Contributions to Mathematics

Klein disc model in hyperbolic
geometry.

As I said before, I would love to tell you more about my contributions to math and what I worked on when I was alive. I made achievements in almost all areas of mathematics. My achievements in geometry is something that I have slightly touched on in the previous posting. I created the projective foundation of the non-Euclidean geometries and the Erlanger program. Before my works were published in 1871 and 1873, non-Euclidean geometries were not common knowledge among mathematicians. I was the first mathematician to recognize that hyperbolic, elliptic, and Euclidean geometry can be constructed purely projectively. I based my work on the work of Karl Georg Christian von Staudt. To his work, I added a continuity postulate and sans the use of the distance and angles. The geometries and mathematics have already been discovered and recognized by previous mathematicians, however I created projective models for these geometries. When one speaks of the models, they speak of Cayley-Klein metric or the Beltrami-Klein model, named after me.

I should discuss more about the Erlanger Program. This was named after the university I was at at the time. More specifically than the last post, it is based upon the idea that every geometry is based on a certain group and the undertaking of that geometry involves setting up invariants of the group. That might sound like a bunch of gibberish to the common person, so for example, topology is geometry with the most general group. It is the geometry of the invariants of the group of all continuous transformations like the plane. My Erlanger Program was used for many years after its creation, until late in the twentieth century. It was even translated into six languages. My work in geometry is not all that I did however.

I also had work in function theory, which I believe is the peak of my work in mathematics. I discovered a relationship between Riemann’s ideas about surfaces and the concepts of my own fields elliptic modular functions and automorphic functions. I also related Riemann’s ideas other disciplines of mathematics. This work is found in my 1882 publication, Riemanns Theorie der algebraischen Funktionen und ihre Integrade. I worked with spatial intuition and concepts from physics to make the relationships between the different ideas (remember I wanted to at first be a physicist). I created my own argument that intertwined Riemann’s and Weierstrass’ concepts. It proved to work out and no one has been able to say that what I found is wrong

 

Something fun that I also did in 1882, was create a 3-D object that is one-sided, has a closed surface, and does not self-intersect. It is now called a “Klein bottle” named after me and interestingly cannot be put into three-dimensional Euclidean space, but can be put into higher dimensions. It has similar properties to the famous Möbius strip. Above are pictures of the Klein bottle. 

In 1884, I wrote a book on the icosahedron, which is a polyhedron with 20 faces. In this, I proposed the theory of automorphic functions that is a connection between algebra and geometry. This theory is actually what caused my health to decrease because of the competition I was up against and the work that was put into it. My contributions after this time where not the same mathematically, but I did want to do more. 

 I had a dream to rebuild the great mathematics that once lived in the city of Göttingen. I wanted to create a center that focused on German mathematics and physics. Thus, my focus with teaching mathematics and the education behind it began. I will discuss this journey and my relation with the ICMI (International Commission on Mathematical Instruction) in my next posting. 

References:

Felix Klein. (2012). Famous-Mathematicians.com. Retrieved from http://www.famous-mathematicians.com/felix-klein/

Halstead, G. (1894). Biography: Professor Felix Klein. The American Mathematical Monthly. Retrieved from http://www.jstor.org/stable/2969034

"Klein, Christian Felix." Complete Dictionary of Scientific Biography. Retrieved from Encyclopedia.com: http://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/klein-christian-felix

O’Connor, J and Robertson, E. (2003). Felix Christian Klein. MacTutor History of Mathematics Archive. Retrieved from http://www-history.mcs.st-andrews.ac.uk/Biographies/Klein.html

Schubring, G. (2000). Felix Klein. The First Century of the Internet Commission Mathematical Instruction (1908-2008): History of ICMI. Retrieved from

http://www.icmihistory.unito.it/portrait/klein.php#bib