5. Foundations of geometry

What is geometry
The use of real numbers
Dimensions and semantic completeness
Multiple automorphisms and models
The double meaning of "invariance"

Intuitive descriptions of affine maps
Affine subspaces
Straight lines and algebraic structure
Directions
Affine forms
Other affine structures

Duality systems
Coordinate systems
...

(For a short simplified version without reference to the prerequisites, just read the text in green)

Beyond affine geometry

Some geometric spaces, such as vector spaces, Euclidean spaces, and both space-times without gravitation: the classical one (the Galilean space-time), and the one of Special Relativity (the Minkowski space), are "richer" than affine geometry, i.e. "contain an affine structure", in the following (roughly) equivalent senses:

• They contain, or at least can define and prove, the language and axioms of affine geometry
• They can be obtained as affine geometry with a choice of one (or two...) more structure(s) that (as the dimension is finite) is definable from affine geometry with a tuple of fixed parameters among points (subject to a finite list of further axioms)
  
• Their automorphism group is included in the affine group.

But other geometries do not contain an affine structure, either because they have no notion of straight line, or because these lines do not satisfy the axioms of affine geometry. For example

• Spheres (n-dimensional spheres inside (n+1)-dimensional Euclidean spaces) and other curved spaces
  
• Automorphism groups of diverse spaces (affine, Euclidean and others)

Projective geometry

Like affine spaces, projective spaces are also based on the structure of alignment, but have no other affine structures from the above list. Parallels cannot be defined there as any pair of straight lines in a plane has one intersection point. Such spaces can be described as affine spaces together with extra points "at infinity" in the role of intersection points of parallel lines, while ignoring which points are at infinity and which are not. These points "at infinity" form the line of "horizon" that is a straight line like others. Projective transformations of the plane (automorphisms of the projective plane) are those involved for perspective representation. Among them, affine transformations are those keeping the horizon to itself, so that affine geometry is equivalent to projective geometry with a constant symbol named "horizon" with type "straight line".

Inversive geometry

It has a notion of circle (or sphere) but cannot distinguish straight lines among them.
Follow the link for details.

Differential geometry

Still, affine geometry is not far away from the above, as small regions of these spaces are approximately affine too: the smaller a region, the better its description can be approximated by affine geometry (with possibly more structures). Usual geometric spaces will have this property of being approximately affine in small regions : we will say they are smooth.

Differential geometry is the "geometry" whose only structure is the notion of smoothness, and smooth curves.
In particular, smooth spaces have an approximation for ratios of small volumes as they become smaller and closer to each other.

The smoothness structure cannot be restricted to a structure of "being approximately projective in small regions" because the "approximately affine structure" of small regions would anyway be definable from it, as the horizon relatively to a small region, has its needed approximative definition as "what is not near this small region".

Topology

This is even weaker than differential geometry, as it has a notion of curve requesting their continuity, that is less restrictive than smoothness. For example the Koch snowflake is continuous but not smooth, so that it is distinguished from curves by differential geometry, but not by topology.

Introduction to topology

Euclidean geometry

Let us first only present the structures, assumed to obey the properties of the intended spaces. The axiomatic specification of these properties will only be completed later.

Euclidean geometry (with any dimension >1) consists in affine geometry together with one structure, thus named the Euclidean structure, that can be expressed (represented) in the following equivalent forms (any one suffices to define the others) :

• The notion of circle, and (in dimension >2) the notion of sphere
  
• The notion of rotation
  
• The measure of angles (between straight or intersecting lines, planes or spheres).
• Orthogonality (as a relation between straight lines or intersecting lines)
  
• Ratios of distances, i.e. distances with values in a type of quantities
  
• Data of the eccentricities, foci and axis of ellipses and hyperbolas
• Dot product between vectors (or inversely, between coordinates), with values in a type of quantities

Optionally, Euclidean geometry may also admit one more structure from the previous list : the unit of area (or unit of volume, in the case of higher dimension), which finally (thanks to the above Euclidean structure) can be called a choice of unit of distance. The choice to include or not this structure in the definition of "Euclidean geometry" is debatable, with motivations for or against it, as follows

• For physics, we have a different situation depending on the theory of physics being considered, and more precisely depending on the "framework vs specific theories" hierarchy. Namely, as a framework, Quantum Field Theory does not include a choice of unit of distance, but each specific substance (type of wave/particle) that it describes, usually comes with its own physical constant(s) (its mass and the intensity of interaction between substances) that break(s) the similarities between scales (making things behave differently at different distances). So, dilations may be automorphisms for the framework of quantum field theory itself, but not for specific laws of physics associated to specific substances inside this framework. The result is that there is not one best unit of distance for all situations, but there are diverse physical phenomena involving some physical constants, by which it is possible to absolutely express some possible choices for a unit of distance (such as the size of a given molecule, though this one may be too fuzzy to be used as a reference). In other words, there is not only one natural unit of length, but there are several ones available from diverse physical definitions.
  As for General Relativity, it relates distances with densities, so that we may see it as dissimilar for distances insofar as we view densities as having an absolute unit. Of course, once put together, General Relativity and Quantum physics define an absolute unit of distance (the Planck length 1.616×10⁻³⁵ metres) ; but this has no practical consequence. 
• For mathematics, admitting both structures has the advantage that they are together expressible as one structure : an operation of distance, or equivalently a dot product, taking values in ℝ (instead of a type of quantities).
  

For any dimension, the operation of distance d(A,B) between any two points A and B (with values either among real numbers or in a set of quantities, depending on the above choice) can be seen as the fundamental structure of Euclidean geometry, as it is completely sufficient to define all other structures of this geometry in a rather natural way.

Indeed, distance suffices to define the affine structures: the betweenness relation is defined as

B∈[AC] ⇔ d(A,C) = d(A,B) + d(B,C)

and others can be defined from it. Therefore, the transformations preserving distance (even if its values are mere quantities), called isometries, preserve all other structures of Euclidean geometry as well.

Other presentations of the Euclidean structure, assume a priori structures:

• Affine geometry is needed for introducing angles between straight lines (using the notion of straight line) or the dot product (using the notion of vector);
• The notion of smooth curve, thus some differential geometry, is assumed for introducing intersection angles.

As for the notions of circle (or sphere) and intersection angles, they may suffice to define affine geometry but only

• either in a way that can be criticized as unnatural (as it needs to distinguish straight lines from circles, which requires measurements near infinity (to know "where the infinity point exactly is"),
  
• or by using the ratio of volumes.

The details of these correspondences between different formulations of the Euclidean structure, will be explained in the introduction to inversive geometry.

The isometries of an Euclidean plane or space are called Euclidean moves (to be distinguished from the isometries of other spaces such as a sphere, with also an operation of distance but that does not satisfy all the same axioms). The space of isometries of the Euclidean plane is 3-dimensional, and split in 2 pieces (each of which is also 3-dimensional):

• Those preserving the orientation, are rotations and translations. Indeed a rotation can be specified by its center (dim=2) and its angle (dim=1). Or by choosing the image of a given point (dim=2) and then the angle (dim=1). (Translations are limit cases of rotations with small angles around faraway points)
  
• Those reversing the orientation, are reflections with respect to any axis (= composites of rotations with the reflection by a fixed axis), composed with translations.

Set theory and foundations of mathematics main page