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Thread: Construction of Pascal's pyramid?

  1. #1 Construction of Pascal's pyramid? 
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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)


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    Pascal's pyramid - Wikipedia, the free encyclopedia

    did you look here?


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    Quote Originally Posted by mathman View Post
    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?
    Last edited by brody; February 21st, 2012 at 04:35 PM.
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    Brassica oleracea Strange's Avatar
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    Surely the clue is in this bullet point?
    Each number in any layer is the sum of the three adjacent numbers in the layer above
    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?
    ei incumbit probatio qui dicit, non qui negat
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    Quote Originally Posted by Strange View Post
    Surely the clue is in this bullet point?
    Each number in any layer is the sum of the three adjacent numbers in the layer above
    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?
    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 (?).
    Last edited by brody; February 21st, 2012 at 08:20 PM. Reason: Silly mistake. Generally distributed not foiled.
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    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.
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  8. #7  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by brody View Post
    So it would be reasonable to assume that for a polynomial , the expansion construction is the corresponding simplex to the "th" dimension?
    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.
    ei incumbit probatio qui dicit, non qui negat
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    Quote Originally Posted by Strange View Post
    Quote Originally Posted by brody View Post
    So it would be reasonable to assume that for a polynomial , the expansion construction is the corresponding simplex to the "th" dimension?
    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
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  10. #9  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by brody View Post
    Quadrinomial: Hyper-tetrahedron (4 dimensional simplex)?
    No, it is just a packing thing. Get some fruit. Put four oranges in a square:

    OO
    OO

    Now, you can balance another orange in the center. Which means you can take the sum of four adjacent values. It is still a 3D pyramid.
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  11. #10  
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    Quote Originally Posted by Strange View Post
    Quote Originally Posted by brody View Post
    Quadrinomial: Hyper-tetrahedron (4 dimensional simplex)?
    No, it is just a packing thing. Get some fruit. Put four oranges in a square:

    OO
    OO

    Now, you can balance another orange in the center. Which means you can take the sum of four adjacent values. It is still a 3D pyramid.
    Ah, I see now. Thanks for your answer and the neat analogy.

    So I'll go ahead and top it off with a generalization. The number of terms in the polynomial is equal to the number of sides in the base of the pyramid construction.
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