Pascal's Triangle and Its Relatives

Background

Other Questions

Other Groups

Can we modify the construction of Pascal's triangle to represent groups other than Zn?

One way to do this is to select two group elements, call them a and b.

Fill each square of the left edge of the triangle template with a, fill each square of the right edge of the triangle template with b, and fill in each remaining square by applying the group operation to the element in the square immediately above and the square immediately above and to the left of the given square.

Can we find fractals in these patterns?

If so, do they tell us anything about the symmetries?

Other Polynomials

Pascal's triangle was built from the coefficients of (x + y)n.

Can we build relatives of Pascal's triangle from powers of other polynomials?

Do these contain fractal patterns?

One simple approach is to recast the original Pascal's triangle as the coefficients of (1 + x)n.

A straightforward variation is to take p(x)n where p(x) is some polynomial other than 1 + x.

Slightly more generally, take p(x) q(x)n for two polynomials p(x) and q(x).

Cellular Automata and Pascal's Triangle

Finally, the rules generating Pascal's triangle may remind you of some cellular automata.

For example, with the standard Pascal's triangle to get an odd number in the generation n box in location i, exactly one of the generation n-1 boxes in locations i and i-1 must be odd.

Thinking of even numbers corresponding to dead cells and odd numbers corresponding to live cells, this means the generation n cell at position i is alive if exactly one of the generation n-1 cells at positions i and i-1 is alive.

In terms of the an N=3 binary CA, regardless of the state of the right neighbor, the (left, center) configurations (live, dead) and (dead, live) give a live cell in the next generation.

That is, a live cell is produced by any of these configurations:

(live, dead, dead), (live, dead, live), (dead, live, dead) and (dead, live, live)

Can you find cellular automata that generate some of these other Pascal triangle patterns?

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