On 2-Generator Subgroups of SO(3)
HTML articles powered by AMS MathViewer
- by Charles Radin and Lorenzo Sadun PDF
- Trans. Amer. Math. Soc. 351 (1999), 4469-4480 Request permission
Abstract:
We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $\pi$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $\pi$ and $\pi /2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.References
- John H. Conway and Charles Radin, Quaquaversal tilings and rotations, Invent. Math. 132 (1998), no. 1, 179–188. MR 1618635, DOI 10.1007/s002220050221
- Charles Radin and Lorenzo Sadun, Subgroups of $\textrm {SO}(3)$ associated with tilings, J. Algebra 202 (1998), no. 2, 611–633. MR 1617675, DOI 10.1006/jabr.1997.7320
- C. Radin and L. Sadun: An algebraic invariant for substitution tiling systems, Geometriae Dedicata 73 (1998), 21–37. [Obtainable from the electronic archive: mp_arc@math.utexas.edu]
- Stan Wagon, The Banach-Tarski paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge, 1985. With a foreword by Jan Mycielski. MR 803509, DOI 10.1017/CBO9780511609596
Additional Information
- Charles Radin
- Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
- MR Author ID: 194238
- Email: radin@math.utexas.edu
- Lorenzo Sadun
- Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
- Email: sadun@math.utexas.edu
- Received by editor(s): October 13, 1997
- Published electronically: June 10, 1999
- Additional Notes: Research of the first author was supported in part by NSF Grant No. DMS-9531584.
Research of the second author was supported in part by NSF Grant No. DMS-9626698. - © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 4469-4480
- MSC (1991): Primary 51F25, 52C22
- DOI: https://doi.org/10.1090/S0002-9947-99-02397-1
- MathSciNet review: 1624202