Everything about Erlangen Programme totally explained
An influential research program and manifesto was published in
1872 by
Felix Klein, under the title
Vergleichende Betrachtungen über neuere geometrische Forschungen. This
Erlangen Program (
Erlanger Programm) — Klein was then at
Erlangen — proposed a new kind of solution to the problems of
geometry of the time.
At the time, geometry contained a very large number of theorems. Under the influence of
synthetic geometry, the emphasis was still on proving theorems from sets of
axioms, on the model of
Euclidean geometry that had held good for two millennia. What Klein suggested was innovative in two ways:
» Firstly, he proposed that
group theory, a branch of mathematics that uses algebraic methods to abstract the idea of
symmetry, was the most useful way of organizing geometrical knowledge; at the time it had already been introduced into the
theory of equations in the form of
Galois theory.
» Secondly, he made much more explicit the idea that each geometrical language had its own, appropriate concepts, so that for example
projective geometry rightly talked about
conic sections, but not about
circles or
angles because those notions were not invariant under
projective transformations (something familiar in
geometrical perspective). The way the multiple languages of geometry then came back together could be explained by the way
subgroups of a symmetry group related to each other.
The problems of nineteenth century geometry
Was there one 'geometry' or many? Since
Euclid, geometry had meant the geometry of
Euclidean space of two dimensions (
plane geometry) or of three dimensions (
solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of
four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the
Parallel Axiom from the others, and
non-Euclidean geometry had been born; and in
projective geometry new 'points' (
points at infinity) had been introduced.
The solution in abstract terms was to use
symmetry as an underlying principle, and to state first that different geometries could co-exist, because they dealt with different types of propositions and invariances related to different types of symmetry and transformation. The distinction between
affine geometry and
projective geometry lies just in the fact that affine-invariant notions such as parallelism are the proper subject matter of the first, while not being principal notions in the second. Then, by abstracting the underlying
groups of symmetries from the geometries, the relationships between them can be re-established at the group level. Since the group of affine geometry is a
subgroup of the group of projective geometry, any notion invariant in projective geometry is
a priori meaningful in affine geometry; but not the other way round. If you add required symmetries, you've a more powerful theory but fewer concepts and theorems (which will be deeper and more general).
Homogeneous spaces
In other words, the "traditional spaces" are
homogeneous spaces; but not for a uniquely determined group. Changing the group changes the appropriate geometric language.
In today's language, the groups concerned in classical geometry are all very well-known as
Lie groups: the
classical groups. The specific relationships are quite simply described, using technical language.
Examples
For example the group of
projective geometry in
n dimensions is the symmetry group of
n-dimensional
projective space (the matrix group of size
n+1, quotiented by scalar matrices). The
affine group will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen
hyperplane at infinity. This subgroup has a known structure (
semidirect product of the matrix group of size
n with the subgroup of
translations). This description then tells us which properties are 'affine'. In Euclidean plane geometry terms, being a parallelogram is affine since affine transformations always take one parallelogram to another one. Being a circle isn't affine since an affine shear will take a circle into an ellipse.
To explain accurately the relationship between affine and Euclidean geometry, we now need to pin down the group of Euclidean geometry within the affine group. The
Euclidean group is in fact (using the previous description of the affine group) the semi-direct product of the orthogonal (rotation and reflection) group with the translations.
| Geometric Group |
Number of copies of othogonal group (rotations and reflections) |
Number of copies of additive group (translations) |
Number of copies of multiplicative group (dilations) |
| 2-D Euclidean |
1 |
2 |
0 |
| 2-D Hyperbolic |
1 |
1 |
1 |
| Elliptic |
|
|
|
| Affine |
|
|
|
| Projective |
|
|
|
Influence on later work
The long-term effects of the Erlangen program can be seen all over pure mathematics (see tacit use at
congruence (geometry), for example); and the idea of transformations and of synthesis using groups of symmetry is of course now standard too in
physics.
When
topology is routinely described in terms of properties
invariant under
homeomorphism, one can see the underlying idea in operation. The groups involved will be infinite-dimensional in almost all cases - and not
Lie groups - but the philosophy is the same. Of course this mostly speaks to the pedagogical influence of Klein. Books such as those by
H.S.M. Coxeter routinely used the Erlangen program approach to help 'place' geometries. In pedagogic terms, the program became
transformation geometry, a mixed blessing in the sense that it builds on stronger intuitions than the style of
Euclid, but is less easily converted into a
logical system.
In his book
Structuralism (1970)
Jean Piaget says, "In the eyes of contemporary structuralist mathematicians, like
Bourbaki, the Erlangen Program amounts to only a partial victory for structuralism, since they want to subordinate all mathematics, not just geometry, to the idea of
structure."
For a geometry and its group, an element of the group is sometimes called a
motion of the geometry. For example, one can learn about the
Poincaré half-plane model of
hyperbolic geometry through a development based on
hyperbolic motions. Such a development enables one to methodically prove the
ultraparallel theorem by successive motions.
The Erlangen Program is carried into mathematical logic by
Alfred Tarski in his analysis of
propositional truth.
Abstract returns from the Erlangen program
Quite often, it appears there are two or more distinct
geometries with
isomorphic automorphism groups. There arises the question of reading the Erlangen program from the
abstract group, to the geometry.
One example:
oriented (for example,
reflections not included)
elliptic geometry (for example, the surface of an
n-sphere with opposite points identified) and
oriented spherical geometry (the same
non-Euclidean geometry, but with opposite points not identified) have
isomorphic automorphism group,
SO(n+1) for even
n. These may appear to be distinct. It turns out, however, that the geometries are very closely related, in a way that can be made precise.
To take another example,
elliptic geometries with different
radii of curvature have
isomorphic automorphism groups. That doesn't really count as a critique as all such geometries are
isomorphic. General
Riemannian geometry falls outside the boundaries of the program.
Some further notable examples have come up in physics.
Firstly,
n-dimensional
hyperbolic geometry,
n-dimensional
de Sitter space and (
n−1)-dimensional
inversive geometry all have isomorphic
automorphism groups,
» ,
the
orthochronous Lorentz group, for
n ≥ 3. But these are apparently distinct geometries. Here some interesting results enter, from the physics. It has been shown that physics models in each of the three geometries are "dual" for some models.
Again,
n-dimensional
anti de Sitter space and (
n−1)-dimensional
conformal space with "Lorentzian" signature (in contrast with
conformal space with "Euclidean" signature, which is identical to
inversive geometry, for 3 dimensions or greater) have
isomorphic automorphism groups, but are distinct geometries. Once again, there are models in physics with "dualities" between both
spaces. See
AdS/CFT for more details.
More intriguingly, the covering group of SU(2,2) is isomorphic to the covering group of SO(4,2), which is the symmetry group of a 4D conformal Minkowski space and a 5D anti de Sitter space AND a complex four dimensional
twistor space.
The Erlangen program can therefore still be considered fertile, in relation with dualities in physics.
Further Information
Get more info on 'Erlangen Programme'.
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