Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!uwm.edu!zaphod.mps.ohio-state.edu!mips!apple!motcsd!xdos!doug
From: doug@xdos.UUCP (Doug Merritt)
Newsgroups: comp.graphics
Subject: Re: 4D Visualization (If you think you do it, you probably don't)
Message-ID: <655@xdos.UUCP>
Date: 10 Feb 90 16:57:31 GMT
References: <99@emtek.UUCP> <16033@well.sf.ca.us> <6162@eos.UUCP> <1263@polari.UUCP>
Reply-To: doug@xdos.UUCP (Doug Merritt)
Organization: Hunter Systems, Mountain View CA (Silicon Valley)
Lines: 85

This stuff is really the domain of sci.physics, but it has been
*thoroughly* hashed and rehashed over there, so I don't think that
it is appropriate to move the discussion. But it also does not
directly concern graphics, so most of you will want to skip the 
following!

Feel free to criticize the following all you like, but I suggest
that email would be the best way to do so, rather than cluttering
comp.graphics with non-graphics stuff.

In article <1263@polari.UUCP> tima@polari (tim anderson) writes:
>
>If I remember correctly, someone proved the existence of two 4th dimensions.
>This solution makes all of the Physicists happy, because they can say that
>time is the fourth dimension, and it makes the Mathematicians happy because 
>they can say it isn't...

Phrased this way, this is pretty much meaningless. To laymen, a "dimension"
always means "a spatial dimension that's part of our universe such as
up/down, right/left etc". In both math & physics, however, the term
just refers to how many variables you've got in an equation. If the
equation happens to refer to space (volume), for instance a Euclidean
equation of distance d = sqrt(x^2 + y^2 +z^2), then these three dimensions
are the intuitive sort.

But the variables (same as dimensions, remember) could just as well
be referring to color, day of the week, and the reader's birthday.
Or to nothing in particular...in pure math it is the relationships
that are of interest, and no particular mapping between the variables
and anything in the real world is assumed. In applied math, some particular
abstract set of relationships developed in the realm of pure math may
often be applied to any one of many totally different kinds of physical
phenomena.

In terms of relativity, it's true that there are four dimensions
assumed, three of which are nominally considered to be "space-like"
and the fourth of which is (again nominally) "time-like". So there
are nominally only three spatial dimensions even in relativistic physics.
Of course, it turns out that at relativistic speeds, that fourth
dimension can appear space-like, loosely speaking. Which just means that
the naive definition of dimension is misleading.

Still, space-time can be visualized geometrically as a 4 dimensional
space. It's not a Euclidean space, though, so intuition tends to give
the wrong answers. And one of those dimensions has different properties
than the others...the distance formula is sqrt(x^2 + y^2 + z^2 - t^2),
and the change in sign for the time dimension gives that dimension different
geometric properties than the three space-like dimension.

This is further complicated by the fact that any given object always
has a constant velocity of C (speed of light) in four-space. An object
apparently at rest has a velocity of C along the time axis. "Accellerating"
that object actually has the effect of rotating its trajectory away from
the time axis, and toward one of the spatial axis, lessening its velocity
component in the time dimension and increasing it in a spatial dimension.
Light itself always has a component of zero along the time axis, and so
its velocity of C is always apparent purely in the spatial dimensions.

The various bizarre and well publicized relativistic effects of high
speed travel are not present in the reference frame of the object itself.
They are visible only to an outside observer, whose time dimension is
pointing in a different direction in four space...the two systems have
different metrics of time, which causes various measurements to behave
non-intuitively.

There have been other geometric models for space time that have been
proposed that add still more dimensions. Although these other dimensions
can be visualized as space-like in an abstract geometric sense, they do not
manifest themselves *macroscopically* (i.e. in intuitive terms) as space-like.

There continues to be questions of exactly what we mean by "space" and
"time" and what that may have to do with the geometric structure of
the universe, and these questions are by no means completely resolved.

Back to the original point: although it may sound impressive to declare
that "time is the fourth dimension" (or like the graffiti in Berkeley,
"gravity is the fifth dimension :-), this is more misleading than anything.
A more neutral phrasing is that "time is *a* fourth dimension", because
then it's more clear that it's just an arbitrary variable. You could
say the same thing about mass or charge or color or whatever...they could
all be *a* fourth dimension in some system of equations.
	Doug
-- 
Doug Merritt		{pyramid,apple}!xdos!doug
Member, Crusaders for a Better Tomorrow		Professional Wildeyed Visionary