Name of these Sierpinski-like fractals?

Question

A Sierpinski triangle can be created by

1. Starting with a row consisting of a single 1
2. Each row below is horizontally offset by a half-cell
3. Each cell is the sum-mod-2 of the two cells above it

Now follow the same rules, but starting with a row of several 1s. If the number of 1s is a power of 2, you get the Sierpinski pattern again (as in the first picture). Otherwise, you get something else.

Do these patterns have names?

Answer 1

It's an additive elementary cellular automaton of dimension 1. "Rule 60" or "Rule 102" for starting values $2^n-1$ ($n$ ones).

In such automata, one tracks the fate of an 1-dimensional array of cells, each of which may be populated with 0 or 1. The cells' contents evolve over time in discrete steps, where the new value of a cell depends only of it's previous content and the contents of it's immediate left and right neighbors.

A 2-dimensional version is Conway's Game of Life where the new value of a cell is determined by the values of the 3×3 grid in which the cell sits. However 2-dimensional automata are usually displayed as animations, wheres the 1-dimensional ones are represented as (color-coded) image where the cells are displayed horizontally and time increases top-down.

In your specific case of "Each cell is the sum-mod-2 of the two cells above it", we can write the evolution law down as:

   old:  111 110 101 100 011 110 001 000 
   new:   0   0   1   1   1   1   0   0   = 0x3f = Rule 60

where we just sum the left neighbor's value and the cell's value mod 2 to get the new value. The value of the right neighbor is ignored, but the general setup of this kind of cellular automata includes the 3 previous values.

We get an equivalent automaton if we ignore the left value and sum the lower two bits:

   old:  111 110 101 100 011 110 001 000 
   new:   0   1   1   0   0   1   1   0   = 0x66 = Rule 102

The top triangle is Pascal's triangle mod 2 (OEIS A001317 for "Rule 60", OEIS A117998 for "Rule 102", OEIS A047999 for bit-wise reading Sierpiński's triangle) with coloring red=odd blue=even and where the tip "1" is missing.

Answer 2

Your patterns are actually the cell-by-cell sums, modulo 2, of one or more "discrete Sierpinski triangles" (as generated from a single starting cell) offset horizontally from each other.

This happens because the recurrence rule that generates your patterns is shift-invariant and linear modulo 2, and thus obeys a form of the superposition principle. Indeed, the patterns generated by any other recurrence of the form: $$x_i(t) = \left( \sum_j a_j \, x_{i+j}(t-1) \right) \bmod m$$ for some modulus $m$ and constant factors $a_j$ will also exhibit similar shift-additivity modulo $m$.