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Topic: Discrete Math for Quaker schools
Replies: 2   Last Post: Apr 30, 2007 8:32 PM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Discrete Math for Quaker schools
Posted: Feb 24, 2007 2:52 PM
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So what I'm pushing in Quaker schools, given my weighty
status, is attention to cumulative totals, even *before*
we monitor differences.

I think it comes earlier, in the Piagetian sense, to want
to accumulate a running total of numbers, to get a
sequence of partial sums. Haim was asking about this
earlier, and how to avoid the calculus right off (i.e.
all that limits stuff, not so intrinsic to discrete math,
where we don't even have a continuum).

The obvious sequences have to do with growing a shape
using similar pieces, gnomons, as when we grow a square
using L shapes, or a half-octahedron with successively
larger squares. These come as terms in a sequence, but
then we want those partial sums, i.e. that square, cube,
tetrahedron, half-octahedron, cuboctahedron, as devel-
oped "so far" from the layers we've committed.

def squarebase(n): return n*n

somelayers = [ squarebase(x) for x in range(1, 21) ]

halfoctas = [sum(somelayers[:y]) for y in range(1, 21) ]

At this point in the module's loading, halfoctas will
have become incarnate (note religious terminology) as:

>>> halfoctas
[1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870]

Next we go to Sloane's On-Line Encyclopedia of Integer
Sequences, a kind of mecca for discrete mathematicians,
and cut 'n paste some of those opening numbers. And we
get:

http://www.research.att.com/~njas/sequences/A000330

OK, so Sloane's has a 0. We couldn've had that too.

Let's do a tetrahedron:

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

>>> somelayers = [ tribase(x) for x in range(1, 21) ]
>>> tetras = [sum(somelayers[:y]) for y in range(1, 21) ]


>>> tetras
[1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540]

Cutting and pasting takes us to:

http://www.research.att.com/~njas/sequences/A000292

And now the cuboctahedrals, which you might think
obscure, but not in our Quaker academies, were we see
this as the CCP, ground zero for crystallography, as
well as the same as the icosahedrals, in terms of balls
in a layer (owing to our Jitterbug Transformation):

>>> def icosas(n):
if n<=1: return 1 # dismissing irrelevant cases
return 10*(n-1)**2 + 2

>>> somelayers = [ icosas(x) for x in range(1, 21) ]
>>> somelayers

[1, 12, 42, 92, 162, 252, 362, 492, 642, 812, 1002, 1212, 1442, 1692, 1962, 2252, 2562, 2892, 3242, 3612]
>>> cuboctas = [sum(somelayers[:y]) for y in range(1, 21) ]
>>> cuboctas

[1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, 10179, 12431, 14993, 17885, 21127, 24739]

Sloane's:
http://www.research.att.com/~njas/sequences/A005902

Kirby



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