From: kirby urner <kirby.urner@gmail.com> Date: Wed, Sep 30, 2009 at 8:00 PM Subject: Re: Geometries, irrationals (was Re: [Edu-sig] thought re graphing calculators ...) To: Edward Cherlin <echerlin@gmail.com> Cc: Gregor Lingl <gregor.lingl@aon.at>, edu-sig@python.org On Wed, Sep 30, 2009 at 5:21 PM, Edward Cherlin <echerlin@gmail.com> wrote: << trim >>

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A feature we're looking for is "accessible to grade schoolers" i.e. we don't want you already out the other end of some lengthy pipeline wherein brainwashing has already occurred.

Spherical geometry. The trig gets a bit complicated, but even in kindergarten children can see that an octant of a sphere is a triangle with three right angles. As long as you don't bring in the big words.

Yes, given our geometry - geography tie-in, with GIS/GPS front and center in a lot of course work, it makes sense to return to the good old days.  Joe Clinton, one of the chief geodesic dome guys out of Carbondale, showed me some old high school text books from before the calculus tsunami.  Looks like that might be the tough final topic before college.  And with Python to help... takes the drudgery out of all that omni-triangulation. Caveats apply though, in that there's nothing especially non-Euclidean about a beach ball, unless you zoom out to show that these "infinite planes" these Euclid folks scribed on, proving their theorems, were only locally flat, took advantage of the relatively small size of humans relative to Planet Earth.  Once you start defining infinite planes out of your geometry, then you're actually reality checking creaky old greek metaphysics, which is what I'm suggesting we *should* be doing. There's room for a math without continua.  There's room for a math closer to following Karl Menger's suggestion that we just think of points, lines and planes as finite blobs, in a geometry of blobs, no distinction as to dimension, more topological in some ways, as this same blob of dough stretches to become this that or the other (it's a cloud of points though, not a continuum, with no ultimate resolution to anything more primitive (than blobs, always more subdivisible)). It's not like we need to start over from scratch to scrounge together a geometry of this nature, accessible to children.  The work is already done, and has many more features and practical applications to boast of, including those geodesic domes I mentioned. http://worldgame.blogspot.com/2009/09/random-posting.html

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I understand that some elite schools get into jiggering with the fifth postulate (Euclid's) even pre-college,

That's doing it the hard way.

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<debate>

Resolved:  "Irrational numbers are of course morally superior to the rationals as all the best constants (e, phi, pi) are irrational, even transcendental if we're lucky."

I'll see your e, phi, and pi, and raise you 0, 1, -1, and i.

e^i*pi + 1 = 0

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.

The integers were made by God. All else is the work of man.--Kronecker

Kronecker had a big problem with Cantor throwing in all those monkey wrenches. Whereas people glommed on to Cantor's inventions around infinity, fewer are aware of Cantor's polemics against established dimension concepts. Did Menger comment on Cantor's thinking on this topic?  I have no idea. The fact that we have a whole branch with defined fractional dimensions proves this language is tractable (amenable to innovation).  What we want students to appreciate is that maths is something of a playground, an amusement park, not some oppressive monolith that weighs on one and drags one under, like some dead albatross around one's neck. Kirby

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Pro, con or stand aside?  Come prepared next Tuesday.

</debate>

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Kirby

Or you could say Phi (golden mean) is the Phirst Phractal (certainly the recursivity is there in the algebra).

Kirby Urner ндсжег воss

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-- Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin Silent Thunder is my name, and Children are my nation. The Cosmos is my dwelling place, the Truth my destination. http://earthtreasury.org/

-- Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin Silent Thunder is my name, and Children are my nation. The Cosmos is my dwelling place, the Truth my destination. http://earthtreasury.org/