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Similarity Submodules and Root Systems in Four Dimensions

Published online by Cambridge University Press:  20 November 2018

Michael Baake
Affiliation:
Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany
Robert V. Moody
Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1
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Abstract

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Lattices and $\mathbb{Z}$-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain $4D$ examples that are related to $4D$ root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Apostol, T. M., Introduction to Analytic Number Theory. Springer, New York, 1976.Google Scholar
[2] Baake, M., Solution of the coincidence problem in dimensions d ≤ 4. In: The Mathematics of Long-Range Aperiodic Order (ed. Moody, R. V.), NATO ASI Series C 489, Kluwer, Dordrecht, 1997, 944.Google Scholar
[3] Baake, M., Combinatorial aspects of colour symmetries. J. Phys. A30(1997), 31713182.Google Scholar
[4] Baake, M., Joseph, D., Kramer, P. and Schlottmann, M., Root lattices and quasicrystals. J. Phys. A23(1990), L1037L1041.Google Scholar
[5] Baake, M. and Moody, R. V., Similarity submodules and semigroups. In: Quasicrystals and Discrete Geometry (ed. Patera, J.), Fields InstituteMonographs 10, Amer.Math. Soc., Providence, 1998, 113.Google Scholar
[6] Baake, M. and Moody, R. V., Invariant submodules and semigroups of self-similarities for Fibonacci modules. In: Aperiodic ‘97 (eds. Boissieu, M. de, Verger-Gaugry, J.-L. and Currat, R.), World Scientific, Singapore, 1998, 21–27; see math-ph/9809008 for a corrected version.Google Scholar
[7] Brown, H., Bülow, H., Neubüser, R., Wondratschek, J. and Zassenhaus, H., Crystallographic Groups of Four-Dimensional Space. Wiley, New York, 1978.Google Scholar
[8] Chen, L., Moody, R. V. and Patera, J., Non-crystallographic root systems. In: Quasicrystals and Discrete Geometry (ed. Patera, J.), Fields InstituteMonographs 10, Amer.Math. Soc., Providence, 1998, 135178.Google Scholar
[9] Conway, J. H., Rains, E. M. and Sloane, N. J. A., On the Existence of Similar Sublattices. Preprint, 1998; Canad. J. Math. 51(1999), 13001306.Google Scholar
[10] Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups. 3rd edn, Springer, New York, 1999.Google Scholar
[11] Coxeter, H. S. M., Quaternions and reflections. Amer.Math. Monthly 53(1946), 136–46.Google Scholar
[12] Coxeter, H. S. M., Regular Polytopes. 3rd edn, Dover, New York, 1973.Google Scholar
[13] Coxeter, H. S. M., Introduction to Geometry. 2nd edn,Wiley, New York, 1980.Google Scholar
[14] Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups. 4th edn, Springer, Berlin, 1980.Google Scholar
[15] Deuring, M., Algebren. 2nd edn, Springer, Berlin, 1968.Google Scholar
[16] du Val, P., Homographies, Quaternions and Rotations. Clarendon Press, Oxford, 1964.Google Scholar
[17] Elser, V. and Sloane, N. J. A., A highly symmetric four-dimensional quasicrystal. J. Phys. A20(1987), 61616168.Google Scholar
[18] Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers. 5th edn, Clarendon Press, Oxford, 1979.Google Scholar
[19] Humphreys, J. E., Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge, 1990; 2nd corr. printing, 1992.Google Scholar
[20] Hurwitz, A., Vorlesungen über die Zahlentheorie der Quaternionen. Springer, Berlin, 1919.Google Scholar
[21] Janusz, G. J., Algebraic Number Fields. 2nd edn, Amer. Math. Soc., Providence, 1996.Google Scholar
[22] Koecher, M. and Remmert, R., Hamiltonsche Quaternionen. In: Zahlen (eds. Ebbinghaus, H.-D. et al.), 3rd edn, Springer, Berlin, 1992, 155–181; English translation in: Numbers, 3rd corr. printing, Springer, New York, 1995.Google Scholar
[23] Kramer, P. and Papadopolos, Z., Symmetry concepts for quasicrystals and noncommutative crystallography. In: The Mathematics of Long-Range Aperiodic Order (ed. Moody, R. V.), NATO ASI Series C 489, Kluwer, Dordrecht, 1997, 307330.Google Scholar
[24] Lang, S., Algebraic Number Theory. 2nd edn, Springer, New York, 1994.Google Scholar
[25] Lifshitz, R., Theory of color symmetry for periodic and quasiperiodic crystals. Rev. Mod. Phys. 69(1997), 11811218.Google Scholar
[26] Moody, R. V. and Weiss, A., On shelling Euasicrystals. J. Number Theory 47(1994), 405412.Google Scholar
[27] Reiner, I., Maximal Orders. Academic Press, London, 1975.Google Scholar
[28] Scheja, G. and Storch, U., Lehrbuch der Algebra, Teil 2. Teubner, Stuttgart, 1988.Google Scholar
[29] Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences. Accessible via: http://www.research.att.com/njas/sequences/.Google Scholar
[30] Sloane, N. J. A. and Plouffe, S., The Encyclopedia of Integer Sequences. Academic Press, San Diego, 1995.Google Scholar
[31] Vignéras, M.-F., Arithmétique des Algèbres de Quaternions. Lecture Notes in Math. 800, Springer, Berlin, 1980.Google Scholar
[32] Schwarzenberger, R. L. E., N-dimensional Crystallography. Pitman, San Francisco, 1980.Google Scholar
[33] Schwarzenberger, R. L. E., Colour symmetry. Bull. LondonMath. Soc. 16(1984), 209240.Google Scholar
[34] Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory. Cambridge University Press, Cambridge, 1995.Google Scholar
[35] Washington, L. C., Introduction to Cyclotomic Fields. 2nd edn, Springer, New York, 1997.Google Scholar
[36] Zagier, D. B., Zetafunktionen und quadratische Körper. Springer, Berlin, 1981.Google Scholar