Re: infinity

Virgil said:
> In article <MPG.1dc05554d78c02f798a4ec@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1dbefa3f98322bc798a4ce@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > >
> > > > Virgil said:
> > > > > In article <MPG.1dbdd071596c8b3798a4be@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > >
> > > > > > It has some interesting implications, even if mostly of a
> > > > > > philosophical nature. It does exactly mirror the 2's complement
> > > > > > system, in the limit as the number of bits goes to oo. Now, doesn't
> > > > > > the addition of the imaginary dimension, also circular, create a
> > > > > > toroidal topology, rather than spherical?
> > > > >
> > > > > Not with a one-point compactification. If TO knew a bit more about
> > > > > mathematics, before setting himself up as genius mathematician, he
> > > > > would
> > > > > fall flat on his face less often.
> > >
> > > > A one-point compactification as in both axes meeting in more than one
> > > > location?
> > >
> > >
> > > As in all axes and both ends of each axis all meeting in one single
> > > point. It is topologically quite sound. That TO does not understand it,
> > > is in no way a drawback to its validity.
>
>
> > But it is topologically different from flat space, whereas toroidal surfaces
> > are homeomorphic to flat space curved on itself.
>
> A spherical surface with one point deleted is homeomorphic the
> Cartesian plane.
>
>
> In other words, a toroidal
> > surface can be unrolled into a rectangle, and the independence of the
> > dimensions within that surface is preserved. On a spherical surface, the
> > dimensions whichare at one point orthogonal are at another point parallel. At
> > least, I understand that much of it, which Virgil seems no to.
>
> There is no particular virtue in preserving an arbitrarily imposed
> system of coordinates in relating one surface to another. If the two
> surfaces are homeomorphic, that is enough, but the plane and the torus
> are not. The plane and the punctured sphere are.
> > >
> > > > That causes points opposite the origin, where the axes again meet, to
> > > > simultaneously have x and y values of both 0 and oo. It doesn't work, if
> > > > that's
> > > > what you mean. If not, explain. A sphere is topologically different than
> > > > a
> > > > torus.
> > >
> > > The one point compactification is much more geometrically obvious for a
> > > plane than a toriodal compactification:
> > >
> > > The geometry is simple, and corresponds to an actual mapmaking technique:
> > >
> > > Put a sphere tangentially on top of a horizontal plane in a 3D space,
> > > and consider lines through the topmost point of the shpere intersecting
> > > the sphere in a second point and and intersecting the plane in a point.
> > > Match the second point on the sphere with point on the plane and vice
> > > versa to get a bijection between the sphere less its topmost point and
> > > the plane. Then every point on the plane matches a point on the shpere
> > > and every point except the topmost one on the sphere matches a point on
> > > the plane.
>
> > Yes, the topmost point maps to an infinite circle.
>
> The topmost point does not "map" to anything.
>
>
> > >
> > > Open sets in the plane map to open sets in the deleted sphere and closed
> > > sets map to closed setsand the reverse,so the bijection is bicontinuous
> > Sure, you can create a bijection there, but it is rather a warped mapping,
> > don't you think?
>
> Any mapping of the plane to any finite surface is warped.
> > >
> > > And the topmost point of the sphere becomes the one point
> > > compactification, as the completed sphere is compact.
>
> For The sphere there is only one anomalous point, on the torus, there
> are two anomalous "circumferences" and their point of intersection.
What is anomalous about the two coordinate axes crossing at a single point?
That is the point of using a toroidal surface, so coordinates are consistent.
The curved 2D cartesian plane cannot exist as the surface of a sphere.
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
    ... > A one-point compactification as in both axes meeting in more than one ... Put a sphere tangentially on top of a horizontal plane in a 3D space, ... and consider lines through the topmost point of the shpere intersecting ...
    (sci.math)
  • Re: a tangency thought experiment
    ... four times your height, which is at rest on the plane, ... Thus the sphere shares exactly one point ... tangency. ... ceiling, in your vicinity, becomes increasingly flat ...
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  • a tangency thought experiment
    ... four times your height, which is at rest on the plane, ... Thus the sphere shares exactly one point ... tangency. ... ceiling, in your vicinity, becomes increasingly flat ...
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  • Re: infinity
    ... Cartesian plane. ... The plane and the punctured sphere are. ... >> and consider lines through the topmost point of the shpere intersecting ... The topmost point does not "map" to anything. ...
    (sci.math)
  • Re: infinity
    ... >> A one-point compactification as in both axes meeting in more than one ... > Put a sphere tangentially on top of a horizontal plane in a 3D space, ... > and consider lines through the topmost point of the shpere intersecting ...
    (sci.math)