Introduction to the foundations of geometry

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

What is geometry

Geometry, which etymologically means “measure of the earth”, developed as the study of mathematical relations (formulas) between possible measures of diverse figures. As these relations are meant to hold for any figures (or some specified figures chosen by convention or imagination) independently of the bodies that may be physically present, they are understood as mathematical properties of the (empty) space inside which these figures can be inserted. Classical studies focused on two geometries : those of “the plane” and “the space” as they naturally appear.
In modern mathematics, geometries are a wide and fuzzy range of mathematical theories describing diverse systems also intuitively thought of as “spaces”, whose basic objects are "points", and other objects (lines, figures...) are constructed from points (i.e. are systems of points).

A geometry, like any other mathematical theory, is made of a list of types, a language (list of structures) and a list of axioms:

Let us clarify this role of "notions" as structures, summarizing the ideas of previous texts without referring to them:
A figure (line, triangle, square...) is a set of points, thus a particular case of relation: a unary relation over points (a relation on only one variable), but usually not found among geometric structures.
Each notion plays the role of a structure by the way it distinguishes some operations, figures or other relations, as belonging to it, as opposed to those which don't. While it is a set of operations or relations, its elements are usually not themselves structures. For example the notion of straight line is a structure by the distinction it makes of which sets of points are straight lines, and which aren't. Still, there, an individual straight line usually is not a structure. It can only be a structure if it is the one named by a given symbol in the language of a theory : it can either be formalized as a constant symbol naming this line (instead of letting it remain a mere possible value of a variable symbol ranging over lines) or as a unary relation symbol over points, (giving the distinction of which points belong to this line, and which points don't). For example, "the sea level" is a structure naming 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 ;)
Or, in a given theory, an element of a notion (e.g. a straight line) may be discovered to be a structure if we find a formula that uniquely defines it (independently of any free variable), i.e. that distinguishes it from all others - which actually does not happen for straight lines in classical geometries.

Among such theories, the classical plane and space geometries are two particular cases, now called 2-dimensional Euclidean geometry (or Euclidean plane geometry), and 3-dimensional Euclidean geometry (Euclidean space geometry). They are named in honor to the Greek geometer Euclid, who around 300 BC, published the Elements, mathematical treaty gathering and methodologically ordering, with logic, axioms and proofs, the main mathematical knowledge of that time, with a special focus on plane and space geometries. It remained the most famous mathematical work until the 19th century, when its axioms and proofs were found incomplete. Only in 1899 a really complete axiomatic expression of Euclidean plane geometry was published by Hilbert.

Other geometries can have any number of dimensions, but this number does not suffice to classify geometries. For each n∈ℕ*, there is one n-dimensional Euclidean geometry, but also diverse other n-dimensional geometries, with different concepts and properties. A general intuitive definition of the dimension n of a space, is : the number of coordinates in any local coordinates system, so that the points in a given (small enough) region of the space, bijectively and continuously correspond to n-tuples of real numbers in some region of ℝn.

The 3-dimensional Euclidean geometry is the first and most obvious theory of physics, describing our physical space, while idealizing it (as always in mathematics): any two points are assumed to be either equal, or different. For it to be a mathematical theory determining the properties of objects, simple properties that seem true by ordinary measures, are taken as exactly true, so that Euclidean geometry is the simplest theory in agreement with ordinary measurements of space.

For example, it states that between any two different points there is another point, and so on, which logically implies that a segment with finite length contains an infinity of points. This is only "approximately" verified, as intermediates between points very close to each other are physically harder and harder to distinguish, with no clear notion of which is the scale where this distinction might become impossible.
Finally, once discovered and verified by experience, General Relativity theory came as a more accurate description of the physical space (or rather, of the physical space-time), that Euclidean Geometry only describes up to a very good accuracy, with known and quite small margins of error.

From a mathematical viewpoint, all (consistent) geometries are equally “true”, as studies of their respective abstract realities, disregarding possible connections to physics. In fact, many aspects of our physical universe are described by geometries; the general study of diverse geometries disregarding which ones play a role in physics and how, helps to understand those which do.

