We have seem beautiful shapes like level-n Sierpinski triangles and repetition patterns.
There are many and many (mathematically speaking, uncountably many) other shapes constructed from these pieces.
Actually, you can form bigger and bigger cluster by connecting pieces as you like and
changing configurations locally when you face deadlocks. They are all shapes with various sizes of triangular holes like the following.
The shapes constructed with TriMata are characterized through an automaton.
First, note that such an infinite shape is a subset of triangles arranged as follows.
In this figure, three triangles are connected at each vertex.
According to the form of our piece, the target shape contains two of them or none of them for every vertex.
On the other hand, any subset that satisfy this condition can be generated in this puzzle.
Therefore, if memberships of triangles on one horizontal line are determined, then that on the next line are determined accordingly; a triangle belongs if and only if exactly one of the two triangles above it belongs. We would call it a triangular cellular automaton with the rule "exclusive or", which is equivalent to the rule 18 elementary cellular automaton. By repeating this procedure, memberships of all the triangles blow that line are determined.
For the line above, if one determines whether one piece belongs or not, then
the memberships of all the triangles on that line are determined. Thus, there are two possibilities for the line above, which are complements of each other.
Note that this observation through a triangular cellular automaton is applicable in three directions. Therefore, if we are given a collection of connected TriMata pieces, there is a unique extension inside the smallest opposite-directed triangle that contains the pieces.
After this, a player can choose any open connector and add a piece in one of the two possible ways. Then, it expands the opposite-directed triangle and
determines the pieces in it.
In this way, one can proceed the construction infinitely.