# Foundations of geometry

This part is under construction

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

Introduction to the foundations of geometry

Structures in the language qualify or measure figures or other objects. Example of structure: the ternary relation of alignment between points. It can also take the form of a second-order structure: the notion of straight line (this notion plays the role of a structure in the language of geometry by the distinction it makes of between sets of points are straight lines, and those which aren't). The binary relation of orthogonality between straight lines is another structure. The notion of rotation may be considered as a type. Other examples involve the type of real numbers (qualifying figures, operations or relations among points, or relating points with real numbers): "The barycenter of points ... with coefficients ... "; "the image of point a by the rotation with center o with angle x".
For example, "the sea level" is the name of a plane, thus distinguishing some points from others, but usually not accepted in the language of Euclidean geometry, especially when describing the internal geometry of an horizontal plane ;) it is not part of the language of Euclidean geometry, so, languages accepting it form a different geometry (even if that is the only difference).

## Structures and permutations in the plane

In the following table, every permutation in a line (named then defined using coordinates, with an amplitude depending on a parameter a∈ℝ), preserves (is an automorphism for) every structure named in a different line (in the last column).
The third column counts the dimension of the permutation group described in each line.
For the sake of naming things, the plane is thought of as a human-size vertical plane with a north-south direction, crossed by the Earth's equator. The coordinates are (x,y) where x = latitude, and y = altitude.

 Permutations names Image of (x,y) Dim. Structures Vertical translation Horizontal translation (x , y+a) (x+a , y) 2 Origin (constant point). (its coordinates = (0,0)) Shear mapping w.r.t. the horizontal axis (x+a.y , y) 2 Euclidean structure: circularity, angles... Squeeze mapping w.r.t. the vertical and horizontal axis (x/a, a.y) with a>0 Rotation around origin (x.cos a − y.sin a, y.cos a + x.sin a) 1 Altitudes comparison ("to be higher than") Dilation [from/to] origin (a.x, a.y) with a>0 1 Unit of area Reflection w.r.t. vertical axis (in pair with Id) (-x , y) 0 Orientation (left/right, sign of angles)

## Affine geometry

Affine geometry, is the geometry whose structures (the affine structures), are preserved by all of the above listed permutations.

Thus, these do not include any of the above listed structures, but the following ones.
Some fundamental affine structures (each of which suffices to define all others), are

• The notion of segment; equivalently, the 3-ary relation of betweenness B∈[AC], and the notion of (straight) half-line.
• The notion of straight line, or equivalently, the 3-ary relation of alignment
• Barycenter
• Coordinate (i.e. affine form)
Other affine structures (definable from the above) are
• Parallels (as an equivalence relation between lines or segments), thus the notion of parallelogram
• Ratio of lengths of parallel segments; middles of segments
• Vector (i.e. translation)
• Ratio of areas (with the same dimension as the space, i.e. filling a region; thus "areas" if the dimension is 2): they can be defined by filling a surface with small parallelograms and counting their numbers.
• Ellipse, parabola, hyperbola; center of an ellipse or hyperbola (but no data on foci, axis and eccentricity)

This list of structures still applies in higher dimension n>2, except that ratios of areas only apply to parallel planes; instead, we have ratios of n-dimensional volumes.
(Precisely, for any 0<k<n we can also compare the k-dimensional volumes in parallel k-dimensional subspaces, so that the ratio of lengths is the particular case k=1, and the comparison of volumes is the case k=n where they are always parallel).

### Properties the affine group

A transformation of a plane, is called an affine transformations if it satisfies the following properties, which are equivalent:
• It is an automorphism of affine geometry (i.e. it preserves the above affine structures)
• It can be obtained as a composite of permutations from the above table
• It can be written with parameters a,b,c,d,e,f such that a.eb.d, as
(x,y)↦(a.x+b.y+c , d.x+e.y+f)
• It can be intuitively understood as the approximation of a transformation obtained by looking at a figure in perspective from a large distance compared to the size of the figure (as much larger as we need the approximation to be accurate), and eventually through a mirror.
The set of these transformations (the automorphism group of affine geometry) is called the affine group.

This study can be generalized to higher dimensions : for each value of the dimension n there is one n-dimensional affine geometry, with only one isomorphism class of models (n-dimensional affine spaces), and an (n2+n)-dimensional affine group for each model.

Let us further comment the above table of diverse affine transformations of the plane.

For every affine transformation f obtained as composite of any number of permutations from given lines of this table (with possible repetitions in any order), and for every (possibly different) choice of order between (only) these lines, there is a unique tuple of values of the parameters of permutations in these lines (except adding multiples of 2π for rotation angles) so that their composite in this new order (now without repetition) coincides with f. This tuple of values of parameters can be used as "coordinates" of f (their number is thus the dimension of the space of automorphisms for structures from the rest of lines).

For this, we had to put vertical and horizontal translations together, as they are mixed when composed with rotations, depending on the composition order. Similarly, shear mappings mix vertical translations with horizontal ones (though not vice versa), and rotations mix shear and squeeze mappings together.

The permutations in each line of the table, move the structure in the same line to "all its possible other values" (in the framework of affine geometry) without repetition. This way, a structure of a kind described in a given line of the table, can be chosen (added to the language, with a value among its "other possible values" from affine geometry) independently of choices of other kinds (described in other lines). It brings no information expressible by closed formulas; the only "effect" of a list of choices of structures from given lines of the table, is to reduce the automorphism group of the resulting geometry, to the mere set of composites of permutations from the complementary list of lines of the table.

There may be other ways to split the affine group as the set of composites from a list of subgroups (and each subgroup as a list of 1-dimensional ones) satisfying the above remarks (that it forms a sort of coordinates system for the group... ). The above way has 2 advantages:
• it shows the main diverse kinds of transformations
• it includes the Euclidean structure.

More detailed study of affine geometry in another page.

## 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)
Let us mention some more.

### 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.

### 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.

## 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 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.

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Next text on geometry: Affine geometry

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