[Edu-sig] More cogitations on group theory...

[kirby urner]

> Groupoids, categories, rings (clock time), fields (modular
> arithmetic), vector spaces, and algebras require a bit more thought,
> but I am sure that they can be done.
>

That's perfect Ed.  Good to hear for another die-hard "group
theory for children" dude, a vanishing breed perhaps.

My intended audience might actually be older people, including
so-called "retirement community" students who have grand kids
and want to have inter-generational topics.  Many learned
BASIC as kids (the Bill Gates generation).

Another group you can massage into a field are integers
multiplying modulo N, except not just any integers, only
N's totatives.  Back to permworld and Guido's exceedingly
simple implementation of Euclid's Algorithm.

>>> def gcd(a,b):
	while b:
		a , b = b, a % b
	return a

>>> gcd(12, 4)
4
>>> gcd(12, 5)
1
>>> totatives12 = [m for m in range(12) if gcd(m, 12) == 1 ]
>>> totatives12
[1, 5, 7, 11]
>>> from random import choice

>>> (choice(totatives12) * choice(totatives12)) % 12
11
>>> (choice(totatives12) * choice(totatives12)) % 12
1
>>> (choice(totatives12) * choice(totatives12)) % 12
5

Asserting closer (group property):

>>> if (choice(totatives12) * choice(totatives12)) % 12 in totatives12: print (True)

True
>>> if (choice(totatives12) * choice(totatives12)) % 12 in totatives12: print (True)

True
>>> if (choice(totatives12) * choice(totatives12)) % 12 in totatives12: print (True)

True

If the target number is prime instead of composite (e.g. 23
instead of 12), then you have field properties, not just group
properties i.e. + is closed as much as * is.

You'll find me ranting on mathfuture how high schools bleep
over any opportunity to introduce "totative" or "totient" in
favor an an exclusive "factor tree" based approach to
gcd.  That made more sense before RSA was in every
web browser.  In a "how things work" curriculum, one
would wish for more computer literacy.

http://groups.google.com/group/mathfuture/msg/11005d0c9dc9eba2
(I've gotten more correspondence from Milo -- he wants to
make sure we all know that Turing at Bletchley Park did
*not* solve the German U-boat 5-rotor puzzle, doesn't like
how much credit Turing gets).

I'd like want to use John Zelle's graphics.py in the module
where we draw some Wolfram checkerboard of black
and orange rectangles ala New Kind of Science (NKS).
We were doing that back in February 2007.

http://mail.python.org/pipermail/edu-sig/2007-February/007736.html

The new version gets Conway's Game of Life from the
same "turtle" (called a "tractor" in farmworld, but the
same idea, transferred to all-ASCII waves of grain).
Even Mandelbrot is rendered in ASCII "tractor art":

http://mybizmo.blogspot.com/2011/05/lesson-planning.html

These are ancient threads as far as edu-sig is concerned.
We've always been trendy around here. :)

> Of course, in every case I am talking about extracting and presenting
> the fundamental ideas, and leaving proofs, notations, and all but the
> simplest calculations for later.
>

Of course.  I've got an older bunch but this isn't a course
about Group Theory, it's a course about learning to
program in the Python computer language, with a backdrop
of standard Computer Science courses (Euclid's Algorithm
chief among them, at least here on edu-sig).

Kirby