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Topic: Revamping Algebra
Replies: 1   Last Post: Jul 23, 2007 10:56 AM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Revamping Algebra
Posted: Jul 23, 2007 10:56 AM
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I'm thinking our writers' group will indeed adopt Glenn Stockton's
suggestion and think harder about how to use spherical
imagery when discussing data storage/retrieval.

His solution, a spaceship of concentric hulls or bulkheads,
with the "Z axis" elevators ascending from the center of
the ship, along disparate radials, is a fine proposal.

It's advantage is strong isomorphism to XYZ (Glenn is quite
fluent about this). Each floor of the ship has an XY plane
of hexagons and 12 pentagons, connecting around in all
circumferential directions.

Those correspond to XY floors in a normal XYZ rectilinear
apartment building, by whatever mapping convention. The
mapping might not even be one to one, yet every apartment
with a unique address in the building, would correspond to
a unique address in the spaceship, even with the same Z-floor
number if we like.

A reason for keeping spherical data structures in view is
our real world mappings so often anchor to a geographic
dimension. IP numbers correspond to physical hosts for the
most part, and hosts have latitude and longitude. Anything
to do with transportation around the Earth's surface has
this implied structure, including the Z floor business (which
applies to mining inward and submarines, not just to space
needles or orbiting ).

Don has no trouble following all this, given his long career
in office automation. IBM Selectric typeheads were of a
spherical nature, even if not true hexapents.

Links to Python:

Our idea of a mapping is the dictionary data structure,
a builtin. Key:value pairs form that most primitive
domain/range pairing, where the range is a subset within
the codomain, reachable by the mapping.

We've traditionally introduced "onto" "into" and "one to
one" at this point, or "surjective" "injective" and
"bijective". However, unlike traditionalists, we don't
believe in over-stressing numeric rule-based functions as
the only worthy delegates. Bit string addressing of Unicode
data points, or IP addressing of physical hosts under the
tcp/ip layer, are every bit as "real" as {x:f -> x**2}.

Put another way: we don't neglect functions that eat
strings, other functions. Nor do we over-stress using a
rectilinear model for the View of every Model (MVC design
pattern), especially where geography is involved (per my
HP4E initiative and Glenn's 'global matrix' idea).

Anyway, a simple first pass within Gnu Algebra would be:
{}-dictionary as mapping. def f(x) notation (function
defs), [(x,f(x)) for x in domain] list comprehensions, with
domain any kind of iterable, maybe a generator.

From there: simple vector arithmetic and xyz-plotting
of polynomials and polyhedra. I take this all up in
(depends on vpython) if you want some scaffolding for
your classes.

There's a link to it from my Resources page (Python for
Math Teachers on Showmedo):

HP4E track chair

Date Subject Author
Read Revamping Algebra
Kirby Urner
Read Re: Revamping Algebra
Kirby Urner

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