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Re: Euclidean, Hyperbolic and Elliptic Geometry
Posted:
Mar 4, 2009 2:33 PM
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Right, no universal rules, just have to make up a game
that's self-consistent enough to have some appeal.
Whereas fiddling with the 5th postulate (about parallel
lines) has been a standard way to branch to non-Euclidean
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 sub-branch of Gnu Math, i.e. teach it
under that category, running FOSS on commodity hardware.
(FOSS = free and open source software).
More on math-thinking-l (open archive), also you can read
about my meeting with Dr. Livio on this topic at our
recent get-together @ Linus Pauling House in Portland,
Oregon.
http://coffeeshopsnet.blogspot.com/2009/02/glass-bead-game.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
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