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)