Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-05T01:41:57.951Z Has data issue: false hasContentIssue false

Epilogue. Gateways Towards Quasicrystals

Published online by Cambridge University Press:  26 October 2017

Peter Kramer
Affiliation:
Institut für Theoretische Physik Universität Tübingen Germany
Michael Baake
Affiliation:
Universität Bielefeld, Germany
Uwe Grimm
Affiliation:
The Open University, Milton Keynes
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Aperiodic Order , pp. 363 - 380
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[AO1] Baake, M. and Grimm, U. (2013) Aperiodic Order. Vol. 1: A Mathematical Invitation (Cambridge University Press, Cambridge).
[1] Baake, M., Kramer, P., Schlottmann, M. and Zeidler, D. (1990) Planar patterns with fivefold symmetry as sections of periodic structures in 4-space, Int. J. Mod. Phys. B 4, 2217–2286.CrossRefGoogle Scholar
[2] Bohr, H. (1925) Zur Theorie fastperiodischer Funktionen I & II,Acta Math. 45, 29–127 and 46, 101–214.
[3] Bravais, A. (1850) Mèmoire sur les systèmes formès par des points distribuès règulièrement sur un plan ou dans l'espace, J. ´Ecole Polytech. 19, 1–128.Google Scholar
[4] Brown, H., Bülow R., Neubüser J., Wondratschek, H. and Zassenhaus, H. (1978) Crystallographic Groups of Four-Dimensional Space (Wiley, New York).
[5] de Bruijn, N.G. (1981) Algebraic theory of Penrose's non-periodic tilings of the plane. I & II, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84, 39–52 and 53–66.CrossRefGoogle Scholar
[6] de Wolff, P.M. and van Aalst, W. (1972) The four-dimensional space group of γ-Na2CO3,conference abstract, Acta Cryst. A 28, S111.Google Scholar
[7] Fedorov, E.S. (1891) Symmetry of regular systems of figures, Proc. Imp. St. Petersburg Mineral. Soc. (Ser. 2) 28, 1–146.Google Scholar
[8] Fourier, J.B.J. (1822) Thèorie analytique de la chaleur (F. Didot, Paris).
[9] Gähler, F., Kramer, P., Trebin, H.-R. and Urban, K. eds. (2000) Proceedings of the 7th International Conference on Quasicrystals, published as Mat. Sci. Eng. A 294–296.Google Scholar
[10] Grünbaum, B. and Shephard, G.C. (1987) Tilings and Patterns (Freeman, New York).
[11] Henry, N.F.M. and Lonsdale, K. eds. (1952) International Tables for X-ray crystallography, vol. 1 (Kynoch Press, Birmingham.
[12] Hermann, C. (1949) Kristallographie in Räumen beliebiger Dimensionszahl. I. Die Symmetrieoperationen, Acta Cryst. 2, 139–145.Google Scholar
[13] Gratias, D. and Michel, L. eds. (1986) Proceedings of the International Workshop on Aperiodic Crystals,Les Houches, published as J. Phys. Colloques 47, C3.Google Scholar
[14] Ishimasa, T., Nissen, H.-U. and Fukano, Y. (1985) New ordered state between crystalline and amorphous in Ni-Cr particles, Phys. Rev. Lett. 55, 511–513.CrossRefGoogle Scholar
[15] Janot, C. and Mosseri, R. eds. (1995) Proceedings of the 5th International Conference on Quasicrystals (World Scientific, Singapore).
[16] Janner, A. and Janssen, T. (1977) Symmetry of periodically distorted crystals, Phys. Rev. B 15, 643–658.CrossRefGoogle Scholar
[17] Janner, A. and Janssen, T. (1978) Bravais lattices associated with incommensurate crystal phases. In Group Theoretical Methods in Physics, Kramer, P. and Rieckers, A. eds., LNP 79, pp. 414–416 (Springer, Berlin).
[18] Katz, A. and Duneau, M. (1986) Quasiperiodic patterns and icosahedral symmetry, J. Phys. France 47, 181–196.CrossRefGoogle Scholar
[19] Kepler, J. (1938) Mysterium cosmographicum. In Collected Works, vol. 1, Caspar, M. eds., pp. 5–128 (C.H. Beck, München).
[20] Kepler, J. (1940) Harmonice Mundi. In Collected Works, vol. 6, Caspar, M. eds (C.H. Beck, München).
[21] Kelton, K.F., Gibbons, P.C. and Goldman, A.I. eds. (1993) Proceedings of the 4th International Conference on Quasicrystals, published as J. Non-Cryst. Solids 153–154.Google Scholar
