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. | ||
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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 |
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 |
A straightforward variation is to take |
Slightly more generally, take |
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 |
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 |
In terms of the an |
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) |
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Can you find cellular automata that generate some of these other Pascal triangle patterns? |
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