# G2 and the Rolling Ball

- Published in 2012
- Added on

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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.

## Links

## Other information

- key
- Baez2012
- type
- article
- date_added
- 2012-05-14
- date_published
- 2012-05-01
- arxivId
- 1205.2447
- pages
- 28

### BibTeX entry

@article{Baez2012, key = {Baez2012}, type = {article}, title = {G2 and the Rolling Ball}, author = {Baez, John C and Huerta, John}, 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.}, comment = {}, date_added = {2012-05-14}, date_published = {2012-05-01}, urls = {http://arxiv.org/abs/1205.2447,http://arxiv.org/pdf/1205.2447v4}, collections = {Basically physics}, archivePrefix = {arXiv}, arxivId = {1205.2447}, eprint = {1205.2447}, month = {may}, pages = 28, url = {http://arxiv.org/abs/1205.2447 http://arxiv.org/pdf/1205.2447v4}, year = 2012, primaryClass = {math.DG}, urldate = {2012-05-14} }