Talk:Pascal's pyramid

Why is this called a pyramid? The base is a triangle, so isn't it a tetrahedron?

A tetrahedron is a pyramid. Factitious 00:47, Jun 25, 2005 (UTC)

> Summing the numbers in each column of a layer of Pascal's pyramid gives the nth power of 111 in base infinity (i.e. without carrying over during multiplication), where n is the layer - 1.

Is it really accurate to call this "base infinity"? It's more accurately base 10 (but without said carry-over). Actually, more generally, it's the nth power of 111 in any base. For example, in base 9, 111^2 = 12321:
1 + 2*9 + 3*9^2 + 2*9^3 + 1*9^4 = 8281 = (1 + 9 + 81)^2. --Matthew0028 13:53, 14 February 2006 (UTC)[reply]

Yes, base 9 works for 111^2, however as you get into higher powers of 111, you are forced, according to the rules of multiplication, to carry over. Higher number bases allow for higher nmbers to not be carried over. Base "infinity" makes the rule applicable to all powers of 111. I take the blame for making up the term "base infinity", however in this situation I feel that it is quite a good description of what is actually happening.

What is actually happening is a degeneration, a summing of digits of partial results of the power. You can easily avoid this, even without the "base infinity", just insert some zeroes:

                  1                              1
                 40                            400
               1000                         100000
              16000                       16000000
             190000                     1900000000
            1600000                   160000000000
           10000000                 10000000000000
           40000000                400000000000000
          100000000              10000000000000000
          ---------              -----------------
  111^4 = 151807041    10101^4 = 10410161916100401, or 1 04 10 16 19 16 10 04 01

which is the layer columns sum you were looking for. But again, this is just another degeneration of the original layer itself. Let's insert some more zeroes:

  10000000101^4 = 10000000404000006120600041212040104060401, or
                  1
     00 00 00 04 04
     00 00 06 12 06
     00 04 12 12 04
     01 04 06 04 01

and you get the original layer itself. There's a formula for inserting the zeroes in Pascal's simplex.

(endless.oblivion (talk) 00:24, 4 April 2010 (UTC))[reply]

Dead Link?

http://people.ucsc.edu/~erowland/pascal.html : Not Found The requested URL /~erowland/pascal.html was not found on this server. --NeoUrfahraner 10:00, 15 November 2006 (UTC)[reply]

the numbers in Pascal's pyramid can be found by summing the three numbers in the preceding layer. The same as you can find the numbers of pascal's triangle by summing the two numbers in the preceding row.

Time waits for no man. I started to extend and clarify this page last June, when I was routed to the hospital to have some of my lung removed. I'm back, but I see someone has a different idea of how the article should flow and has rearranged my order. So I thought I would preview how I think the article should proceed.

I am aiming the text at a target audience that has some familiarity with Pascal's Triangle, but is not a "numbers nut" like me. By showing readers the natural source of the numbers, they should come to appreciate their significance and develop an understanding of the most intuitive and simplest ways to generate the numbers. My focus is on how the Tetrahedron fits into the real world (probability theory and algebra), rather than cleaver ways the numbers can be generated and manipulated. The Tetrahedron is not a stand-alone structure, but part of the continuum of mathematics, in which everything is related to everything else.

Here are the sections as I see them. Not seeing any objections, I will continue as planned.

General introduction (as is)

Table of contents

Structure of the Tetrahedron (existing tables of layers (0 to 5); [currently "Construction"])

Overview of the Tetrahedron (brief preview of the following sections; [currently "Features"])

Trinomial expansion connection (use (A+B+C)^4 as example to show expansion coefficients same as numbers of Tetrahedron; [NEW])

Trinomial distribution connection (relationship of expansion exponents to distribution coefficients--algebraic form [NEW] and combinatorial form [currently "Trinomial construction"])

Parallels with Pascal's Triangle and Multinomial Coefficients ([NEW]; table of 3 columns and several rows:

Type of polynomial [bi-, tri-, multi-nomial]

Order of polynomial [2, 3, m]

Example of polynomial [A+B, A+B+C, A+B+C...+M]

Geometric structure [triangle, pyramid, clusters]

Element structure [line, layer, group]

Symmetry of element [2-way, 3-way, m-way]

Number of terms per element [n+1, (n+1) (n+2)/2, (n+1) (n+2) ... (n+m-1)/(m-1)

Sum of exponents, all terms [n, n, n]

Sum of values per element [2^n, 3^n, m^n]

Coefficient equation [n!/(x! y!), n!/(x! y! z!), n!/(x1! x2! x3! ...xm!)

Sum of numbers "above" [2, 3, m]

Ratio of ad--Colin.campbell.27 (talk) 05:55, 9 January 2011 (UTC)jacent numbers [2, 6, (m) (m-1)][reply]

Sum of adjacent terms in prior layer ([currently "Sum of numbers in layer above"]; better illustration with numbers replacing red and black dots)

Ratio of adjacent terms in same layer ([NEW]; ratios related to expansion exponents; example using (A+B+C)^4):

Other Relationships (which need further discussion when we get there)

“Exponentiational” construction

Sum of coefficients of a layer by rows

Sum of coefficients of a layer by columns

--Colin.campbell.27 (talk) 05:55, 9 January 2011 (UTC)[reply]