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Further collaboration with Lie followed and they worked on an investigation of W -curves, curves invariant under a group of projective transformations.

In fact Lie played an important role in Klein's development, introducing him to the group concept which played a major role in his later work.

It is fair to add that Camille Jordan also played a part in teaching Klein about groups. During his time at Göttingen in Klein made major discoveries regarding geometry.

He published two papers On the So-called Non-Euclidean Geometry in which he showed that it was possible to consider euclidean geometry and non-euclidean geometry as special cases a projective surface with a specific conic section adjoined.

This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent.

The fact that non-euclidean geometry was at the time still a controversial topic now vanished. Its status was put on an identical footing to euclidean geometry.

Cayley never accepted Klein's ideas believing his arguments to be circular. Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm , profoundly influenced mathematical development.

This was written for the occasion of Klein's inaugural address when he was appointed professor at Erlangen in although it was not actually the speech he gave on that occasion.

The Erlanger Programm gave a unified approach to geometry which is now the standard accepted view. Transformations play a major role in modern mathematics and Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties.

In this way the Erlanger Programm defined geometry so that it included both Euclidean geometry and non-Euclidean geometry.

However Klein himself saw his work on function theory as his major contribution to mathematics. As W Burau and B Schoenberg write in [ 1 ] :- Klein considered his work in function theory to be the summit of his work in mathematics.

He owed some of his greatest successes to his development of Riemann 's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations , especially in his own fields, elliptic modular functions and automorphic functions.

By considering the action of the modular group on the complex plane, Klein showed that the fundamental region is moved around to tessellate the plane.

He also used physical ideas in this work, especially those of fluid dynamics. Klein considered equations of degree greater than 4 and was particularly interested in using transcendental methods to solve the general equation of the fifth degree.

After building on methods due to Hermite and Kronecker , producing similar results to Brioschi , he went on to completely solve the problem using the group of the icosahedron.

This work led him to consider elliptic modular functions which he studied in a series of papers.

He developed a theory of automorphic functions, connecting algebraic and geometric results in his important book on the icosahedron.

Working under great stress, Klein succeeded in formulating such a theorem and in sketching a strategy for proving it.

However it was during this work that Klein's health collapsed as mentioned above. With Robert Fricke who came to Leipzig in , Klein wrote a major four volume classic on automorphic and elliptic modular functions produced over the following 20 years.

We should also mention the Klein bottle, a one-sided closed surface named after Klein. A Klein bottle cannot be constructed in Euclidean space.

It is best pictured as a cylinder looped back through itself to join with its other end. However this is not a continuous surface in 3 -space as the surface cannot go through itself without a discontinuity.

It is possible to construct a Klein bottle in non-Euclidean space. In the s Klein became interested in mathematical physics, although throughout his career he showed he was never far from this area in attitude.

Following from this interest, he wrote an important work on the gyroscope with A Sommerfeld. Later in his career Klein became interested in teaching at school level.

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