[22] Kowalewski, W. (1938) Der Keplersche Körper und andere Bauspiele (Köhlers Antiquarium, Leipzig).
[23] Kramer, P. (1982) Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells, Acta Cryst. A 38, 257–264.Google Scholar
[24] Kramer, P. (1985) On the theory of a quasilattice associated with the icosahedral group. I, Z. Naturforsch. 40a, 775–788.Google Scholar
[25] Kramer, P. (1986) On the theory of a quasilattice associated with the icosahedral group. II, Z. Naturforsch. 41a, 897–911.Google Scholar
[26] Kramer, P. (2006) Quasiperiodic systems. In Encyclopedia of Mathematical Physics, Francoise, J.-P., Naber, G.L. and Tsun, T.S. eds., pp. 306–315 (Elsevier, Amsterdam).
[27] Kramer, P. and Kramer, L. (1985) Diffraction and layer structure of a quasilattice, Z. Naturforsch. 40a, 1162–1163.Google Scholar
[28] Kramer, P. and Neri, R. (1984) On periodic and non-periodic space fillings of Em obtained by projection,Acta Cryst. A 40, 580–587 and Acta Cryst. A 41 (1985), 619 (erratum).
[29] Kramer, P. and Schlottmann, M. (1989) Dualisation of Voronoi domains and Klotz construction: A general method for the generation of proper space fillings, J. Phys. A: Math. Gen. 22, L1097–L1102.CrossRefGoogle Scholar
[30] Levine, D. and Steinhardt, P.J. (1984) Quasicrystals: A new class of ordered structures, Phys. Rev. Lett. 53, 2477–2480.CrossRefGoogle Scholar
[31] Lothaire, M. (1983) Combinatorics on Words (Addison-Wesley, Reading, MA).
[32] Mackay, A.L. (1981) De nive quinquangula: On the pentagonal snowflake,Kristallografiya 26, 910–919; reprinted as Sov. Phys. Cryst. 26, 517–522.
[33] Mackay, A.L. (1982) Crystallography and the Penrose pattern, Physica A 114, 609–613.Google Scholar
[34] Meyer, Y. (1970) Nombres de Pisot, nombres de Salem et analyse harmonique, LNM 117 (Springer, New York).
[35] Meyer, Y. (1972) Algebraic Numbers and Harmonic Analysis (North Holland, Amsterdam).
[36] Moody, R.V. eds. (1997) The Mathematics of Long-Range Aperiodic Order, NATO ASI Series C 489 (Kluwer, Dordrecht).
[37] Mosseri, R. and Sadoc, J.F. (1982) Two and three dimensional non-periodic networks obtained from self-similar tiling. In The Structure of Non-Crystalline Materials, Gaskell, P.H., Parker, J.M. and Davis, E.A. eds., pp. 137–150 (Taylor and Francis, London).
[38] Penrose, R. (1974) The rˆole of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10, 266–271.Google Scholar
[39] Penrose, R. (1989) Tilings and quasi-crystals: A non-local growth problem? In Introduction to the Mathematics of Quasicrystals, Jaric, M.V. eds., pp. 53–79 (Academic Press, New York.
[40] Platon (1991) Timaios, dialogue. In Collected Works, vol. 8, Hülser, K. eds., pp. 197–425 (Insel, Frankfurt).
[41] Schönflies, A.M. (1886) Über Gruppen von Bewegungen. I, Math. Ann. 28, 319–342.Google Scholar
[42] Schönflies, A.M. (1887) Über Gruppen von Bewegungen. II, Math. Ann. 29, 50–80.Google Scholar
[43] Schwarzenberger, R.L.E. (1980) N-Dimensional Crystallography (Pitman, London).
[44] Shechtman, D., Blech, I., Gratias, D. and Cahn, J.W. (1984) Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951–1953.CrossRefGoogle Scholar
[45] Takeuchi, S. and Fujiwara, T. eds. (1998) Proceedings of the 6th International Conference on Quasicrystal (World Scientific, Singapore).
[46] von Goethe, J.W. (1808) Faust part I (J.G. Cotta, Tübingen).
[47] Yacaman, M.J., Romeu, D., Castano, V. and Gomez, A. eds. (1990) Quasicrystals and Incommensurate Structures in Condensed Matter (World Scientific, Singapore).
[48] Ye, H.Q. and Kuo, K.H. (1984) High-resolution images of planar faults and domain structures in the σ phase of an iron-base superalloy, Philos. Mag. A 50, 117–132.CrossRefGoogle Scholar
[49] Zassenhaus, H. (1948) Uber einen Algorithmus zur Bestimmung der Raumgruppen, Comment. Math. Helv. 21, 117–141.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×