Hello. I'm confused as to how Pascal's pyramid is constructed from the expansion of trinomials.
But first, just to clear it up, it's a tetrahedron, and not a 4-sided pyramid, correct? ("Pyramid" seems misleading)
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Hello. I'm confused as to how Pascal's pyramid is constructed from the expansion of trinomials.
But first, just to clear it up, it's a tetrahedron, and not a 4-sided pyramid, correct? ("Pyramid" seems misleading)
Pascal's pyramid - Wikipedia, the free encyclopedia
did you look here?
Yes. Unfortunately, my confusion isn't clear. Upon rereading though, I now see that it is a tetrahedron. But I don't see why a 3-dimensional construction is necassary to show the coefficients by expanding a trinomial. I just don't see how it's 3 dimensional; wouldn't that mean that for a positive natural degree there are several expansions?
Surely the clue is in this bullet point?
If it is 2-dimensional (P's triangle) then there are only two adjacent numbers. In order to have three (or more) adjacent numbers then it needs to be three dimensional. Doesn't it?Quote:
Each number in any layer is the sum of the three adjacent numbers in the layer above
I still don't understand. In Pascal's triangle, when you expanded a binomial, it would've been like ... which is 1 2 1.
But if I were to expand a trinomial, it'd be like (if I distributed correctly).
And that comes out to be:
-1 - 2 - 1-
-- 2 - 2 --
---- 1 ----
So a better way, I'd say, to make my question is ... Why does each degree have its own triangle? How would one expand something like the above into a 2-dimensional form? Unless I'm completely misunderstanding trinomial expansion (?).
Okay, I understand why now. I did know that the constructions involve adding from the above layers/rows in a bricked-pattern. But I noticed that I cannot add the coefficients in any layer in trinomial expansions to produce a correct sum in layer . So I guess it's not a matter of how one wants it to look, but more like one has to.
So it would be reasonable to assume that for a polynomial , the expansion construction is the corresponding simplex to the "th" dimension?
BTW, thank you mathman and Strange for bearing with my questions.
I may be wrong, but I don't think it is a matter of , rather it is the number of terms in (a+b+...). A binomial (a+b) can be represented in 2D (triangle). A trinomial (a+b+c) requires a tetrahedron - a 3-sided pyramid; a quad (a+b+c+d) requires a 4-sided pyramid, etc. I guess you can think of the triangle as simply a 2 sided pyramid.
The expressions with n in the exponent are simply ways of calculating the value after n rows.
Hmm... Seems so, but maybe not. I was wrong. I mistakenly stated the degree determines the construction; I meant the number of terms corresponds to that number dimension. As in...
Monomial: Line (since the coefficient is always 1)?
Binomial: Triangle
Trinomial: Tetrahedron
Quadrinomial: Hyper-tetrahedron (4 dimensional simplex)?
I'll have to see how I can arrange these quadrinomial coefficients...
1
1 1 1 1
1 2 2 2 1 2 2 1 2 1
1 3 3 3 3 6 6 3 6 3 1 3 3 3 6 3 1 3 3 1