# 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:

• Types (names of pairwise disjoint sets of objects): in usual formal expressions of geometries, there are 2 basic types
• "The space", i.e. the set of "points" being described
• The type ℝ of real numbers (to serve as convenient tools in descriptions)
• Other types are usually also involved but will rather be placed instead in the list of structures, as "notions", since they are second-order structures, and are usually contained in those of a full second-order theory over points and real numbers.
• A language (vocabulary), that is a list of names (symbols) designating structures : here, by "structure", we shall mean either
• An operation or a relation between types of objects.
Examples : "Points a,b,c are aligned" ; "The barycenter of points ... with coefficients ... "; "the image of point a by the rotation with center o with angle x"; the orthogonality relation between lines (but the mere operations and relations between real numbers only, are not counted as structures).
• A "notion" (for example the notion of straight line, or the notion of rotation), i.e. a name that measures or qualifies figures, operations or relations among points, or relating points with real numbers.
• Axioms (claims expressed in this language, specifying some required properties of these structures).
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
• Educational purposes : its intermediate level of complexity. It is easy to imagine but complex enough for doing interesting mathematics.
• As a part of space geometry, it can describe the contents of any fixed flat physical surface (images on screens or papers, pieces of land).
• To represent complex numbers, which play key roles both for pure mathematics, and for physics (especially in the formalisms of waves and quantum physics).

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:

• 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.

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) 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.

## 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.

• While formalizations with the only type of points are possible, the type ℝ can anyway be restored from them by means of a construction.
• Conversely, geometrical spaces can also be produced from ℝ by constructing a type of "points" (as tuples of real numbers for affine spaces, or other ways for projective spaces...) and then forgetting some structures.
• The "hard" part of the axioms system of geometry beyond a finite list of first-order axioms, that is its topological aspect, can be expressed by a second-order axiom on the order relation in ℝ.
• In other words, for each of most classical geometries (Euclidean, affine... of a given kind with a fixed dimension), a finite list of structures and first-order axioms relating points to real numbers, suffice to completely specify the spaces of that kind relatively to each interpretation of ℝ, as falling in the same isomorphism class:
Take a pair of 2 spaces of a kind, described by the doubled version of that first-order theory with 2 types of points connected the same way to the same type ℝ. In this new theory, an isomorphism between both types of points keeping each real number fixed, can be defined with parameters (typically by choosing coordinate systems and identifying points with the same tuple of values of coordinates) and its property of being an isomorphism is provable there.
• Presentations directly using the type ℝ are rather natural and useful ways to formalize geometries.
• 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.

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:
• Like with arithmetic, a second-order theory can determine ℝ up to isomorphism from a set theoretical viewpoint ; or we can build ℝ from P(ℕ) in the theory of second-order arithmetic.
• Taking the second-order axiom needed for the specification of ℝ (namely, the axiom saying its order is complete) and using its weak translation as a schema of first-order axioms, we get the first-order theory of real closed field that is complete, in the sense that any closed first-order formula of this theory is decidable (provable or refutable). In other words, its models are all elementarily equivalent, though not isomorphic to each other.
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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