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Re: Euclidean, Hyperbolic and Elliptic Geometry
Posted:
Mar 4, 2009 2:33 PM


Right, no universal rules, just have to make up a game that's selfconsistent enough to have some appeal.
Whereas fiddling with the 5th postulate (about parallel lines) has been a standard way to branch to nonEuclidean geometry, another route was suggested by Karl Menger, dimension theorist, in his The Theory of Relativity and Geometry (1949).
Per this essay, we might define points, lines and planes to be "lumps" (as in "of clay"), not distinguished by their "dimension number" (standard ideas of "linear independence" between height and breadth and width, per French res extensa philosophy (ala Descartes)) is what's being undermined (swapped out) in this picture (we keep an idea of energy though, say "energy has shape").
http://en.wikipedia.org/wiki/Res_Extensa
By this time, we have a pretty coherent geometry built up, including lots of input from Euler's topology, sometimes share it with kids under the marketing label of 'Claymation Station' (for obvious reasons).
This way of thinking about geometry doesn't really interfere with most Euclidean proofs, so is hardly that radical a departure, just a different way of thinking and talking (alternative nomenclature, e.g. MITE for "minimum tetrahedron"). At Saturday Academy, we think of this as a subbranch of Gnu Math, i.e. teach it under that category, running FOSS on commodity hardware. (FOSS = free and open source software).
More on maththinkingl (open archive), also you can read about my meeting with Dr. Livio on this topic at our recent gettogether @ Linus Pauling House in Portland, Oregon.
http://coffeeshopsnet.blogspot.com/2009/02/glassbeadgame.html
Karl Menger's daughter Eve is a member of this group, although she wasn't able to make this particular meeting.
Kirby Urner Chief Marketing Officer Coffee Shops Network



