An influential research programme 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. 1872 (MDCCCLXXII) was a leap year starting on Monday (see link for calendar) of the Gregorian calendar or a leap year starting on Wednesday of the 12-day-slower Julian calendar. ... Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ... Erlangen around 1915 Erlangen is a German city in Middle Franconia. ... Table of Geometry, from the 1728 Cyclopaedia. ...

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, an algebraic approach that encapsulates the idea of symmetry, was the correct way of organising geometrical knowledge; 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. Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... Group theory is that branch of mathematics concerned with the study of groups. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, the theory of equations comprises a major part of traditional algebra. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ... Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... This article is about angles in geometry. ... A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ... Perspective projection is a type of drawing, or rendering, that graphically approximates on a planar (two-dimensional) surface (e. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

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, points with complex number co-ordinates) had been introduced. Euclid Euclid of Alexandria (Greek: ) (ca. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical geometry. ... In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ... Higher dimension in mathematics refers to any number of dimensions greater than three. ... If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-euclidian geometry) describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ... In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

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 have a more powerful theory but fewer concepts and theorems (which will be deeper and more general). Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

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 mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

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. In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a group of Lie type G(k) is a (not necessarily finite) group of rational points of a linear algebraic group G with values in the field k. ...

Examples: affine geometry

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 is not affine since an affine shear will take a circle into an ellipse. Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ... In mathematics, a projective space is a fundamental construction from any vector space. ... In mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. ... In mathematics, in particular projective geometry, the hyperplane at infinity, also called ideal hyperplane, is a projective (n − 1) -space added to Euclidean n-space — — in order to give it closure of incidence properties, thereby converting into the projective n-space . ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

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. In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...

Influence on later work

The long-term effects of the Erlangen programme 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. An example of congruence. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ...

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 programme approach to help 'place' geometries. In pedagogic terms, the programme 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. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, an invariant is something that does not change under a set of transformations. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... H(arold). ... In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. ... Euclid Euclid of Alexandria (Greek: ) (ca. ...

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." Jean Piaget (August 9, 1896 – September 16, 1980) was a Swiss natural scientist and developmental psychologist, well known for his work studying children and his theory of cognitive development. ... Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ...

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. In non-Euclidean geometry, the Poincaré model is a model of two-dimensional hyperbolic geometry as a homogeneous space for the group of Möbius transformations. ... A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... In physics, hyperbolic motion is the motion of an object with constant acceleration in special relativity. ... In hyperbolic geometry, the Ultraparallel theorem states that every pair of ultraparallel lines in the hyperbolic plane has a unique common perpendicular hyperbolic line. ...

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. Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

One example: oriented (i.e., reflections not included) elliptic geometry (i.e., 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. This article or section should be merged with Orientable manifold. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... For other uses, see sphere (disambiguation). ... This article or section should be merged with Orientable manifold. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

To take another example, elliptic geometries with different radii of curvature have isomorphic automorphism groups. That does not really count as a critique as all such geometries are isomorphic. General Riemannian geometry falls outside the boundaries of the program. Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...

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, A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... In mathematics and physics, n-dimensional de Sitter space, denoted , is the maximally symmetric, simply-connected, Lorentzian manifold with constant positive curvature. ... In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

,

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. The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ...

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. In mathematics and physics, n-dimensional anti de Sitter space, denoted , is the maximally symmetric, simply-connected, Lorentzian manifold with constant negative curvature. ... In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space... In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space... In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ... Table of Geometry, from the 1728 Cyclopaedia. ... In physics, the AdS/CFT correspondence is the equivalence between a string theory or supergravity defined on some sort of Anti de Sitter space and a conformal field theory defined on its boundary whose dimension is lower by one. ...

The Erlangen program can therefore still be considered fertile, in relation with dualities in physics.

See also

• homogeneous space
• Klein geometry

In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. ...

References

Guggenheimer, Heinrich, 1977. "Differential Geometry", Dover, NY, ISBN 0486634337. An inexpensive book that's still in print, not too difficult, with many references to Lie, Klein and Cartan. P. 139, "A Klein geometry is the theory of geometric invariants of a transitive tranformation group (Erlangen program, 1872)".

Felix Klein, 1872. "Vergleichende Betrachtungen über neuere geometrische Forschungen" ('A comparative review of recent researches in geometry'), Mathematische Annalen, 43 (1893) pp. 63-100 (Also: Gesammelte Abh. Vol. 1, Springer, 1921, pp. 460-497). An English translation by Mellen Haskell appeared in Bull. N. Y. Math. Soc 2 (1892-1893): 215--249. German text of the Erlangen Program can be viewed at the University of Michigan online collection. [1] Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ...

Felix Klein, 2004 "Elementary Mathematics from an Advanced Standpoint: Geometry", Dover, NY, ISBN 0486434818 (translation of Elementarmathematik vom höheren Standpunkte aus, Teil II: Geometrie, pub. 1924 by Springer). Not a hard book. Has a section on the Erlangen Program. Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ...