting surfaces. The dihedral angle of intersection is pi/n.

Thirdly we have the hypergraph, which represents many connectivity graphs with the same broad limit-set connectivity. In this diagram the branch orders are not important, just the presence of not of pairwise sphere intersections (an edge), triplet-wise sphere intersections (a semi-translucent face) and quad-wise sphere intersections (a solid volume). 

Fourthly and lastly we have an abstract hypergraph, which represents many types of hypergraph that all map to the same structural class in the 7x7 table. This structure is a schema for generating hypergraphs. The key component is the polyhedral sphere, which represents any polyhedron in the hypergraph that is intersected by a single orthogonal sphere. 

In the above abstract hypergraph, the red polyhedral sphere is the parent and the single curved edge represents one or many child edges in the hypergraph. In the hypergraph above it, there are four such edges and the spherical polyhedron is an irregular tetrahedron.

This particular abstract hypergraphs generates cluster-sponges. These are sponges that would be clusters if boundary points were removed. 

You can see this limit set in the table at the top of the page. These can be viewed in realtime here.

So this new paper is an extension of the previous paper but rather than creating a single family of limit sets for the five main diagonal classes, it is a method for generating a broad family of limit sets for all 49 classes.

Well... nearly, you can see in the top table that five limit sets are missing. These are easy to generate with linear limit-sets (see them rendered here) but hard to find for inversive limit sets. It is not certain whether they even can be generated, at least with simple sphere inversions. Working out whether these last 5 classes is possible with sphere inversions is the last major open question for this structural classification.

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

void-sponges

 Void-sponges are probably the easiest self-siimilar shape to make with inversive geometry, but it isn't obvious whether there is a particular shape that is somehow a better archetype than others; something that is 'the canonical inversive void-sponge'.

It does exist and there are six of them. The trick is to realise that the most symmetric 3D structure is a 3-sphere, and that regular polychora (4D polyhedra) have evenly distributed vertices on the 3-sphere.

We therefore run the iterative inversions in 4D with sphere inversion centres at the polychora vertices, then stereographically project them back into 3D. This projection is conformal and Mobius so spheres remain as spheres. Any such projection will do, but in practice there is one that mimimises the object size by placing the pole on the face centre (farthest from the vertices), this also has rotational symmetry in 3D so is the best choice. Noting that it is structurally the same shape regardless of the Mobius transformation, and these inversive shapes should always be considered as equivalence classes under not just similarity transformations but also Mobius transformations. 

So the six shapes come from the six regular polychora, the 5-cell:

The other free parameter is whole number n where the intersection angle between spheres is 180/n. For large n we get sparser shapes like above. And for smaller n they are thicker like below.

There is also the 8-cell:

16-cell:

and the 24-cell:

and here is as dense as the 24-cell gets before it encloses a sphere:

The 5-cell and 24-cell are special in that they are their own dual. A consequence of this is that you can fit another copy of the shape interwoven in it but not intersecting:

5-cells:

we can manipulate the Mobius transforms to make them the same shape in 3D:

24-cell:

and again transformed so they are the same shape in 3D:

here's a thinner version:

These shapes are not new I'm sure, but I am glad that at least one of the 49 classes of self-similar shape has a definitive family of archetypes.