The interests of Euclidean plane geometry, are

A common feature of many interesting geometrical spaces, is that they have an infinity of automorphisms (bijective transformations that preserve all structures - for example, rotations are automorphisms for Euclidean geometry). Precisely, each one's group of automorphisms is itself a space with several dimensions (or at least one).
This happens for many finite-dimensional spaces of interest, but yet not all:

To compare with set theory, models of the traditional ZF set theory (without pure elements) admit no automorphism, but a set theory including geometrical spaces as systems of pure elements, usually inherits their respective automorphisms.

The presence of many automorphisms brings the following motivation to consider a plurality of spaces isomorphic to each other (instead of taking one to represent them all).

Given 2 systems E and F with an isomorphism f : EF, we have a bijection from Aut(E) to the set of isomorphisms between E and F, given by gfg. Thus, a plurality of automorphisms of E means a plurality of isomorphisms between E and F.
It would not be interesting to consider 2 models with only 1 isomorphism between them, as this isomorphism makes them copies of each other, letting each one play the role of the other in a unique manner. But a plurality of isomorphisms between 2 systems makes their difference meaningful, as an object in one system may correspond to several possible ones in the other, depending on the choice of isomorphism. Thus choosing an object in the one, does not mean choosing an object in the other (in an invariant way).

For this and other reasons, geometry will be viewed from the framework of set theory : each geometry will explicitly admit a diversity of possible models, that are sets where the geometric structures (vocabulary) are interpreted as operations and relations (and sets of figures).

A geometrical space with a large automorphism group, can be described by giving this permutation group instead of its structures. Indeed, this roughly determines the list of structures as those which are invariant (preserved) by these permutations. These invariant structures are the expressions of what is known of any tuple or figure that is seen after undergoing any (unknown) transformation from this group. This approach to geometry by first giving a permutation group on a set of points, and then looking for their invariants, was the basis of Felix Klein's Erlangen Program for the foundations of geometry.

Let us introduce geometry in this way. It will first be more intuitive than rigorous, assuming geometrical spaces and real numbers as known from experience (from secondary-level mathematics). Fully rigorous (set theoretical and axiomatic) foundations will be presented later.

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)
Vertical translation
Horizontal translation
(x , y+a)
(x+a , y)
Origin (constant point).
(its coordinates = (0,0))
Shear mapping w.r.t. the horizontal axis
(x+a.y , y)
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

Other affine structures (definable from the above) are

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: 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:

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:

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

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.
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".


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) :

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 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:

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

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):

The completeness of first-order geometry

Geometries may be formalized either as first-order or second-order theories (even to describe the same space). Let us explain how both options can be seen as relatively equivalent in a satisfying way, though not exactly equivalent (they do not have all the same models). 

In any case, they naturally include the set ℝ of real numbers as a type of objects, for the following reasons.

But, first-order theories cannot specify geometrical spaces into one isomorphism class independently of ℝ, because they cannot do so for ℝ :

Definition of the order relation in ℝ. Theories of "real numbers" that we shall consider, define the order between these numbers as
xy ⇔ (∃z, y = x + zz)
with axioms that require this to be a total order.

Existence of nonstandard models of . Any first-order theory of real numbers (that may include other types, and even be a set theory) admits non-standard models, that is, where there exists "real numbers" larger than any standard natural number.

We may either deduce this from the existence of nonstandard models of any first-order theory of ℕ, including one that relates it to ℝ (such as set theory or second-order arithmetic seen as first-order theories), or adapt the argument which we gave for ℕ to directly apply to the case of ℝ, as follows.

Take a first-order theory of real numbers, with ≤ defined as above, and add to it a constant symbol ω with all axioms of the form 1+...+1≤ω.
Then, models of this consistent theory are nonstandard models of ℝ with an infinitely large element ω.

However, ℝ may still be seen axiomatizable in the following 2 weak senses:
Therefore, we get an expression of any geometry as a complete first-order axiomatic theory, by means of its relation with ℝ together with such an axiomatization of ℝ.
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 "x=0 or x=1 or x=2 or...(and so on to infinity)", well, is not a 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 !

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