The Riemannian (Euclidean) Metric is the .... <snip>
This is strange. Riemannian geometry isn't the same as Euclidean geometry,
I'd be surprised if the Riemannian metric was the same as the Euclidean
> Riemannian geometry is the geometry that resembles that of the surface of a
> sphere, [...]
No. That is an instance of Riemannian geometry, but is by no means the
Riemannian geometry is the geometry in which distances are described
by a Riemannian metric. For details see a good textbook....
Well lets get some basic definitions in here:
1. Both Riemann and Weyl (and Helmhotz) agree that
a "space" is simply an ordered array of
"n-tuple points" P(x1,x2,x3,x4,...xn) where each
of the x's belongs to the continuous domain of
the real numbers. This space is essentially
"formless", to use Weyl's words.
2. An "affine" space is created by defining a "length"
on each x-axis and defining a "congruent translation"
of a vector on each axis so that "parallel" lengths
can be compared. Again the space is essentially
"formless", but now a "length" has been introduced
but only on each axis.
3. A "metrical" space is one step beyond affine space
in that it allows the comparison of "lengths" which
are not necessarily "parallel" to each other. This
amounts to defining a "scalar product" of some kind
between two non-colinear vectors. Apparently the
form of this scalar product determines the "metric"
because it is a form of non-parallel length comparison.
4. If in fact you want to impose the condition that
"congruent rotations" of a vector are possible, as
well as congruent translations, it turns out that
this requirement requires that the metric be a
homogenous quadratic form (Weyl, ibid 1920).
IOW, requiring that the (congruent) infinitesimal
rotation group exist in the space is sufficient to
determine the metric, and that metric is quadratic
(i.e. is Riemannian). In that case the result is
that the space is locally Euclidean and the familiar
scalar product appears. Volumes are also locally
invariant upon rotation.
The upshot of all of this is that only a quadratic (Riemannian)
metric will allow "congruent rotations" and that requirement
alone seems to be sufficient to explain why "real" space is
actually Riemannian. Congruent simply means that when you
rotate a solid object it doesn't change shape, which is the
experience we have of real space.
One important result of the quadratic metric of real space
is that (locally) 3D space is Euclidean. It certainly is on the
surface of the Earth where even modern science cannot detect
the departure of real 3D space from Euclidicty by direct
Since it is known that the simplest coordinate system which
will obey the Euclidean metric is the Cartesian coordinate
system, this has an immediate impact on the visible form
of the World.
This is so because a simple "machine" is nothing more than
a mechanical coordinate system. Of the simple coordinate
systems that may be devised in Euclidean space:
Without any doubt, the Cartesian coordinate system is the
simplest to mechanically construct, therefore most simple
machines are "Cartesian mechanical systems" (T.V., typewriter,
car, airplane). Of course you have exceptions like a
washing machine which is a cylindrical coordinate machine
or a ball bearing which is a spherical machine.
Of particular note is the fact that if "Nature" wanted to
build a machine, it would be a Cartesian machine. And this is
the reason that the "Body Plan" of all Plants and Animals is
in fact 3-Axis Cartesian.
Human beings are in fact nothing more than walking, talking,
3-Axis Cartesian coordinate systems. The skeleton itself
is a 3-Axis Cartesian coordinate system, and believe it or
not, there are 3-semicircular canals in the middle ear which
detect rotation along the 3-Cartesian axes of the skeleton.
So that the bottom line is, that the FORM of the human body
is determined by the METRICAL LAW of real space... not by
Darwinian Natural Selection for instance which is what most
If then, "Man is made in the image of God", Riemannian
geometry must be the description of God.. or more specifically,
Einstein's theory is. See:
-- Be sure to visit my website below, and please ask your news service provider to add alt.sci.proof-of-god ----------------------------------------------------------- George Hammond, M.S. Physics Email: firstname.lastname@example.org Website: http://people.ne.mediaone.net/ghammond/index.html -----------------------------------------------------------
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