The first two geometries classically considered, were those of “the plane” and “the space” as they naturally appear. They are now called Euclidean geometries(*), distinguished by their dimension, respectively 2 (for the plane geometry), and 3 (space geometry).

The 3-dimensional Euclidean geometry is the first and most obvious
theory of physics,
describing our physical space. To be a clear (simple) and complete
mathematical theory, this description must be idealized, accepting the simple properties
that seem to fit with ordinary physical measures, as if they were exactly
true, regardless the physical limitations of any attempt
at accuracy verifications. For example, it must regard any two points as
either equal, or different, and that between any two different points there
are other points and so on, which logically implies that any segment
contains infinitely many points. This infinity so comes as a mere good
approximation, as intermediates between points very close to each
other are physically harder and harder to distinguish, with no clear way
to figure out at which scale may this distinction become impossible or
meaningless (this may be argued to be the Planck length, which is far too
small to be measured anyway, much smaller than nuclei of atoms).

Actually, this geometry is known to be inaccurate as a description of
physical space, since General Relativity theory is established as a more accurate
description of the physical space-time, which comes down to Euclidean
Geometry as a usually quite good
approximation (of its pure space aspect, part of Special
Relativity as a description of space-time).

While not exactly a physical theory, Euclidean plane geometry is useful for
educational purposes (being more directly visible and simpler than Euclidean
space geometry but with some complexity, thus a good intermediate step to
understand it), its relevance to aspects of physical space (describing flat
surfaces: screens, papers, pieces of land), and as an understanding of
complex numbers, which play key roles both for pure mathematics, and for
physics (especially the formalisms of waves and quantum physics).

By its general definition, the automorphism group for any theory with a finite language is a second-order construction. In classical geometries (but not topology), it is even first-order constructible from the space. For example, rotations are automorphisms in Euclidean geometries, and can be specified by just a few parameters (depending on the dimension: center or axis, angle... or the image of one tuple). Anyway geometries generally admit their own automorphism group either as a type, or as a subset of a type (especially the type of endomorphisms, if different).

Like in a set theoretical framework, for geometries with nontrivial automorphisms, multiple models (spaces) may be interestingly considered in each isomorphic class (instead of taking one to represent them all), for the following reason. Any isomorphismsInv (*G*) = Inv (*G*
∪ -*G*) = Inv (*G*) ∩ Inv(-*G*)

∀*L*⊂Rel_{E}, ∀*G*⊂ ⤹*E*, *L*⊂sInv
(*G*) ⇔ *G*⊂Aut_{L}(*E*).

We initially conceived invariant structures as those
defined by expressions without parameters. This implies invariance by
automorphisms. Like proofs explore truths (theorems), that is the closure of the set of axioms,
whose addition to axioms preserves the class of models,
definitions
from a language *L*⊂*R* explore its closure
sInv(Aut_{L}(*E*)) =
Inv(Aut_{L}(*E*)) ⊃ *L*, whose addition to
*L* leaves unchanged the set of automorphisms of each model
(among other isomorphisms between models).
Constructions leave unchanged the groups of automorphisms as abstract
groups, just providing more types on which these groups act.

Our last scheme
of constructions (set of all structures
defined by an expression with all values of parameters), was that of an
invariant second-order structure that is a set of non-invariant first-order
structures. Well-defined by that formula, its invariance by
automorphisms means that automorphisms preserve that set while acting as
transformations on it.

Two tuples can be distinguished by a structure

There are still 2 viewpoints which may make both concepts of invariance somehow coincide :

- In first-order logic, the theory admits non-standard models with automorphisms moving some undefinable real numbers to some infinitely close neighbors (by moving non-standard positions of decimals...); but these automorphisms only exist outside the model.
- In second-order logic, we may admit "second-order" ways of defining structures where any subset of ℕ is "defined" by the infinite data of all its elements(**).

The system of real numbers is essentially unique in the sense that

- Its description in second order logic (which, beyond first-order axioms, uses the second-order axiom that its order is complete) is sematically complete (all its standard models are isomorphic);
- Its first-order version (with weak translation as a schema of first-order axioms) is the logically complete first-order theory of real closed fields.
- Having no nontrivial automorphisms in its standard interpretation, the uniqueness of the isomorphisms between its standard models, makes these models essentially the same

Geometries may be interestingly generalized by replacing ℝ by something else such as the set ℂ of complex numbers, still keeping many things work the same.

ℝ can be built from *P*(ℕ), using binary expansions.
Its non-standard models either contain non-standard numbers, or fail to contain
some undefinable numbers. As the first-order theory of real closed fields only
has algebraic equations as a means to define
numbers, the poorest model is just made of algebraic numbers.

This quality of completeness comes in contrast with the incompleteness theorem of arithmetic. It avoids contradiction with it, by the fact that formulas of arithmetic (and especially the undecidable ones) cannot be expressed as formulas in the first-order theory of real numbers. Because this theory cannot express the predicate over real numbers "to be an integer", as the formula "

The theory of real closed field obliges its models (the real
closed fields) to contain all algebraic numbers (solutions of
algebraic equations with (standard) rational or integral
coefficients). However, it cannot express the existence of
any number that is not algebraic (because any finite list of
first-order properties satisfied by a non-algebraic number, is
also satisfied by some algebraic number).

In particular, such a theory cannot express the number π (which is
not an algebraic number). What a pity for a geometry !

At first, we shall assume geometrical spaces and real numbers as known from experience (from secondary-level mathematics). Fully rigorous (set theoretical and axiomatic) foundations will be presented later.

As some geometries (such as the spherical geometry) have no good global coordinates system, at least we put this as a

By the action of the automorphism group, the space of points usually has only one orbit (but vector spaces have 2 orbits, one being a singleton), so that when regarding the group of automorphisms as a space itself, it has even higher dimension.

This happens for many finite-dimensional spaces of interest, but yet not all:

- Some spaces, such as arbitrary Riemannian manifolds, may have no automorphism except the identity.
- Other ones, such as topological spaces, have a "space" of automorphisms that can best be qualified as infinite-dimensional.

(*) in honor to the Greek geometer Euclid, who around 300 BC, published the

(**)This bridge to Inv(Aut

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