Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: Kaleidoscopic examples

Acta Crystallographica Section A: Foundations of Crystallography

Volumn 65, Issue 2, 2009, Pages 81-108

  Ramsden, S J ^a   Robins, V ^a   Hyde, S T ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Kaleidoscopic hyperbolic tilings; Three dimensional Euclidean nets; Triply periodic minimal surfaces; Two dimensional hyperbolic tilings

Indexed keywords

EID: 60749132004     PISSN: 01087673     EISSN: 16005724     Source Type: Journal    
DOI: 10.1107/S0108767308040592     Document Type: Article

References (63)

1. Adams, C. C. (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. Providence: American Mathematical Society.
2. Balke, L. & Huson, D. H. (1996). Geom. Dedic. 60, 89-106.
3. Banerjee, R., Phan, A., Wang, B., Knobler, C., Furukawa, H., O'Keeffe, M. & Yaghi, O. M. (2008). Science, 319, 939-943.
4. Barabasi, A. L. (2002). Linked: The New Science of Networks. Cambridge: Perseus.
5. Batten, S. R. (2001). Cryst. Eng. Commun. 3, 67-73.
6. Blatov, V. A. (2006). Topos, http:///www.topos.ssu.samara.ru. Described in the online newsletter of the International Union of Crystallography Commission on Crystallographic Computing, No. 7, http://www.iucr.org/resources/ commissions/crystallographiccomputing/newsletters.
7. Blatov, V. A. (2007). Personal communication.
8. Blatov, V. A., Carlucci, L., Ciani, G. & Proserpio, D. M. (2004). Cryst. Eng. Commun. 6, 377-395.
9. Blatov, V. A., Delgado-Friedrichs, O., O'Keeffe, M. & Proserpio, D. M. (2007). Acta Cryst. A63, 418-425.
10. Brunner, G. O. & Laves, F. (1971). Wiss. Z. Tech. Univ. Dresden, 20, 387-390.
11. Castle, T., Evans, M. E. & Hyde, S. T. (2008). New J. Chem. 32, 1484-1492.
12. Castle, T., Robins, V. & Hyde, S. T. (2009). In preparation.
13. Conway, J. H. (1992). Groups, Combinatorics, and Geometry, Vol. 165, pp. 438-447. Cambridge University Press.
14. Conway, J. H. & Huson, D. H. (2002). Struct. Chem. 13, 247-256.
15. Delgado-Friedrichs, O. (1994). Euclidicity criteria for three-dimensional branched triangulations. Doctoral thesis, University of Bielefeld, Germany.
16. Delgado-Friedrichs, O. (2003). Theor. Comput. Sci. 303, 431-445.
17. Delgado-Friedrichs, O. (2006). GAVROG, http://www.gavrog.org.
18. Delgado-Friedrichs, O., Dress, A. W. M., Huson, D. H., Klinowski, J. & Mackay, A. L. (1999). Nature (London), 400, 644-647.
19. Delgado-Friedrichs, O. & Huson, D. H. (1999). Discrete Comput. Geom. 21, 299-315.
20. Delgado-Friedrichs, O. & O'Keeffe, M. (2003). Acta Cryst. A59, 351-360.
21. Dress, A. W. M. (1987). Adv. Math. 63, 196-212.
22. Dress, A. W. M. & Huson, D. H. (1987). Geom. Dedic. 24, 295-310.
23. Durand, M. (2005). Phys. Rev. E, 72, 011114.
24. Durand, M. & Weaire, D. (2004). Phys. Rev. E, 70, 046125.
25. Foster, M. D. (2008). Personal communication.
26. Foster, M. D. & Treacy, M. J. (2008). Atlas of Prospective Zeolite Frameworks, http://www.hypotheticalzeolites.net/.
27. Gibson, L. J. & Ashby, M. F. (1997). Cellular Solids - Structure and Properties, 2nd ed. New York: Cambridge University Press.
28. Grünbaum, B. & Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman.
