(For a short simplified version without reference to the
prerequisites, just read the text in green)
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.
A common feature of many
interesting geometrical spaces, is that they have an infinity of
(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
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
: E↔F, we have a bijection from Aut(E) to
the set of isomorphisms between E and F, given by
g ↦ f०g. Thus, a plurality of automorphisms
of E means a plurality of isomorphisms between E
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
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.
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
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)
|(x , y+a)
(x+a , y)
(its coordinates = (0,0))
|Shear mapping w.r.t. the horizontal axis
||(x+a.y , y)
|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
("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, 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
This list of structures still applies in higher dimension
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).
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
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).
More detailed study of affine geometry
in another page.
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)
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
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".
It has a notion of circle (or sphere) but cannot distinguish
straight lines among them.
Follow the link for details.
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
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
Introduction to topology
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
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) :
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
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
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):
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
x ≤ y ⇔ (∃z, y = x + z⋅z)
with axioms that require this to be a total order.
Existence of nonstandard models of ℝ. 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
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.
However, ℝ may still be seen axiomatizable in the following 2 weak senses:
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.
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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