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<snip>

[Hammond]

The Riemannian (Euclidean) Metric is the .... <snip>

[D.E.]

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

metric.

[S.G.]

*> Riemannian geometry is the geometry that resembles that of the surface of a
*

*> sphere, [...]
*

[T.R.]

No. That is an instance of Riemannian geometry, but is by no means the

entire story.

Riemannian geometry is the geometry in which distances are described

by a Riemannian metric. For details see a good textbook....

[Hammond]

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

measurement.

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:

Cartesian

polar

Cylindrical

Spherical

Elliptical

etc.

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

biologists believe.

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:

http://people.ne.mediaone.net/ghammond/Rie-Helm-Weyl.html

-- 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: ghammond@mediaone.net Website: http://people.ne.mediaone.net/ghammond/index.html -----------------------------------------------------------

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