Advertisement

Nonexistence of 2-Reptile Simplices

  • Jiří Matoušek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)

Abstract

A simplex S is called an m-reptile if it can be tiled without overlaps by simplices S 1,S 2,...,S m that are all congruent and similar to S. The only m-reptile d-simplices that seem to be known for d ≥ 3 have m=k d , k ≥ 2. We prove, using eigenvalues, that there are no 2-reptile simplices of dimensions d ≥ 3. This investigation has been motivated by a probabilistic packet marking problem in theoretical computer science, introduced by Adler in 2002.

Keywords

Characteristic Polynomial Single Cycle Receive Packet Theoretical Computer Science Eigenvalue Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, M.: Tradeoffs in probabilistic packet marking for IP traceback. In: Proc. 34th Annu. ACM Symposium on Theory of Computing, pp. 407–418 (2002)Google Scholar
  2. 2.
    Adler, M., Edmonds, J., Matoušek, J.: Towards asymptotic optimality in probabilistic packet marking. In: Proc. 37th Annu. ACM Symposium on Theory of Computing, pp. 450–459 (2005)Google Scholar
  3. 3.
    Alexanderson, G.L., Wetzel, J.E.: Dissections of a simplex. Bull. Amer. Math. Soc. 79, 170–171 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bandt, C.: Self-similar sets. V. Integer matrices and fractal tilings of \({\bf R}\sp n\). Proc. Amer. Math. Soc. 112(2), 549–562 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burch, H., Cheswick, B.: Tracing anonymous packets to their approximate source. Contribution at the conference Usenix LISA, New Orleans (2000)Google Scholar
  6. 6.
    Debrunner, H.E.: Tiling Euclidean d-space with congruent simplexes. In: Discrete geometry and convexity (New York, 1982). Ann. New York Acad. Sci., vol. 440, pp. 230–261. New York Acad. Sci., New York (1985)Google Scholar
  7. 7.
    Doeppner, T.W., Klein, P.N., Koyfman, A.: Using router stamping to identify the source of IP packets. In: Proc. 7th ACM conference on Computer and communications security, pp. 184–189 (2000)Google Scholar
  8. 8.
    Gelbrich, G.: Crystallographic reptiles. Geom. Dedicata 51(3), 235–256 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gelbrich, G.: Self-affine lattice reptiles with two pieces in \({\bf R}\sp n\). Math. Nachr. 178, 129–134 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hertel, E.: Self-similar simplices. Beiträge Algebra Geom. 41(2), 589–595 (2000)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Lehmer, D.H.: A machine method for solving polynomial equations. J. Assoc. Comput. Mach. 8, 151–162 (1961)zbMATHGoogle Scholar
  12. 12.
    Ngai, S.-M., Sirvent, V.F., Veerman, J.J.P., Wang, Y.: On 2-reptiles in the plane. Geom. Dedicata 82(1-3), 325–344 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Savage, S., Wetherall, D., Karlin, A., Anderson, T.: Practical network support for IP traceback. In: Proc. of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication (ACM SIGCOMM), pp. 295–306 (2000)Google Scholar
  14. 14.
    Snover, S.L., Waiveris, C., Williams, J.K.: Rep-tiling for triangles. Discrete Math. 91(2), 193–200 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Zaslavsky, T.: Maximal dissections of a simplex. J. Combinatorial Theory Ser. A 20(2), 244–257 (1976)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jiří Matoušek
    • 1
  1. 1.Department of Applied Mathematics, and Institute of Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic

Personalised recommendations