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

[Edward Cherlin]

On Sun, May 15, 2011 at 00:35, kirby urner <kirby.urner at gmail.com> wrote:
> More cogitations on group theory...

The fundamental idea of a group can be introduced to preschoolers. I
am writing and programming lessons on this topic, which I will test on
children, and report to you.

My first example will be a triangle with sides in three colors. How
many ways can you place that in a triangular frame? How many moves are
possible? Can you reverse any move? Do the moves associate? (That is,
is doing A and B, and then C, the same as doing A, and then B and C?)
Well, there you are, all of the group axioms. There is an associative
function with inverses.

Then we look at other symmetries of other shapes, and find that the
positions and the moves match up, and we always get a group, even if
it is the trivial group for a completely asymmetric shape: the group
with only one element.

Then infinite groups, like the symmetries of a circle or sphere.

Then permutations. Every finite group is a subset of a permutation
group. I'll leave matrices for a later grade.

There is a very pleasant book on group theory for non-mathematicians,
which uses Rubik's cube for examples throughout. I have it bookmarked
on Amazon, but I am going to finish this e-mail and go to bed rather
than look it up tonight.

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.

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.

> You get these fads, gripping at the time, that leave legacy traces.
> In New Math, we got imprinted with some Theory of Sets.
>
> Some scoff at that now, say it was just notation:  intersection,
> union, subset, element of... taught to grade schoolers.  Parents
> were agog.  This was 1960s or so.
>
> A lot of it stayed.  Singapore Math, favored by many conservatives
> (e.g. Mathematically Correct with ties to Cal State) still has it.
>
> Converting among bases suffered though, during the Great
> Retaliation.  So-called "space age math" was hated and feared
> by the peasantry.  NASA was given no interesting missions,
> boost phase was outsourced.  Orlando engineering families
> went to work for The Mouse instead.
>
> http://www.google.com/search?hl=en&biw=1920&bih=948&gbv=2&tbm=isch&aq=f&aqi=&oq=&q=venn%20intersection%20%22new%20math%22
>
> http://bit.ly/kWlclB  (compressed version of above)
>
> (I scooped up several of mine in the above in this swoop through
> Google's data space, might be fun to pick out which ones... **).
>
> In Python, we have the set type, and it's important.  It has to
> share the limelight though, and comes off almost as a subspecies
> of dict, a dict with keys, but no values.
>
> What a concept.
>
> set tends to come later in the Pythonista's learning career, though
> at OST it's still pretty early, definitely in the first unit.  But then
> we're pretty steep, with MySQL and GUI programming in unit 2.
>
> Group Theory, like Clifford Algebra, is one of those things a
> small cadre thought might be taught to high schoolers, in
> some Jetsons-style science fiction future sponsored by Quaker
> Oats or one of those Breakfasts of Champions.
>
> I was an ardent member of this cast and to this day know
> how to wave the flag when called upon.
>
> Core to the idea of a group is a set of permutations, easily
> modeled as a Python dict.  A permutation, or perm for short,
> is a complete swap of some finite set (no dupes), such that
> the original list maps to a random.shuffle of a mirror.
>
> That's what we implement in permworld.PermDNA, one of
> the early bird perm factories in this particular Level 4 curriculum.
>
>>>> ================================ RESTART ================================
>>>> import OST
>>>> from OST.permworld import PermDNA, PermCell
>>>> from OST.permutils import cyclic, anticyclic
>>>> p = PermDNA()
>>>> thecell = PermCell(p)
>>>> antibody = ~thecell
>>>> voter = thecell("Eating transcriptionase for breakfast and spitting back collogen")
>>>> voter # enciphered ballot
> 'Elndpscnglpkvgdjndbplkrcfbgcagrleflknclpxckjdnndpscalvecvbzzbsrp'
>>>> antibody(voter)  # decryption key per ~ operator, __invert__ in PermCell
> 'Eating transcriptionase for breakfast and spitting back collogen'
>
> Why go to all this trouble, why not just work with integers
> (which form various finite groups in upcoming lessons)?
> Answer:  operator overloading.  There's no point overloading
> +, *, ** etc. where integers are concerned.  The operators are
> already defined.  With perm objects, or PermCells as we
> call them (permDNA incepted) you still need to write that
> __mul__ operator, and that __div__ which is "multiply by
> the multiplicative inverse of."
>
>>>> thecell
> PermCell: (('a', 'l', 'z', 'w', 'h', ' ', 'c', 'v', 'q', 'y', 't',
> 'n', 'p', 'j', 'm', 'i', 'd', 'x', 'o', 'b'), ('e', 'r', 'g', 's',
> 'k'), ('f',), ('u',))
>>>> antibody
> PermCell: (('a', 'b', 'o', 'x', 'd', 'i', 'm', 'j', 'p', 'n', 't',
> 'y', 'q', 'v', 'c', ' ', 'h', 'w', 'z', 'l'), ('e', 'k', 's', 'g',
> 'r'), ('f',), ('u',))
>>>> identity = thecell * antibody
>>>> identity
> PermCell: (('a',), (' ',), ('c',), ('b',), ('e',), ('d',), ('g',),
> ('f',), ('i',), ('h',), ('k',), ('j',), ('m',), ('l',), ('o',),
> ('n',), ('q',), ('p',), ('s',), ('r',), ('u',), ('t',), ('w',),
> ('v',), ('y',), ('x',), ('z',))
>>>> print(OST.permutils.__doc__)
>
> permutils.py  1.1
> (gpl) OST / Python 4
> http://www.gnu.org/software/gsl/manual/html_node/Permutations-in-cyclic-form.html
>
> is the kind of thing we're thinking.  __div__ has not been
> implemented and so would be a student project, just as
> __setattr__ and __getarr__ might be defined on a
> reworked farmworld.py (free and open source, just
> like MIT courseware -- what all of the better schools
> are doing it seems, putting their cards on the table).
>
> Sorry for so much technical jargon, but I'm thinking to
> attract others with experience in this area.  We've got
> cyclic notation going, modeled on the J language and one
> of the GNU projects.  The isomorphism of a perm expressed
> as a dict, and a perm expressed as a tuple of tuples,
> and the algorithms for going both directions, is one of
> the pillars of permworld, as you might expect.
>
> Kirby
>
>
> (sm) ghost ship productions, usa.or.pdx.4d (dba pending)
>
> ** http://controlroom.blogspot.com/2007/10/portland-radio.html
> (because of surrounding posts I suppose -- the link was to
> these pictures from Portland's KBOO radio station).
>
> Another couple:
> http://www.grunch.net/synergetics/ncmtmemo.html
>
> More obviously (totally about the New Math):
> http://controlroom.blogspot.com/2007/10/recalling-sputnik.html
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-- 
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/