Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 58479, 7 pages
http://dx.doi.org/10.1155/IJMMS/2006/58479

Mappings which preserve regular dodecahedrons

Mathematics Section, College of Science and Technology, Hong-Ik University, Chochiwon 339-701, South Korea

Received 4 June 2006; Accepted 5 July 2006

Copyright © 2006 Byungbae Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. D. Aleksandrov, “Mappings of families of sets,” Soviet Mathematics - Doklady, vol. 11, pp. 116–120, 1970. View at Google Scholar · View at Zentralblatt MATH
  2. F. S. Beckman and D. A. Quarles, Jr., “On isometries of Euclidean spaces,” Proceedings of the American Mathematical Society, vol. 4, no. 5, pp. 810–815, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. W. Benz, “Isometrien in normierten Räumen,” Aequationes Mathematicae, vol. 29, no. 2-3, pp. 204–209, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. W. Benz, “An elementary proof of the theorem of Beckman and Quarles,” Elemente der Mathematik, vol. 42, no. 1, pp. 4–9, 1987. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. R. L. Bishop, “Characterizing motions by unit distance invariance,” Mathematics Magazine, vol. 46, pp. 148–151, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. D. Greenwell and P. D. Johnson, “Functions that preserve unit distance,” Mathematics Magazine, vol. 49, no. 2, pp. 74–79, 1976. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. Mielnik and Th. M. Rassias, “On the Aleksandrov problem of conservative distances,” Proceedings of the American Mathematical Society, vol. 116, no. 4, pp. 1115–1118, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Th. M. Rassias, “Is a distance one preserving mapping between metric spaces always an isometry?,” The American Mathematical Monthly, vol. 90, no. 3, p. 200, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Th. M. Rassias, “Mappings that preserve unit distance,” Indian Journal of Mathematics, vol. 32, no. 3, pp. 275–278, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Th. M. Rassias and P. Šemrl, “On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings,” Proceedings of the American Mathematical Society, vol. 118, no. 3, pp. 919–925, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Th. M. Rassias and C. S. Sharma, “Properties of isometries,” Journal of Natural Geometry, vol. 3, no. 1, pp. 1–38, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. E. M. Schröder, “Eine Ergänzung zum Satz von Beckman and Quarles,” Aequationes Mathematicae, vol. 19, no. 1, pp. 89–92, 1979. View at Google Scholar · View at MathSciNet
  13. C. G. Townsend, “Congruence-preserving mappings,” Mathematics Magazine, vol. 43, pp. 37–38, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S.-M. Jung and B. Kim, “Unit-circle-preserving mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 66, pp. 3577–3586, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S.-M. Jung and B. Kim, “Unit-sphere preserving mappings,” Glasnik Matematički. Serija III, vol. 39(59), no. 2, pp. 327–330, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. B. Kim, “Mappings which preserve regular tetrahedrons,” in press.
  17. A. Oosterom and J. Strackee, “The solid angle of a plane triangle,” IEEE Transactions on Biomedical Engineering, vol. 30, no. 2, pp. 125–126, 1983. View at Google Scholar