What is Euclid's Euclidean space?

[Rasalhague]

The word Euclidean space is applied to various distinct mathematical objects. One, kind of Euclidean space is the affine space (general sense of "affine space") defined by the Euclidean group of isometries, which don't including scaling. But wouldn't Euclid's axioms apply equally well if we scaled our units by any real number? How would Euclid's Euclidean space be classed in, say, the language of manifolds?

 

[lavinia]

The word Euclidean space is applied to various distinct mathematical objects. One, kind of Euclidean space is the affine space (general sense of "affine space") defined by the Euclidean group of isometries, which don't including scaling. But wouldn't Euclid's axioms apply equally well if we scaled our units by any real number? How would Euclid's Euclidean space be classed in, say, the language of manifolds?

Euclidean space has many meanings now a days.

In topology, it refers to R^n with the topology inherited from the Euclidean metric.
But the metric is forgotten and only the topological space is retained.

On this topological space, metrics can be added as extra structure. Often the metric is Riemannian which means that it arises from an idea of angle and length of tangent vectors at each point. For a Riemannian metric, one needs to view R^n as a differentiable manifold and not just a topological space. So this is the second meaning of Euclidean space, the topological space together with a notion of calculus on it.

Once this manifold has a metric then as a Riemannian manifold it is only called Euclidean space when the metric is flat, this when its curvature tensor is identically zero. In this case, the Euclidean group will be the group of isometries of this flat Riemannian manifold.

Scale expansions will not be isometries on this flat manifold but will preserve the sum of the angles of a triangle and this invariance of the sum of the angles can be viewed as the defining feature of Euclidean geometry.

underlying Euclidean geometry is R^n, the topological space. Flat geometries can be put on other manifolds with different topologies and locally these flat manifolds will be indistinguishable from flat Euclidean space. For instance, a Klein bottle can be given a flat metric.

Klein's Erlangen program - which I know nothing about - tried to classifly geometries by their isometry groups. The group of Euclidean motions defines Euclidean geometry and other groups defined other geometries. Since I can not explain this point of view I include the Wikipedia link.

http://en.wikipedia.org/wiki/Erlangen_program

 

[Rasalhague]

Thanks, lavinia. Could Euclid's Euclidean space (EES), for a given dimension, be defined as an equivalence class of Riemannian manifolds (equivalent up to scaling of the metric) with a certain topology, and vanishing curvature at every point, and differentiable structure too as required? (Or similarly of affine spaces whose translation space is equipped with an inner product, equivalent up to scaling of the inner product.)

Wikipedia limits the definition of a symmetry group to isometries. Would it be possible to define--or does there already exist--a more general kind of group which would include scaling?

 

[lavinia]

Could Euclid's Euclidean space (EES), for a given dimension, be defined as an equivalence class of Riemannian manifolds (equivalent up to scaling of the metric) with a certain topology, and vanishing curvature at every point? (Or similarly of affine spaces whose translation space is equipped with an inner product, equivalent up to scaling of the inner product.)

I think so.

I think that originally Euclidean geometry was a plane geometry and did not use coordinates for its definition. It used simple postulates about lines and planes such as a line separates a plane into two identical half planes, two lines intersect in at most one point and so on. The parallel postulate was added but was long believed to be provable from the simple separation postulates. I do not know whether there are other models of these postulates that are unlike R^2 with a flat metric.

Wikipedia limits the definition of a symmetry group to isometries. Would it be possible to define--or does there already exist--a more general kind of group which would include scaling?

I am not sure about this - but there is the idea of a conformal map. This is a map which in the infinitesimal preserves angles but scales lengths. For instance stereographic projection is a conformal map from the sphere to the plane.

 

[Bacle]

To add my 2 cents, my understanding was that a Euclidean space is any model

of Euclid's axioms, i.e., any collection of objects with specific objects named

point, line, etc., that satisfy the axioms. And standard R^n is one such a model. And

some maps on standard R^n to other spaces may preserve the satisfaction of the

axioms, giving us other Euclidean spaces. But my comment is bare bones.

 

[Rasalhague]

I was thinking that EES must be something more than a topological space (since it has a notion of angles and distance), but less than a metric space (since there's no natural choice of units, i.e. given any choice, there are infinitely many equally good ones). The principle of symmetry groups seems to be to define a space by functions which leave it unchanged. Since scaling leaves EES unchanged, I'm curious as to why scaling has been given this special status and not included along with rotation, reflection and translation as part of the Euclidean group of symmetries. Is there a mathematical reason for this, or a motivation that comes from physical applications?

In terms of manifolds, could scale be made a property of individual charts, rather than an intrinsic property?

 

[disregardthat]

Does this touch upon what you had in mind?

 

[Rasalhague]

Thanks, Jarle. I'm afraid I don't understand what the author means by "synthetic geometry":

Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.

I thought all mathematics, including every branch of geometry, was based on "axioms, theorems and logical arguments".

That said, the links to "synthetic differential geometry" and the axioms of "synhthetic spacetime" by Wilson and Lewis look interesting. Incidentally, regarding the latter, do you know what "open order" means, of points?

"Axiom 4: The line shall be regarded as a continuous array of points in open order."

This text is the only result Google gives for expression "points in open order".

 

[lavinia]

Mathematics is not based on axioms. It is based upon geometric and analytic structures.Much of Physics though seems to be axiomatized - for instance non-relativistic quantum mechanics and the Special Theory of Relativity.

 

[disregardthat]

What I think lavinia means is that mathematics use axioms as logical foundations, but is not itself, as an activity so to say, based on axioms. One can freely pick ones axioms, or leave them out altogether, and still do mathematics. He could probably correct me on this if I'm wrong.

 

[lavinia]

What I think lavinia means is that mathematics use axioms as logical foundations, but is not itself, as an activity so to say, based on axioms. One can freely pick ones axioms, or leave them out altogether, and still do mathematics. He could probably correct me on this if I'm wrong.

Yes. You said it better than I did.

mathematicians think about structures not axioms.

Many people seem to think that mathematical reasoning is just deduction from initial assumptions or axioms. In fact, mathematics relies more on visualization, intuition and insight. Understanding comes unexpectedly and is rarely the result of deduction.

 

[disregardthat]

To illustrate that I like to think of simple combinatorial arguments, e.g. solutions to combinatorial puzzles. Such arguments often make very little use of the type of formal arguments normally used in mathematics, and rather make frequent appeal to intuition. But the arguments are undoubtedly mathematical in nature.

Sure, one could at any stage axiomatize the argument and deduce on the grounds of this, but I would say that the process of accepting one type of axiomatization for the existing intuition would be more or less logically equivalent to accepting the intuitive argument in the first place. It's just that axiomatization sometimes can be useful, and other times not.