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Topic: Geometry and GIS
Replies: 2   Last Post: Aug 21, 2006 6:29 PM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Geometry and GIS
Posted: Aug 21, 2006 2:26 PM
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So kids are quite fascinated with Google Earth and the genre (Terraserver etc.), with their zoom out, in and around type flyabilities.

Here's an opportunity to capitalize on that fascination by building segues to connected math topics, such as the geometry of latitude and longitude, a grid turned on or off with a check box or other binary toggle.

My approach is to contrast this "orange peel grid" (lines of longitude parallel internal orange slices) with a more "science fiction" (alien, unfamiliar, futuristic) vision, of a high frequency hexapent:
http://www.4dsolutions.net/ocn/hexapent.html

In this futuristic twilight zone world, all your beef cattle now wander between invisible hexagons (they're unlikely to meet up with a pentagon, as there're only 12 of those), tracked by sensor and fed back to some consortium of industries concerned with complete record-keeping regarding the precise history of all meat on the hoof in our food chain.

Such tracking projects are already in the works, we're just adding a visual (part of an SBIR grant we're still waiting on).[1]

Once in a hexapent, you've got that American Scientist article to go for, with its simple classification into soccer balls and fullerenes (buckyballs).[2]

According to the authors' rule book, SBs (soccer balls) must have all hex/pent borders and get pretty strange looking.

BBs (buckyballs) need the minimum 12 pentagons, then go wild with the hexagons to very high frequencies.

Both SBs and BBs are hexapents. It's one of these latter types I think we'd use with the cows. We have a further classification scheme (not covered in the cited article) for labeling some of our BBs as Class I, II or III.[3]

Or you could just stick with the traditional lat/long grid (there's an algorithm to/from the hexapent, which you might treat as buried-yet-operational machinery for the time being, a game engine service).

However, there's still the problem of flagging particular points and communicating that information.

A lat/long scribbled on a paper napkin might be intelligible, but what if you need to transmit several thousand points of interest, as when listing car dealerships or restaurant chains. There you'd need a standard format, as machine parsability is definitely implied (thanks to IBM's "keeping tabs" in the old days). Enter XML.

That's right, we consider XML a math topic? Why?

Because it's a tree structure (not binary -- lots of siblings OK), and a tree structure, like a network, is a data structure. And data structures are command-line accessible in many free open source gnu languages like Python, Ruby, Perl.

Consider how a tree might be implemented, or a network.

This is math that we're doing. Can't have algorithms without data structures.

Like how will our lat/long to hexapent machine know how to eat 10,000 MacDonalds locations and pop them into a Fuller Projection, as dots of light? Answer: XML again (maybe a Python data structure on the client and/or server end, but sent as XML in between, perhaps via xml-rpc, to make the map server do its thing according to its own strict rules for packaging input).

In sum, we can use student fascination with global browsers (more fun than the old classroom globes, though they were fun too), to get into topics of coordinate system (lat/long, spherical, other) and data handling.

What's easy to do a few times, as a manual process, and without strict rules, may not scale well. For industrial strength processing, you need much higher levels of precision.

Not only do our GIS/GPS systems zoom in more tightly than ever before (thanks to very sensitive time stamping, in which relativity must be taken into account), but our record-keeping has been honed to high standards (and we're still getting better at it).

Kids should learn about this in school, so they have a better understanding of how mundane mathematics keeps the wheels of our global economy turning, keeps the meat safe -- or doesn't, when mathematical integrity breaks down, as sometimes happens.

Learning to do this multi-culturally has meant making XML a unicode-aware technology. If your character data needs to be in Thai or Laotian, no problem. The Internet's engineers did not commit it to serving only the English or Latin characterset worlds.

Even mathematical expressions may be captured in an XML (e.g. MathML) and displayed in your browser.

So here's another segue into specific and partially overlapping mathematics notations. Segue from here to music notation, for many helpful analogies and parallels.

Anyway, I think you get the idea: from the global browsers, you've got entry points into a large network of connected math topics. So why not leverage this opportunity?

Kirby
4D Studios
Portland, Oregon

[1] per remarks by Glenn Stockton aboard Meliptus recently.

[2] citing the American Scientist article:
http://mathforum.org/kb/message.jspa?messageID=4988588&tstart=0

[3] re Class I, II, III see:
http://packinon.sourceforge.net/py_progs/pg_geo.html



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