Re: infinity

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.

> 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.

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

And the topmost point of the sphere becomes the one point
compactification, as the completed sphere is compact.
.

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: 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
    ... 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
    ... >>> 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. ... the topmost point maps to an infinite circle. ...
    (sci.math)
  • Re: Cube Planet
    ... >>> Which has me wondering what sort of shape would work, ... > would fly (one point, one face, no edges - think of the sphere as the ... > one-point compactification of the plane). ...
    (rec.arts.sf.written)
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