[Edu-sig] Geometries, irrationals (was Re: thought re graphing calculators ...)
echerlin at gmail.com
Thu Oct 1 02:21:46 CEST 2009
>>> Could we develop a geometry which does not depend on the metaphysics
>>> of real numbers, continuity, infinity? Or still have infinity, but
>>> make it more like Poincare's, a direction (like a time axis).
>> There are vast realms of such geometries, going back to projective
>> geometries over finite fields and the like, and to general topology.
>> Lie groups (including the one-point compactification of the complex
>> plane) and Lie algebras. Banach spaces. Measure theory. Spaces of
>> fractal dimension. Minkowski space. Differential geometry of
>> Einsteinian spacetime. Non-commutative geometries in quantum mechanics
>> (von Neumann algebras over Hilbert space). Many more.
> 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.
> 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.
> 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
> Pro, con or stand aside? Come prepared next Tuesday.
> Or you could say Phi (golden mean) is the Phirst Phractal (certainly
> the recursivity is there in the algebra).
> Kirby Urner
> ндсжег воss
>> 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.
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.
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