Abstract
Until 1982 it was considered an irrefutable paradigm in crystallography that crystals cannot have rotational symmetry of the order of five or higher than six. This limitation of rotational symmetry has already been mentioned several times in this book (see ► Sects. 3.2 and 5.2). This limitation applies to both the order of axes of rotation as well as screw axes: only axes with the order 1–4 and 6 occur in each case. However, we will discover in ► Sect. 8.4 that this is not the whole truth.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This term is equivalent to the term tiling and describes in mathematics the gapless and overlap-free covering of the (Euclidean) plane by uniform sub-areas.
References
Reinhardt K (1918) Über die Zerlegung der Ebene in Polygone. Dissertation. Johann Wolfgang Goethe Universitat, Frankfurt a.M
Kershner RB (1968) On paving the plane. Am Math Mon 75:839–844. https://doi.org/10.2307/2314332
James RE (1975) New pentagonal tiling. Reported by Gardner, M. Sci Am 233:117–118
Stein R (1985) A new pentagon tiler. Reported by Schattschneider, D. Math Mag 58:308
Mann C, Mc-Loud-Mann J, Von Deroux D (2015) Convex pentagons that admit i-block transitive tilings. eprint arXiv:1510.01186. https://arxiv.org/abs/1510.01186. Accessed 30 Sept 2019
Shechtman D, Blech I, Gratias D, Cahn JW (1984) Metallic phase with long-range orientational order and no translational symmetry. Phys Rev Lett 53:1951–1954. https://doi.org/10.1103/PhysRevLett.53.1951
Mackay AL (1982) Crystallography and the Penrose pattern. Physica A 114:609–613. https://doi.org/10.1016/0378-4371(82)90359-4
Levine D, Steinhardt PJ (1984) Quasicrystals: a new class of ordered structures. Phys Rev Lett 53:2477–2480. https://doi.org/10.1103/PhysRevLett.53.2477
The Nobel Prize in Chemistry 2011. http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/. Official web site of the Nobel prize. Accessed 30 Sept 2019
Steurer W (2018) Quasicrystals: what do we know? What do we want to know? What can we know? Acta Cryst A 74:1–11. https://doi.org/10.1107/S2053273317016540
Prof. Dan Shechtman 2011 Nobel Prize Chemistry Interview with ATS. https://www.youtube.com/watch?v=EZRTzOMHQ4s. Accessed 30 Sept 2019
Kuhn TS (1996) The structure of scientific revolutions.3rd edn. University of Chicago Press, Chicago
Bindi L, Steinhardt PJ, Yao N, Lu PJ (2009) Natural quasicrystals. Science 324:1306–1309. https://doi.org/10.1126/science.1170827
Bindi L, Yao N, Lin C, Hollister LS, Andronicos CL, Distler VV, Eddy MP, Kostin A, Kryachko V, Mac-Pherson GJ, Steinhardt WM, Yudovskaya M, Steinhardt PJ (2015) Natural quasicrystals with decagonal symmetry. Sci Rep 5:Article number 9111. https://doi.org/10.1038/srep09111
Penrose R (1974) The role of aesthetics in pure and applied mathematical research. Bull Inst Math Appl 10:266–271
Takakura H, Gomez CP, Yamamoto A, de Boissieu M, Tsai AP (2007) Atomic structure of the binary icosahedral Yb-Cd quasicrystal. Nat Mater 6:58–63. https://doi.org/10.1038/nmat1799
Wagner T, Schönleber A (2009) A non-mathematical introduction to the superspace description of modulated structures. Acta Cryst B 65:249–268. https://doi.org/10.1107/S0108768109015614
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hoffmann, F. (2020). “Forbidden” Symmetry. In: Introduction to Crystallography. Springer, Cham. https://doi.org/10.1007/978-3-030-35110-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-35110-6_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35109-0
Online ISBN: 978-3-030-35110-6
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)