Abstract
In 1988 Danzer [3] constructed a family of four tetrahedra which allows—with certain matching conditions—only aperiodic tilings. By analogy with the Ammann bars of planar Penrose tilings we define Ammann bars in space in the form of planar Penrose tilings we define Ammann bars in space in the form of plane sections of the four tetrahedra. If we require that the plane sections continue as planes across the faces of the tilings, we obtain an alternative matching condition, thus answering a question of Danzer.
Article PDF
Similar content being viewed by others
References
L. Bendersky, Quasicrystals with one-dimensional translational symmetry and a tenfold rotational axis,Phys. Rev. Lett. 55 (1985), 1461–1463.
N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings,Nederl. Akad. Wetensch. Proc. Ser. A 84 (1981), 39–66.
L. Danzer, Three-dimensional analogs of the planar Penrose tilings and quasicrystals,Discrete Math. 78 (1989), 1–7.
Y. Fukamo, T. Ishimasa, and H.-U. Nissen, New ordered state between crystalline and amorphous in Ni-Cr particles,Phys. Rev. Lett. 55 (1985), 511–513.
B. Grünbaum and G. C. Shephard,Tilings and Patterns, Freeman, New York, 1987.
P. Kramer and R. Neri, On periodic and non-periodic space fillings on\(\mathbb{E}^m \) obtained by projections,Acta Cryst. A 40 (1984), 580–587.
R. D. Nelson, Quasicrystals,Sci. Amer. 255 (1986), 32–41.
R. Penrose, Pentaplexity,Eureka 39 (1978), 16–22.
M. Senechal and J. Taylor, Quasicrystals: the view from Les Houches,Math. Intelligencer 12 (1990), 54–64.
D. Shechtman, I. Bleck, D. Gratias, and J. W. Cahn, Metallic phases with long-range orientational order and no translational symmetry,Phys. Rev. Lett. 53 (1988), 105–112.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stehling, T. Ammann bars and quasicrystals. Discrete Comput Geom 7, 125–133 (1992). https://doi.org/10.1007/BF02187830
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02187830