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Thursday, April 7, 2011

And more!



So the answer to the question in my previous post turns out to be: it depends. It depends on what you mean by "the same idea as Pascal's triangle, but in three dimensions," which boils down to how you want to define adjacency. Pascal's triangle adds the two numbers that are "above" it in the two-dimensional triangle (which is skewed sideways if you write it in a grid). So if you're making a pyramid on a grid, what counts as the numbers "above" a number? is it all nine (ie: n,s,e,w, nw,ne,sw,se, and directly above)? Or is it just the four at the cardinal directions plus the one directly above?

If you choose the former, the sum of each level of the pyramid produces the powers of nine. If you choose the latter, its the powers of five. This is unsurprising in retrospect; the series of powers you will produce depends on the number of neighbors you define as adjacent to the central cell. Ie: 9 way adjacency produces the powers of nine.

Come to think of it, making the pyramid on a square grid isn't necessarily the next logical step from the original Pascal's triangle. You could make a triangular grid (though not with a spreadsheet; not easily anyway), which would give you four neighbors, presumably resulting in the powers of four.

Or you could do an octagonal grid, which would give you 9 neighbors and the powers of 9 again. Come to think (further) of it, that's essentially what you're doing when you define adjacency in the first way above (ie; nine-way adjacency).

(note: each square is intended to be stacked on top of the one below it, forming the pyramid. )







9-way adjacency













5-way adjacency










1

sum





0 0 0




sum





0 0 0



2

1





0 1 0




1





0 1 0



3








0 0 0











0 0 0



































1







0 0 0 0 0









0 0 0 0 0


2







0 1 1 1 0









0 0 1 0 0


3

9




0 1 1 1 0



5




0 1 1 1 0


4







0 1 1 1 0









0 0 1 0 0


5







0 0 0 0 0









0 0 0 0 0


































1






0 0 0 0 0 0 0







0 0 0 0 0 0 0

2






0 1 2 3 2 1 0







0 0 0 1 0 0 0

3






0 2 4 6 4 2 0


25



0 0 2 2 2 0 0

4

81



0 3 6 9 6 3 0







0 1 2 5 2 1 0

5






0 2 4 6 4 2 0







0 0 2 2 2 0 0

6






0 1 2 3 2 1 0







0 0 0 1 0 0 0

7






0 0 0 0 0 0 0







0 0 0 0 0 0 0

































1





0 0 0 0 0 0 0 0 0





0 0 0 0 0 0 0 0 0
2





0 1 3 6 7 6 3 1 0





0 0 0 0 1 0 0 0 0
3





0 3 9 18 21 18 9 3 0





0 0 0 3 3 3 0 0 0
4





0 6 18 36 42 36 18 6 0

125


0 0 3 6 12 6 3 0 0
5

729


0 7 21 42 49 42 21 7 0





0 1 3 12 13 12 3 1 0
6





0 6 18 36 42 36 18 6 0





0 0 3 6 12 6 3 0 0
7





0 3 9 18 21 18 9 3 0





0 0 0 3 3 3 0 0 0
8





0 1 3 6 7 6 3 1 0





0 0 0 0 1 0 0 0 0
































9





0 0 0 0 0 0 0 0 0





0 0 0 0 0 0 0 0 0

































What better use of Spreadsheet mastery could there be but to solve curiosities about mathematical constructs? I know, right? Oh and look at the beeeauuutiful surface it makes in 3d!


One final question: what powers can be produced in this way; ie by the concept of "adjacency"? So far we've seen 2, 5, 9, and conjecturally 4. Can three be done? How bout six, seven, and eight?
I suppose it depends on how wedded you are to the concept of a regular geometric grid? Could a more abstract geometry accomplish those other power series?

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