Search found 12 matches

Fri Dec 13, 2013 4:44 pm UTC
Forum: Mathematics
Topic: Idea for using pi approximations to get close to 1
Replies: 24
Views: 6306

Re: Idea for using pi approximations to get close to 1

There is no geometry where the circumference:diameter ratio is constant but not equal to the usual value. Yes there is. There are geometries that aren't hyperbolic, eulcidean, or elliptic. For example the 1-norm on R 2 gives pi = 4. pi=1 violates the triangle inequality though. It'd be interesting ...
Tue Nov 26, 2013 2:42 am UTC
Forum: Mathematics
Topic: Measure-theory-ish thingamajig
Replies: 38
Views: 6826

Re: Measure-theory-ish thingamajig

I still don't think you can avoid the Cantor set problem with anything you've mentioned so far: Let C be the "half-open Cantor set". That is, we start with [0,1), then remove [1/3, 2/3), and on and on. C is an uncountable set. But q(C) = x - 1/3 sum(x(2/3)n) = 0.
Sat Nov 23, 2013 3:40 pm UTC
Forum: Mathematics
Topic: Measure-theory-ish thingamajig
Replies: 38
Views: 6826

Re: Measure-theory-ish thingamajig

I think the question to ask isn't "can this be made rigorous?" but instead "once this is rigorous, what subsets does it apply to, and what properties does it have?" It seems that so far, the properties we want are (1) invariance under isometries and (2) finite additivity of disjo...
Fri Oct 18, 2013 8:17 pm UTC
Forum: Mathematics
Topic: n sided polygon, n < 3
Replies: 35
Views: 6532

Re: n sided polygon, n < 3

There's a good generalization if you look at polygons that don't have straight edges. At any point on a smooth curve, there is a unique circle which approximates the curve up to second order. The curvature at that point is defined to be the inverse of the radius of that circle. (A straight line is a...
Sun Mar 17, 2013 11:38 pm UTC
Forum: Mathematics
Topic: Beautiful proofs/mathematical properties
Replies: 16
Views: 3751

Re: Beautiful proofs/mathematical properties

Along these lines, I always hated the proof of Euler's formula using Taylor series. Much better is this: z=e x is a holomorphic function satisfying z' = z (either by definition using existence & uniqueness of solutions to diff eqs, or by the Taylor series definition). Then d/dx(e x+y /e x e y )...
Sun Mar 17, 2013 11:20 pm UTC
Forum: Mathematics
Topic: Beautiful proofs/mathematical properties
Replies: 16
Views: 3751

Re: Beautiful proofs/mathematical properties

I quite like how all of the exterior angles of a simple polygon add up to one full revolution of a circle. There's lots of other neat geometric rules which show which angles are equal to which other angles that are also pretty neat. (eg: Angles in the same arc are equal .) They're obvious enough (w...
Fri Nov 09, 2012 3:42 am UTC
Forum: Mathematics
Topic: Periodic points of Mose-Smale diffeomorphisms
Replies: 7
Views: 2759

Re: Periodic points of Mose-Smale diffeomorphisms

My guess is that you can't smooth such a function (locally) in such a way that you are still left with only hyperbolic periodic points. An exercise in a previous section (Section 1.4, page 25, exercise 3 here ) asks us to show that if f is a diffeomorphism, then any hyperbolic periodic point is iso...
Thu Nov 08, 2012 10:44 pm UTC
Forum: Mathematics
Topic: Periodic points of Mose-Smale diffeomorphisms
Replies: 7
Views: 2759

Re: Periodic points of Mose-Smale diffeomorphisms

Wait...so what if your diffeomorphism has the piecewise definition f(x) = 1-2x when x<1/3, and f(x) = 1/2(1-x) when x > 1/3 So strictly speaking we need to glue in some kind of interpolating function to make f smooth near x = 1/3, but I'm not interested in the behavior of f near that point anyways. ...
Sun Jun 03, 2012 3:49 am UTC
Forum: Mathematics
Replies: 9
Views: 2044

I think it's good to start as simple as we possibly can, and only get more complicated as we need to. Let's really think about the equation q = b^x. The only thing we really know from the start is how to calculate q if b is a real number and x is a positive integer: multiply b times itself, x times....
Wed Mar 21, 2012 2:22 am UTC
Forum: Mathematics
Topic: Distance on a sphere
Replies: 14
Views: 4340

Re: Distance on a sphere

Suppose you have a geodesic g passing through the point p with tangent vector v. If H is the plane containing 0, p, and v, then reflection through H is an isometry which fixes 0, p, and v, therefore g' = refl_H(g) is another geodesic passing through p with tangent vector v. By existence and uniquene...
Wed Jul 13, 2011 3:09 am UTC
Forum: Mathematics
Topic: Wrong on the Internet II: Norman J. Wildberger
Replies: 85
Views: 49515

Re: Wrong on the Internet II: Norman J. Wildberger

Humans, from an extremely young age, think of the number line as an order-complete field. They don't realize they think about it this way, but they do. And personally, I think the definition is very straight forward. Not at all arbitrary. Also, the reals are uncountable. Computers can't handle that....
Thu Jul 15, 2010 10:48 am UTC