%0 Generic %1 baez2012rolling %A Baez, John C. %A Huerta, John %D 2012 %K ball coxeter g2 mirrors rolling %T G2 and the Rolling Ball %U http://arxiv.org/abs/1205.2447 %X Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again when the ratio of radii is 1:3. We explain this ratio in simple terms using the incidence geometry of G2, and show how a form of geometric quantization applied to this system gives the imaginary split octonions.

@misc{baez2012rolling, abstract = {Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again when the ratio of radii is 1:3. We explain this ratio in simple terms using the incidence geometry of G2, and show how a form of geometric quantization applied to this system gives the imaginary split octonions.}, added-at = {2012-06-04T05:24:43.000+0200}, author = {Baez, John C. and Huerta, John}, biburl = {https://www.bibsonomy.org/bibtex/278af581b75e649e7022ad72fa88ea74f/axblount}, description = {G2 and the Rolling Ball}, interhash = {bc548b25e3f1f13720836dba8f5b3378}, intrahash = {78af581b75e649e7022ad72fa88ea74f}, keywords = {ball coxeter g2 mirrors rolling}, note = {cite arxiv:1205.2447Comment: 28 pages, 2 png figures}, timestamp = {2012-06-04T05:24:43.000+0200}, title = {G2 and the Rolling Ball}, url = {http://arxiv.org/abs/1205.2447}, year = 2012 }