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Bravais colourings of planar modules with N-fold symmetry Michael Baake*, I and Uwe GrimmII I Institut für Mathematik, Universität Greifswald, Jahnstr. 15a, D-17487 Greifswald, Germany II Applied Mathematics Department, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK Received August 13, 2003; accepted October 20, 2003 Colourings / Planar modules / Cyclotomic fields / Dirichlet series / N-fold symmetry Abstract. The first step in investigating colour symme- tries for periodic and aperiodic systems is the determina- tion of all colouring schemes that

On color groups of Bravais colorings of planar modules with quasicrystallographic symmetries Enrico Paolo C. Bugarin*, I, Ma. Louise Antonette N. De Las PeñasI, Imogene F. EvidenteII, Rene P. FelixII and Dirk FrettloehIII I Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines II Institute of Mathematics, University of the Philippines, Diliman, Quezon City, Philippines III Faculty of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany Received June 15, 2008; accepted July 27, 2008 Color groups / Lattices

Multiple planar coincidences with N-fold symmetry Michael Baake*; I and Uwe GrimmII I Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany II Department of Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK Received October 21, 2005; accepted December 14, 2005 Lattices / Coincidence ideals / Planar modules / Cyclotomic fields / Dirichlet series / Asymptotic properties Abstract. Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formula- tion based on cyclotomic

parameter, because different polytypes may offer differ- ent bonding configurations to the interlayer guest. The interplaying of (planar) modules described above is often referred in literature as modular crystallography and the compounds sharing one or more modules normally can be grouped according to modular families obeying to properly established combi- natorial rules, e.g.: polytypes; homologous, polysomatic, merotype and plesiotype ser- ies. The combinatorial rules can be exploited to solve crystal structures but can also inspire new paths to the synthesis of new

on CSLs, and CSLs which are not related to a colour symmetry. References [1] Baake, M.: Combinatorial aspects of colour symmetries. J. Phys. A: Math. Gen. 30 (1997) 2687–2698; mp arc/02-323. [2] Baake, M.; Grimm, U.: Bravais colourings of planar modules with N-fold symmetry. Z. Kristallogr. 219 (2004) 72–80, math.CO/0301021. [3] Baake, M.; Grimm, U.; Heuer, M.; Zeiner, P.: Coincidence rota- tions of the root lattice A4. Europ. J. Comb., in press, 0709.1341. [4] De las Pẽnas, M. L. A. N.; Felix, R. P.; Laigo, G. R.: Colorings of hyperbolic plane crystallographic

and deter- mined periodic colorings of a lattice. Baake and Grimm in [1], determined all coloring schemes that are compatible with the symmetry group or a subgroup of the symmetry group of all planar modules with N-fold symmetry. In [2], De Las Peñas and Felix obtained a list of all color groups associated with color- ings of a square or hexagonal lattice L arising from its sublattices. The results were obtained using a canonical representation of a sublattice L of L by a 2 2 matrix and no restriction on whether L is compatible or not with L was imposed. The

is described by a vector point group (rigorously speaking, by a poly- chromatic vector point group: Nespolo, 2004). Those op- erations that do not belong to the point group of the indi- viduals are the twin operations (the chromatic operations). In relatively recent times, the term “twin” has been also applied, with or without modifiers, to indicate build- ing mechanisms of modular structures. This use may be misleading. The modular structures we are referring to are composed of planar modules whose mappings are opera- tions defined in the point space, i.e. the

[1] Baake, M.: Combinatorial aspects of colour symmetries. J. Phys. A: Math. Gen. 30 (1997) 2687–2698; mp arc/02–323. [2] Baake, M.; Grimm, U.: Bravais colourings of planar modules with N-fold symmetry. Z. Kristallogr. 219 (2004) 72–80; math.CO/0301021. [3] Coxeter, H. S. M.: Regular Polytopes. Metheun & Co Ltd., Lon- don (1948). [4] de las Peñas, M. L. A. N.; Felix, R. P.; Laigo, G. R.: Colorings of hyperbolic plane crystallographic patterns. Z. Kristallogr. 221 (2006) 665–672. [5] The GAP Group, GAP – Groups, Algorithms, and Program- ming. Version 4.4.10 (2007

additional stage between privacy and public life. It seems reason- able to examine these forms of housing with respect to their suitability for layered models (Fig. 14 b). 15 a 16 b c 22 17 Design and Typology Combinations of spatial and planar modules in hotel construction Hotel buildings or similar construction tasks, such as residential homes or hospitals, are characterised by recurring sequences of rooms with anterooms and sanitary areas. In combinations comprising room modules and planar elements, the sanitary areas are always prefabricated, while on