Natural Science Forum / Physics / Research / September 2004

You would be interested in Hilbert's axiomatization of geometry, which
supercedes Euclid's.  I think Hilbert was an associate of Klein.

If you assume the group and look for algebraic invariants of degree d
in k variables for small d and k you find none for d=1 and k=1, but
a unique one for d=k=2 (up to a constant factor).

Does this derivation satisfy you?

Arnold Neumaier

One way that you might like is to start with a geometry in which the
group of transformations is the general linear group. By Klein's
erlanger programme, this is projective geometry. Then find the duality
between points and hyperplanes in projective space. From this duality, there
turns out to be a natural map (correlation) from points to hyperplanes which
we denote by I. If this map is important it must be the same for all
observers. Since a correlation transforms as I-> fIf^{-1} under the action
of a element f of the transformation group, then the correlation must commute
with the elements of the transformation group. The resulting geometry, given
by the congruence group which commutes with the natural correlation, turns out
to be elliptic geometry and the group is SO(n). In projective geometry, there
is a natural product pH of points p and hyperplanes H which is number-valued
and a number transforms as a scalar. So, the absolute correlation
(or polarity) I can make a number out of points as pIp. The equation pIp=0
defines a hypersurface in the elliptic space (called the absolute quadric).
Now an element f of the congruence group maps the quadric into itself because
if pIp=0 then it turns out that since f and I commute, fp.Ifp=0, so if p
is on the quadric, then so is fp. So, the quadric hypersurface can be
considered as the arena for a geometry in its own right. If we pick an
arbitrary point on the quadric and call it the point-at-infinity and further
restrict the congruence group so that this point is the same to all observers,
then it turns out that the resulting geometry is that of a general flat-space.
The elliptic quadric is not real, so by imposing some restrictions on the
reality or otherwise of certain reference points (rather like the way
Minkowski space is handled by using ict for the time coordinate) one can
make the flat space on the quadric Euclidean or Minkowskian.

The usual formulas for distance (e.g. Pythagoras' theorem) are obtained in the
following way. A simple element of the transformation group can depend on
one parameter s as f(s). Distance between two points p and q is then defined
as the parameter s such that f(s) moves p into q as q=f(s)p. Pythagoras'
theorem is a consequence if f is for Euclidean geometry.

The details are in a text at
http://homepage.ntlworld.com/stebla/Whitehead.html

swamijs@neo.tamu.edu (Swami) wrote in message

Sure. Projective geometry's invariant is the cross-ratio of collinear
points (dual, coincident lines). One now asks for all collineations
that leave a certain quadratic form invariant. The locus of points
represented by the quadratic form is a conic. Let two points not on
the conic be given. Produce the unique line they determine. Determine
the two points of intersection with the conic. Form the cross ratio of
these 4 points, and take the log. This gives a distance function in
the projective plane with respect to the conic. Each conic determines
a different distance function. One of them correpsonds to Euclidean
geometry.

-drl