29. Holt, D. F. (1998). KBMAG - Knuth-Bendix for Monoids and Automatic Groups (http://www.maths.warwick.ac.uk/~dfh). Version 2.4. University of Warwick, Coventry, UK.
30. Huson, D. H. (1993). Geom. Dedic. 47, 269-296.
31. Hyde, S. T., Delgado-Friedrichs, O., Ramsden, S. J. & Robins, V. (2006). Solid State Sci. 8, 740-752.
32. Hyde, S. T., Larsson, A.-K., DiMatteo, T., Ramsden, S. J. & Robins, V. (2003). Austr. J. Chem. 56, 981-1000.
33. Hyde, S. T. & Oguey, C. (2000). Eur. Phys. J. B, 16, 613-630.
34. Hyde, S. T., O'Keeffe, M. & Proserpio, D. M. (2008). Angew. Chem. Intl Ed. Engl. 47, 7996-8000.
35. Hyde, S. T., Robins, V. & Ramsden, S. J. (2009). In preparation.
36. Hyde, S. T. & Schroeder, G. E. (2003). Curr. Opin. Collect. Interf. Sci. 8, 5-14.
37. Hyde, S. T. & Schröder-Turk, G. E. (2007). Acta Cryst. A63, 186-197.
38. International Zeolite Association (2008). Database of Zeolite Structures, http://www.iza-structure.org/databases/.
39. Klee, W. E. (2004). Cryst. Res. Technol. 39, 959-968.
40. Li, Y., Yu, J. & Xu, R. (2008). AlPO Database, http://mezeopor.jlu.edu.cn/alpo/.
41. Molnar, E. (2002). Struct. Chem. 13, 267-275.
42. Moulton, B., Abourahma, H., Bradner, M. W., Lu, J., McManus, G. J. & Zaworotko, M. J. (2003). Chem. Commun. pp. 1342-1343.
43. Ockwig, N. W., Delgado-Friedrichs, O., O'Keeffe, M. & Yaghi, O. M. (2005). Acc. Chem. Res. 38, 176-182.
44. Öhrström, L. & Larsson, K. (2005). Molecule-Based Materials: The Structural Network Approach. Amsterdam, Boston: Elsevier Science.
45. O'Keeffe, M. (2008). RCSR: Reticular Chemistry Structure Resource, http://rcsr.anu.edu.au.
46. O'Keeffe, M. & Hyde, B. G. (1996). Crystal Structures I. Patterns and Symmetry. Washington: Mineralogical Society of America.
47. O'Keeffe, M., Peskov, M. A., Ramsden, S. J. & Yaghi, O. M. (2008). Acc. Chem. Res. 41, 1782-1789.
48. Ramsden, S. J., Robins, V. & Hyde, S. T. (2004). 2D Hyperbolic tilings by infinite tiles, http://epinet.anu.edu.au/infinite-tiles.
49. Ramsden, S. J., Robins, V. & Hyde, S. T. (2005). EPINET: Euclidean Patterns In Non-Euclidean Tilings, http://epinet.anu.edu.au.
50. Roberts, A. P. & Garboczi, E. J. (2002). J. Mech. Phys. Solids, 50, 33-55.
51. Robins, V. (2006). Unpublished.
52. Robins, V., Hyde, S. T. & Ramsden, S. J. (2004a). Eur. Phys. J. B, 39, 365-375.
53. Robins, V., Hyde, S. T. & Ramsden, S. J. (2004b). Physica A, 339, 173-180.
54. Robins, V., Ramsden, S. J. & Hyde, S. T. (2005). Eur. Phys. J. B, 48, 107-111.
55. Sadoc, J.-F. & Charvolin, J. (1989). Acta Cryst. A45, 10-20.
56. Sowa, H. & Koch, E. (2006). Acta Cryst. A62, 379-399.
57. Stillwell, J. (1989). Mathematics and Its History. New York, Berlin: Springer-Verlag.
58. Strong, R. T., Pickard, C. J., Milman, V., Thimm, G. & Winkler, B. (2004). Phys. Rev. B, 70, 045101.
59. The GAP Group (2002). GAP - Groups, Algorithms, and Programming, Version 4.3, http://www.gap-system.org.
60. Treacy, M. M. J., Randall, K. H., Rao, S., Perry, J. A. & Chadi, D. J. (1997). Z. Kristallogr. 212, 768-791.
61. Watts, D. J. (2003). Six Degrees: The Science of a Connected Age. New York: Norton.
62. Wells, A. F. (1977). Three-dimensional Nets and Polyhedra. New York: Wiley.
63. Zou, X., Conradsson, T., Klingstedt, M., Dadachov, M. S. & O'Keeffe, M. A. (2008). Nature (London), 202, 69-88.