Evolution in shapes

In my fractal classification I showed how scale-symmetric structures can be classified as a list and, for more expressive power, as a table. The base list is length 3, 5 and 7 for dimensions 1, 2 and 3. The first and last members are just nothing and solidness respectively and can be ignored as trivial cases. In addition the latter half of the list are negations of the opposite end of the list, where negation means emptiness and solidness are swapped. Consequently there are only 1, 2 and 3 unique shapes for dimensions 1, 2 and 3.

In that page I present a classification of moving structures, which is simply stating the range of classes in the above table or lists which the structure changes between during its lifetime. This has a lot of expressive power, but there is another, more constrained description for such dynamic fractals.

We can generalise the initial classification lists by giving them a time dimension. If the time dimension were like any other spatial dimension then the lists generalise easily. You would use the 3D list for the 2+1D case, and a 4D list for the 3+1D case. The 4D list has 9 elements, or 4 unique non-trivial elements.

anyway, time is different from other spatial dimensions. We can model it as having Lorentz rather than rotational symmetry, in which case the shapes are constrained to acting within light cones. The upshot of this is that (in the time direction) nothing grows to too sharp an angle, and objects don't suddenly appear from nothing. Other than this geometric constraint the list is the same for the +1 spatial dimension list.

So what does it look like?

In 1D we have intervals on the line which either grow from and to nothing, or continuously split.

The equivalent happens in 2D where we can have individual blobs growing and shrinking to nothing, and tree-like blobs that continually split into smaller trees and grow. We might call these two behaviours sporadic and asexual reproduction respectively.

There is a third behaviour in 2D, where network-like blobs split apart and change size, but eventually connect up with some other network-like blob. We might label this 'sexual reproduction' as the new child blobs always come from combined blobs.

The 'sexual reproduction' case is in some sense the special case in 2D, because geometrically it is the same class under negation. And 2D is the smallest dimension where it can happen. We note that most life on earth lives in a fairly 2D world due to the constraint of gravity.

However this is less the case in the ocean, and the 3+1D geometry contains a single new behaviour, which is the special case for 3+1D.

This behaviour looks like a 3D sponge (network of connected arms) where the arms split into two arms, thus making the network more complex, basically growing new mini versions of the network, and also growing in size overall. This behaviour is a little like the asexual case above, as it is unidirectional towards more complexity and growth, but it never separates and arms don't disconnect from their ends. So even though small versions of the network are continually generated and grown, there is no separation and the whole structure expands.

This is actually quite similar to the growth of a sea-sponge. The sea sponge is an unusual life form which many think of as a single organism but is in fact made of millions of unspecialised and quite independent cells.

Anyway it might be interesting to consider this third method of reproduction from an evolution standpoint, and whether it applies to sea sponges. Unlike asexual and sexual reproduction the genetic information is never detached, instead it could vary with location on the network, and the sub-network with effective variations might grow more than defective areas.

Another important area that evolves is the universe at large scales, and here there is perhaps an analogy to the sea sponge: Galaxy filaments and Galaxy walls (or sheets). Here the growth is from inflation and the formation into walls and then sheets is approximated by the Zeldovich pancake dynamics.