Abstract
A lattice in a Euclidean space gives rise to facet-to-facet and space-filling convex polyhedral tilings called the Voronoi tiling and its dual Delaunay tiling of the Euclidean space. Given a subspace of the Euclidean space, we develop a systematic way of constructing facet-to-facet and space-filling convex polyhedral tilings of the subspace called the Namikawa tilings, which are generalization of the Delaunay tiling. Our results here amplify the author’s previous work with Seshadri that was obtained as a by-product of Geometric Invariant Theory in algebraic geometry. In the orthonormal setting where the lattice is spanned by an orthonormal basis of the ambient Euclidean space, the method of Namikawa tilings turns out to be the “cut and project” method relevant not only to periodic convex polyhedral tilings hidden in crystals but also to aperiodic convex polyhedral tilings hidden in quasicrystals. In another paper, the author showed that a Voronoi tiling is hidden in the “standard realization”, in the sense of Kotani and Sunada, of crystals in the case of maximal abelian covering of finite graphs. In this connection, we give an affirmative answer to a question raised by Kotani to the effect that the crystal lies in the 1-skeleton of a nondegenerate convex polyhedral tiling that is a subdivision of the Voronoi tiling appearing in the other paper of the author.
Similar content being viewed by others
References
Baake, M.: A guide to mathematical quasicrystals. In: Suck, J.-B., Schreiber, M., Häusster, P. (eds.) Quasicrystals, An Introduction to Structure, Physical Properties and Applications. Springer Series in Materials Science, vol. 55, pp. 17–48. Springer, Berlin (2002)
de Bruijn, N.G.: Algebraic theory of Penrose’s nonperiodic tilings of the plane, I, II. Indagationes Math. 43, 39–66 (1981). http://alexandria.tue.nl/repository/freearticles/597566.pdf
Erdahl R.M.: Zonotope, dicings, and Voronoi’s conjecture on parallelohedra. Eur. J. Combin. 20, 527–549 (1999)
Harris, E., Frettlöh, D.: Tilings Encyclopedia. http://tilings.math.uni-bielefeld.de
Kotani M., Sunada T.: Standard realizations of crystal lattices via harmonic maps. Trans. Am. Math. Soc. 353, 1–20 (2000)
Oda, T.: Voronoi tilings hidden in crystals—the case of maximal abelian coverings. Tohoku Math. J. (2012, to appear)
Oguey C., Duneau M., Katz A.: A geometrical approach of quasiperiodic tilings. Commun. Math. Phys. 118, 99–118 (1988)
Oda T., Seshadri C.S.: Compactifications of the generalized Jacobian variety. Trans. Am. Math. Soc. 253, 1–90 (1979)
Senechal, M.: A critique of the projection method. In: Moody, R.V. (ed.) The Mathematics of Long-Range Aperiodic Order, pp. 521–548. Kluwer, Netherlands (1997)
Senechal M.: What is a quasicrystal?. Notices Am. Math. Soc. 53(8), 886–887 (2006)
Sunada T.: Crystals that nature might miss creating. Notices Am. Math. Soc. 55, 208–215 (2008)
Takakura H., Gómez C.P., Yamamoto A., de Boissieu M., Tsai A.P.: Atomic structure of the binary icosahedral Yb-Cd quasicrystal. Nat. Mater. 6, 58–63 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
In honor of Professor Heisuke Hironaka on his eightieth birthday
Partly supported by JSPS Grant-in-Aid for Scientific Research (S-19104002).
Rights and permissions
About this article
Cite this article
Oda, T. Convex polyhedral tilings hidden in crystals and quasicrystals. RACSAM 107, 123–143 (2013). https://doi.org/10.1007/s13398-012-0078-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-012-0078-2
Keywords
- Convex polyhedral tiling
- Cut and project method
- Crystal
- Quasicrystal
- Strongly connected graph
- Geometric invariant theory