Martin Bartels

John Snygg, retired, USA

Three mathematicians: Carl Friedrich Gauss, Nikoilai Lobachevsky, and János Bolyai are credited with the discovery of non-Euclidean geometry. Martin Bartels clearly had an impact on Gauss (as his tutor) and Lobachevsky (as his teacher). In this paper, I am throwing out the possibility that Martin Bartels had an impact on János Bolyai - - - not directly but through his father Farkas Bolyai. For Farkas, trying to prove the parallel axiom of Euclid from Euclid’s other axioms became an obsession. When János was 16 years old he wrote his father a letter from Vienna where he was enrolled in the Academy of Military Engineers. In that letter, János told his father that he was attempting to prove the parallel axiom. Realizing, that he had wasted much of his life on this endeavor, Farkas wrote back to his son,

Do not lose one hour on that. It brings no reward, and it will poison your whole life. Even through the pondering of a hundred great geometers lasting for centuries it has been utterly impossible to prove the eleventh without a new axiom. I believe that I have exhausted all imaginable ideas.

Dunnington 2004, p. 178

I few years ago, I realized that Martin Bartels completed his studies at Göttingen in the spring of 1794 - - - roughly two years before the arrival of Farkas Bolyai. It was my understanding that Martin Bartels had to wait several years before he got his first teaching job in Switzerland. It was my idea that Bartels may have remained in Göttingen in hopes that notice of a job opening would be sent to one of his Göttingen professors. I could imagine Bartels, Gauss, and Bolyai together in a Göttingen beer hall discussing the possibility that one of them could gain world recognition by proving the parallel axiom.

Proper rotations for cube and tetrahedron

John Snygg, retired, USA

A big advantage of Clifford Algebra is that it can deal with rotations in a more satisfying way than can be done with a vector only formulation. However even with Clifford Algebra, there are limitations. If one computes the composition of two arbitrary rotations in 3 dimensions it may be quite difficult to determine the axis of rotation and the angle of rotation for the resulting rotation. By limiting rotations to those associated with the proper symmetry groups of the cube and the tetrahedron, one can give a low level student a multitude of problems with recognizable solutions. In the process, one can discuss some interesting aspects of the two proper symmetry groups.