Locomotion and self-restructuring
There is one main candidate for a structure that moves into a fixed direction. This is Diggutome saliens:
This shapes, which is stabilized but moves roughly to the left side of the screen, undergoes periodical momentary restructuring, where the outer shell opens and closes again, while the innermost sphere seems to vanish and appear again. It is peculiar that while 4 apparent spheres are visible, they are actually 2 distinct nested structures that are topologically hollow spheres with some 'wall thickness'. Furthermore, while the restructuring usually manifests in the components connecting, there is one instance in the Betti numbers, where the cavities seem to join instead. Further detailed study for this particular shape might prove useful, as it would be insightful to find out where exactly the change in topology takes place.
Tubes and Membranes
When accounting for periodic boundary conditions, there are two emerging types of lifeforms that form by crossing the boundary. These are Stylomembranomes and Planomembranomes. The former connect with through the boundary window in 1 dimension, while the latter do so in 2 dimensions.All Stylomembranomes except for one case are stable shapes with constant Betti numbers. The only candidate for which this does not hold is Stylomembranome limus, shown below:
While this lifeform moves in one direction similar to Diggutome saliens and shows perturbations on its surface, it still has a completely stable topology in terms of Betti numbers:
The Planomembranomes however differ greatly in their behaviour: Planomembranome lithos is completely stable with zero variation in Betti numbers. Here we can count by hand that 30 tunnels exist within the structure. This is represented exactly with Betti1 being of value 33, as there are 3 tunnels that are counted twice due to overlapping over the boundary. b0 is slightly lower than expected, probably due to some 'internal droplets' of the structure not being fully disconnected from the rest. It is noteable that b0 and b2 sum up to 33 again. This is likely due to small variations in connectivity creating 5 instances where a void is detected instead of a connected component.
Furthermore, there is a Planomembranome that acts roughly analogue to Diggutome saliens, moving in one direction while undergoing restructering, but being of similar base shape to Planomembranome lithos.
This lifeform is named Planomembranome saliens. While the original animation was approximately of size 180MB, the resolution in this gif is greatly diminished due to compression. However, a rough cycle is repeated twice during the observed timeframe. In this cycle we see a sharp spike in b1 while the tunnels are momentarily stable (and the shape looks reminiscent of the Planomembranome lithos). While it is hard to keep track of what is happening intuitively, one can observe that b2 first rises, followed by b1 and b0 with an offset in time. They then decrease in the same order before repeating that behaviour. Another interesting finding is that there are two separate topological restructurings happening. Between frames ~5-22 there is approximately the same change as around frame 68-85, while the time between repeats another subcycle twice. So the whole lifeform goes through a larger cycle that is made up of two separately nested cycles, both repeated twice.
The last remaining members of the Planomembranomes are Planomembranome inversus and the similarly looking Planomembranome vagus. Both are shown below and exhibit comparable behaviour, starting with an initial restructuring, until slowly settling into a stable shape (much slower other stable shapes studied yet, taking in excess of 100 frames to reach full stability).
Planomembranome inversus goes through a similar restructuring to Planomembranome saliens, while Planomembranome vagus simply displays a short dis- and re-attachment of the two membrane layers:
Planomembranome inversus is seeing an increase in b2, as small floating orbs manifest close to the tunneling and a decrease as they start to reconnect (not completely visible as reconnection, since the visualization ignores max values to show the interior better):
Planomembranome vagus sees continued increase in b1 as the connecting spaces between the two membranes differentiate into smaller, more fragmented looking shapes.
Unstable Shapes
Most patterns maintain their structure and therefore do not classify as 'lifeforms' in Lenia. However, some patterns could be described as quasistable, taking long timeframes to fall apart while appearing stable on first glance.
The most noteworthy example of this phenomenon is an unnamed pattern with ID 45 in the original Lenia code. This is a pattern that stays somewhat stable for over 500 frames before it vanishes, all while shrinking slowly and diminishing in topological complexity, as shown by the following gif and Betti-graph.
