## Axiomatic expressions of Euclidean and Non-Euclidean geometries

Contrary to traditional works on axiomatic foundations of geometry, the object of this section is not just to show that some axiomatic formalization of Euclidean geometry exists, but to provide an effectively useful way to formalize geometry; and not only Euclidean geometry but other geometries as well.
Precisely, this section aims to combine the following goals:

• Give a nice and useful axiomatic expression of Euclidean geometry
• Also provide nice axiomatic foundations of other (non-Euclidean) geometries
• Show the deep similarities between these diverse geometries, right from their axiomatic foundations
• Let this be a good start towards higher theories of physics
However, while this section tries to be relatively self-sufficient, it is not completely so : it rather comes after a (still incomplete) series of previous studies on the foundations of mathematics (see the main page for the whole table of contents). This already included some other studies of geometry, especially a general introduction to geometry, a study of affine geometry, and of vector spaces in duality.

We may say that affine and vector spaces were already cases of geometrical spaces, however they still did not look so much like our usual Euclidean space. Now finally we are coming to this familiar geometry, as a particular case of a diversity of similar geometries (somehow more similar to the Euclidean one than vector or affine spaces were).

### Inner product spaces

While the value of the dimension n sufficed to classify affine geometries, it does not suffice with inner products. Inner products are classified by an oriented pair of natural numbers (p,q) with p+q=n, called the signature, which roughly means that "scalar squares are positive in p dimensions and negative in q dimensions". The geometry is Euclidean if scalar squares of all vectors have the same sign (usually taken as positive, that is q=0, but the opposite sign convention of negative scalar squares, that is p=0, essentially gives the same geometry).
Notice that the set of "rotations" (transformations which preserve the inner product) has dimension n(n-1)/2 for any signature.

### Common aspects to all these geometries

Remember that like any mathematical theory, a geometry is made of the following hierarchy of components:
• Types
• Structures
• Axioms
Now all geometric theories we are going to consider, will have the same following list of types
• ℝ = the field of real numbers, with its usual structures (0,1, +,-,⋅), and axioms.
• S = a 1-dimensional ℝ-vector space (meant as the type of surface densities, i.e. lengths to the power -2), thus with its structures 0,+,-, and the multiplication by real numbers.
• E = a finite-dimensional ℝ-vector space, whose elements may be called "coordinates", "affine forms"
• M = the set of points

and the following common list of structures and axioms:

• An operation from E×M to ℝ, that makes the pair (P,E) form an ℝ-duality system and identifies M as a subset of the dual E* of E ; equivalently, E is identified as an ℝ-subspace of ℝM. Formally, this means the axioms
• : ∀x,yE, (∀uM, x(u) = y(u)) ⇒ x=y (Separation of E by M)
• u,vM, (∀xE, u(x) = v(x)) ⇒ u=v (Separation of M by E)
• x,yE,∀uM, u(x+y) = u(x) + u(y)
• xE,∀a∈ℝ,∀uM, (ax)(u) = a x(u)
• A bilinear, symmetric operation ⋅ from E×E to S, called the metric :
• x,yE, xy = yxS
• x,y,z∈E, (x+y)⋅z = xz +yz
• x,yE,∀a∈ℝ, (ax)⋅y,=a(xy)
The separation of E by M, i.e. (∀x,yE, (∀uM, x(u) = y(u)) ⇒ x=y), may be seen here as implicit by the functional notation x(u). It may also be seen as a consequence of later axioms, and dispenses us of the need to declare the vector space properties of E as axioms (though I'm not sure both virtues may be added up).

The next structures and axioms will vary between geometries.

### The case of flat geometry

For flat geometries, the last needed structure will be a symbol of constant called the mass, that is mE, with the axioms
• m ≠ 0E
• xE, mx = 0
• M={uE*| m(u)=1}, i.e.∀uE*, m(u)=1 ⇔ uM
Note that through the last axiom, the axiom m ≠ 0E is just equivalent to M≠ ∅ and thus may be seen as redundant with previous axioms. We may also notice that this axiom may serve as a definition of m, which is thus not really a new structure.

We can define the set of vectors, as V={uE*| m(u)=0}.

Except for the geometry of the Galilean space-time, useful geometries (in particular the Euclidean and the Minkowski geometry) also obey the following non-degeneracy axiom:
• xE, (∀a∈ℝ, xam)⇒(∃yE, xy ≠ 0)

that may also be written in the contrapositive form ∀xE, (∀yE, xy = 0)⇒(∃a∈ℝ, x = am)
and may be condensed with the above axiom mx = 0, to form a single axiom

• xE, (∀yE, xy = 0)⇔(∃a∈ℝ, x = am)

The Euclidean geometry is specified by the more precise axiom
xE, (∃a∈ℝ, x = am)⇔( xx = 0)

which more precisely implies that the sign of xx is fixed (for topological reasons : it cannot switch sign without cancelling somewhere), and thus may be held as positive (defining the order in S).

### The below is a draft being reworked from old versions. Not sure when it will be ready...

The work of correct reasoning on incorrect figures (mentioned in the relativity page) to rebuild the different geometrical concepts, is processed as follows:

Orthogonality is such that for any point of a circle (sphere), the tangent line (plane) is orthogonal to the radius (line through the center).
Distances are related to the notion of sphere, in these ways: a sphere is a set of points at equal distance to the center; the measure of any volume given by a geometrical definition depending on 2 points, is proportional to the n-th power of the distance between them; in a straight line, distances just add up.
The notion of rotation can be defined as "The affine transformations that map circles (or spheres) into circles";
The measure of angles is also related to the notions of distance and circles, in different ways: an angle can be defined as the length of an arc of circle (where the length is the limit of the sum of distances for small divisions of the arc of circle). Or, since rotations (as defined by the preservation of circularity) also preserve angles, we can define angles as the number of "small unit angles" they are made of, for an arbitrary given small unit angle that is moved to different directions using rotations. See other ways to define the measure of angles by its relation to the notion of circle that we already expressed. the fact they preserve angles, gives a general way to compare angles.

Now re-apply the same rules : we cannot anymore use the area inside a circle (the volume inside a sphere) for getting the length of its radius, because circles (spheres), being hyperbolas (hyperboloids), no more contain a finite area (volume) but an infinite one; however we can still use almost the same method through the choice of a fixed angle from the center of this "circle" that cuts there a portion of "disk" (ball) with finite area.

#### Using split-complex numbers for Minkowski plane geometry

The system of split-complex numbers are a nice tool to study the 2-dimensional Minkowski geometry (signature (1,1)) in the same way as complex numbers can be used to study Euclidean plane geometry.

### The case of curved geometry

In this case, the mass is replaced by a relative mass, that is a function from M to E, with a constant cS called the curvature of this geometry, with the axioms

• uM, mu(u)=1
• xE, ∀uM, mux = x(u) c
• (one more axiom, to be completed)

Note that flat geometry comes as the particular case where c=0.