Back to Pascal's Tetrahedron

To: synergetics-l@teleport.com

From: Kirby Urner 

Subject: Re: syn-l: Pascal's tetrahedron

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>Subject:      Re: General pascal triangle coefficients

>From:         jnikola@gamma.hut.fi (Juha Nikola)

>Date:         1995/05/09

>Newsgroups:   sci.math,

>which gave a longwinded algebraic construction of the trinomial

>expansion; this must have been dyscovered a few times, although plugging 

>into the IVM is undoubtedly rare(r).



So you're saying trinomial expansion is via the square-based

pyramidal packing, which, as you point out, is also part of

the IVM (VE nuclear sphere outward via any square face).

So that wouldn't be Pascal's Tetrahedron (if there were one)

but some other.



My goal wasn't to work back from algebraic expansion to some

geometrical tool, but to start with the tetrahedral 'brass

monkey' and go from there.  I think it evident that each

sphere's numeric entry will give the number of routes by

which a falling token could have arrived, ala the science

museum demo, so I stick to my 'Bell hill' or 'Bell cone'

picture of what's at the bottom (after we dump a lot of

tokens through the maze).



Throughout these numbering games, I like to stick with 

pool or ping pong balls, one 'spheric' per IVM locus,

so we don't get into 'up/down' triangles on a surface.

The confuses me, especially in light of the complementary

voxels that go into the IVM.  The printed output of

such a tetrahedron would likely involve 'cheese slices'

working outward i.e.:



    1     1     1        1          1

         1 1   2 2      3 3       4  4

              1 2 1    3 6 3     6 12 6

                      1 3 3 1   4 12 12 4

                               1 4  6  4 1



We have Pascal's triangle developing on each face, 

with a central region growing at a faster rate.



A literature search should pull up whatever's been

discovered about this creature already.  I still 

don't know if Pascal ever looked into it himself.

If not, it probably has someone else's name attached.



This just in (as I'm writing this):



>A long time ago, during childhood, I got this really neat idea to

>arrange the corresponding numbers in a magic triangle, that I later

>found out was called Pascal's triangle.  However, my idea also applied

>to higher dimensions and gives you Pascal's Pyramid, Pascal's Hyperpyramid

>and so on.  The 3-dimensional object looks like this:



OK Bri, if it's not goet a name on it yet, lets call it the 

Hutchings Pyramid.



Kirby