Skip to main content

Wild tiles in ℝ3 with spherical boundaries

Abstract

Wildly embedded tiles in ℝ3 with spherical boundary are discussed. The construction of the topologically complicated, crumpled cube tiles is reviewed. We construct an infinite family of wildly embedded, cellular tiles with Fox-Artin-type wild points. Finally, a condition on the set of wild points on a cellular tile is given to show that certain wild cells cannot be tiles. Several observations are recorded for further investigations.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    C. C. Adams, “Tilings of space by knotted tiles,” Math. Intelligencer, 17, No. 2, 41–51 (1995).

    MATH  Article  MathSciNet  Google Scholar 

  2. 2.

    C. C. Adams, “Knotted tillings,” in: R. V. Moody, ed., The Mathematics of Long-Range Aperiodic Order, Kluwer Academic (1997), pp. 1–8.

  3. 3.

    W. R. Alford and B. J. Ball, “Some almost polyhedral wild arcs,” Duke Math. J., 30, 33–38 (1963).

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    W. A. Blankinship and R. H. Fox, “Remarks on certain pathological open subsets of 3-space and their fundamental groups,” Proc. Amer. Math. Soc., 1, 618–624 (1950).

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    C. E. Burgess and J. W. Cannon, “Embeddings of surfaces in \(\mathbb{E}^3 \)Rocky Mountain J. Math., 1, 259–344 (1971).

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    H. E. Debrunner, “Tiling three-space with handlebodies,” Stud. Sci. Math. Hungar., 21, 201–202, 52–54 (1986).

    MATH  MathSciNet  Google Scholar 

  7. 7.

    R. H. Fox and E. Artin, “Some wild cells and spheres in three-dimensional spaces,” Ann. Math., 49, 979–990 (1948).

    Article  MathSciNet  Google Scholar 

  8. 8.

    B. Grunbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Company, New York (1987).

    Google Scholar 

  9. 9.

    W. Kuperberg, “Knotted lattice-like space fillers,” Discrete Comput. Geom., 13, 561–567 (1995).

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    W. Kuperberg, “Tiling the solid torus, the 3-cube and the 3-sphere with congruent knotted tori, ” in: Intuitive Geometry (Budapest, 1995), Bolyai Soc. Math. Stud., Vol. 6, Janos Bolyai Math. Soc., Budapest (1997), pp. 399–406.

  11. 11.

    S. Oh, “Knotted solid tori decomposition of B 3 and S 3,” J. Knot Theory Ramifications, 5, No. 3, 405–416 (1996).

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    P. Schmitt, “Another space-filling trefoil knot,” Discrete Comput. Geom., 13, 603–607 (1995).

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    P. Schmitt, “Space filling knots,” Beitr. Algebra Geom., 38, No. 2, 307–313 (1997).

    MATH  Google Scholar 

  14. 14.

    E. Schulte, “Space fillers of higher genus,” J. Combin. Theory Ser. A, 68, 438–453 (1994).

    MATH  Article  MathSciNet  Google Scholar 

  15. 15.

    T.-M. Tang, “Crumpled cube and solid horned sphere space fillers,” Discrete Comput. Geom., 31, No. 3, 421–433 (2004).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 203–211, 2005.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tang, TM. Wild tiles in ℝ3 with spherical boundaries. J Math Sci 144, 4504 (2007). https://doi.org/10.1007/s10958-007-0289-9

Download citation

Keywords

  • Tame
  • Solid Sphere
  • Spherical Boundary
  • Adjacency Relation
  • Triangular Block