Date: Jan 14, 2001 2:27 PM
Author: Kirby Urner
Subject: Pool Hall Math


    
                 POOL HALL MATH
             Oregon Curriculum Network


You can start with 15 pool balls (bring 'em to class)
to kick off a discussion of Triangular Numbers.

Links to Gauss:  Tn = 1 + 2 + 3... + n

Which he solved (age 7) by going:

   1     2     3 ...   n
   n   n-1   n-2 ...   1
  ----------------------
 n+1   n+1   n+1 ... n+1  = n(n+1) 

which is twice the sum he wanted so: Tn = n(n+1)/2

Then stack smaller triangles on the 15 to an apex
ball (good if you've got the pool balls, need 35 
in all -- switch to ping pong balls if pool balls
too spendy).  That's a tetrahedron.  If you're 
lucky enough to have a CLI, you can do things like:

  >>> def tri(n): return n*(n+1)/2

  >>> def tetra(n):
	 if n==1: return 1
	 return tri(n) + tetra(n-1)

  >>> tetra(5)
  35

Now if you're dealing with somewhat older students,
it's useful to say "as you know, just as area increases
as a 2nd power of linear increase, so does volume 
increase as a 3rd power, so we'd expect to find an
expression in the 3rd degree for the Tetrahedral 
Number Ttn, just as Gauss did for Tn".

The solution involves writing:

    A n^3 + B n^2 + C = Ttn

and then substituting known pairs (n, Ttn), to get 
3 linear equations in 3 unknowns.  Could go with 
matrix methods if that's what's up in class.

Kirby

Links:

Ttn to CCP (cube centric packing = fcc = ivm)
CCP and Alexander Graham Bell (kites + truss)
CCP and R. Buckminster Fuller (octet truss)
Sequences Tn and Ttn in Pascal's Triangle (Math Forum)
Pascal's Triangle as itself figurate (a triangle)
Pascal's Triangle and Binomial Theorm
Pascal's Tetrahedron (advanced: trinomial theorem)