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Revamping Algebra
Posted:
Jul 23, 2007 10:56 AM


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 Zfloor 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 overstressing numeric rulebased 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 overstress 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 xyzplotting of polynomials and polyhedra. I take this all up in stickworks.py (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):
http://www.4dsolutions.net/ocn/cp4e.html
Kirby HP4E track chair wwwanderers.org



