Re: [Phys-l] Another t trivial question

[John Denker <jsd@av8n.com>]

On 01/04/2011 12:34 PM, ludwik kowalski wrote:
> On Jan 4, 2011, at 2:20 PM, John Denker wrote:
>
> > On 01/04/2011 12:10 PM, ludwik kowalski wrote:
> > > Suppose I create a 2D coordinates system that has three axes, at
> > > 60 degrees with respect to each other. It has three triants (what
> > > is a better name three regions in such space?). Suppose I use
> > > this system to describe rooms. Suppose the axes are labeled (a)
> > > for the absolute pressure, (b) for the absolute temperature, and
> > > (c) for the floor area. Is there anything heretical in doing
> > > this?
> >
> > Yes (in almost all cases).
> >
> > There is a theorem that says you cannot map 3 dimensions onto 2
> > dimensions in a way that is one-to-one and continuous.
> >
> > So ... except for the unusual case where your data was somehow 
> > confined to a 2-dimensional subspace of the nominally
> > 3-dimensional space (pressure, temperature, area) ... something bad
> > is going to happen if you try to plot 3 axes in the plane.
>
> I deliberately selected a case in which three coordinates are always
> positive. Each positive axis begins at the origin, not at infinity.
> Plotting three positive axes in 2D space is actually easier than in
> plotting three positive perpendicular axes in 3 D space.

That doesn't repeal the theorem that says you can't have a mapping 
from M dimensions to N dimensions that is one-to-one and continuous.

Try it.