Geometrical aspects of the algebraic number related to quasicrystals

Main Article Content

G Nagendra Kumar
E Keshava Reddy


A quasicrystal is a mathematical term for an infinite point or tiling space. It possesses several intersecting features, including Delone, relative discreteness, and self-similarities. The model set is the basic form in which the physical quasicrystals are represented. Pisot numbers are used to identify the one-dimensional model sets' adjacent points. The quadratic irrational numbers are related to all one-dimensional model sets that have been experimentally discovered the paper offers mathematical models of quasicrystals with particular attention given to cut and projection sets for the eight folded symmetry and discuss about one dimensional cut and projection set.

Article Details

How to Cite
G Nagendra Kumar and E Keshava Reddy, “Geometrical aspects of the algebraic number related to quasicrystals”, Int. J. Comput. Eng. Res. Trends, vol. 10, no. 11, pp. 66–69, Nov. 2023.
Research Articles


P. Bak, Symmetry, stability, and elastic properties of icosahedral incommensurate crystals,Phys. Rev. B 32, 5764 (1985).

S. Berman, R. V. Moody, The algebraic theory of quasicrystals with five-fold symmetry, J. Phys. A: Math. Gen. 27 (1994) 115–130

E. Bombieri, J. E. Taylor, Quasicrystals, tilings, and algebraic number theory: some preliminary connections, Contemp. Math., 64, Amer. Math. Soc., Providence RI, (1987) 241–264

N. G. de Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane, Nederl. Akad.Wetensch. Indag. Math. 43, (1981) 39-66

F. G¨ahler, R. Klitzing, The diffraction pattern of self-similar tilings, in Mathematics of Long Range Aperiodic Order, Proc. NATO ASI, Waterloo, 1996, ed. R. V. Moody, Kluwer (1996)


L.-S. Guimond, Z. Mas´akov´a, J. Patera, E. Pelantov´a, The Periodicity of Linear Congruential Generators Broken deterministically using Quasi-Crystals, Preprint CRM-2624, (1999)

C. Janot, Quasicrystals: A primer, Oxford Univ. Press, Oxford, UK (1994)

P. Kramer and R. Neri, On Periodic and Non-periodic Space Fillings of E m Obtained by Projection, Acta Cryst. A 40, 580 (1984)

J.C. Lagarias, Meyer’s concept of Quasicrystal and quasiregular sets, Comm. Math. Phys. 179, (1996) 365–376.

Z. Mas´akov´a, J. Patera, E. Pelantov´a, Inflation centers of the cut and project quasicrystals, J. Phys. A: Math. Gen. 31 (1998) 1443–1453

Z. Mas´akov´a, J. Patera, E. Pelantov´a, Minimal distances in quasicrystals, J. Phys. A: Math. Gen. 31 (1998) 1539–1552

Z. Mas´akov´a, J. Patera, E. Pelantov´a, Substitution rules for cut and project sequences, preprint (1999)

Z. Mas´akov´a, J. Patera, E. Pelantov´a, s-convexity, model sets and their relation, Preprint MSRI 1999-040; Z. Mas´akov´a, E. Pelantov´a, M. Svobodov´a, Characterization of model sets using a binary operation, preprint (1999)

Y. Meyer, Nombres de Pisot, nombres de Salem et analyse harmonique, Lecture Notes in Mathematics 117, Springer (1970); Algebraic numbers and harmonic analysis, North-Holland (1972)

Y. Meyer, Quasicrystals, Diophantine approximations and algebraic numbers, Proc. Les Houches, March 1994, Beyond Quasicrystals, Les Editions de Physique, eds. F. Axel and D. Gratias, Springer, 1995, pp.3–16

R. V. Moody, Meyer sets and their duals, in Mathematics of Long Range Aperiodic Order, Proc. NATO ASI, Waterloo, 1996, ed. R. V. Moody, Kluwer (1996) 403–441

R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane, Math. Intelligencer, 2, (1979/80) 32–37

M. Senechal, Quasicrystals and Geometry, Cambridge Univ. Press, Cambridge, UK (1995)

D. Shechtman, I. Gratias, J.W. Cahn, Metallic phase with long range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951–1953

N. B. Slater, Gaps and steps for the sequence nθ mod 1, Math. Proc. Camb. Phil. Soc. 63 (1967) 1115–1123

H. Weyl, Uber die Gleichungverteilung von Zahlen mod, ¨ Eins. Math. Ann. 77 (1926) 313–352

Proceedings of the 6th International Conference on Quasicrystals, ed. T. Fujiwara, World Scientific, Singapore (1997); Proceedings of the 5th International Conference on Quasicrystals, eds. C. Janot and R. Mosseri, World Scientific, Singapore (1996)

The Physics of Quasicrystals, eds. P. J. Steinhardt and S. Ostlund, World Scientific, Singapore (1987)

J.C. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom. 21 (1999), 161–191.

J.C. Lagarias, Mathematical quasicrystals and the problem of diffraction, in Directions in mathematical quasicrystals, CRM Monogr. Ser. 13, Amer. Math. Soc., Providence, RI, (2000), 61–93