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Zeitschrift für Kristallographie, Bd. 142, S. 1—23 (1975) Point groups and color symmetry By MaRJORIE SENECHAL Smith College, Northampton, Massachusetts* (Received 15 May 1974 and in revised form 13 January 1975) Auszug Auf der Grundlage elementarer Betrachtungen wird eine Theorie der Mehr- farben-Symmetrie entwickelt. Es wird gezeigt, daß jede Farbgruppe durch eine Symmetriegruppe 67 und eine Untergruppe H, welche einen Homomorphis- mus von 67 zu einer Gruppe P von Farbpermutationen bestimmt, charakterisiert ist. Die verschiedenen Wege, auf welchen die

Zeitschrift für Kristallographie, Bd. 142, S. 1—23 (1975) Point groups and color symmetry B y MABJORIE SENECHAL Smith College, Northampton, Massachusetts* (Received 15 May 1974 and in revised form 13 Janua ry 1975) Anszug Auf der Grundlage elementarer Betrachtungen wird eine Theorie der Mehr- farben-Symmetrie entwickelt. Es wird gezeigt, daß jede Farbgruppe durch eine Symmetriegruppe β und eine Untergruppe H, welche einen Homomorphis- mus von G zu einer Gruppe Γ von Farbpermutat ionen bestimmt, charakterisiert ist. Die verschiedenen Wege, auf welchen die

, The symmetry of the complete twin. Amer. Mineralogist 41 (1959) 1067—1070. 4 V. L. Indenbom, N. V. Belov and N. N. Neronova, The color-symmetry point groups. Soviet Physics—Crystallography 5 (1961) 477—481. 154 Oscar Wittke tschek and Niggli5.6 avIio introduced the names 'simple crypto- symmetry'and 'multiple cryptosymmetry'. They also discussed various criteria for enumerating the corresponding groups. Van derWaerden and Burckhardt7 related the colour-symmetry groups to the repre- sentations by means of permutations. In this article the cryptosymmetry point groups

Zeitschrift für Kristallographie, Bd. 117, S. 153-165 (1962) The colour-symmetry groups and cryptosymmetry groups associated with the 32 crystallographic point groups B y OSCAR W I T T K E Centro de Investigaciones de Cristalografia, Inst i tuto de Fisica y Matematicas, Universidad de Chile, Santiago, Chile (Received January 22, 1962) Auszug Die Kryptosymmetrie- und Farbsymmetrie-Punktgruppen, die sich von den 32 kristallographischen Punktgruppen herleiten lassen, können durch 139 Sym- bole dargestellt werden. Deren Ableitung und tabellarische

Ten colours in quasiperiodic and regular hyperbolic tilings Reinhard LückI and Dirk Frettlöh*, II I Weilstetter Weg 16, 70567 Stuttgart, Germany II Universität Bielefeld, Universitätsstr. 25, 33501 Bielefeld, Germany Received June 13, 2008; accepted July 26, 2008 Hyperbolic tilings / Regular tilings / Colour symmetry / Colour group Abstract. Colour symmetries with ten colours are pre- sented for different tilings. In many cases, the existence of these colourings were predicted by group theoretical meth- ods. Only in a few cases explicit constructions were

); http://www.gap-system.org. [6] Grünbaum, B.; Shephard, G. C.: Tilings and patterns. Freeman, New York (1987). [7] Humphreys, J. E.: Reflection Groups and Coxeter Groups. Cam- bridge University Press (1990). [8] Lifshitz, R.: Theory of color symmetry for periodic and quasi- periodic crystals. Rev. Mod. Phys. 69 (1997) 1181–1218. [9] Moody, R. V.; Patera, J.: Coloring of quasicrystals. Can. J. Phys. 72 (1994) 442–452. [10] Schwarzenberger, R. L. E.: Colour symmetry. Bull. London Math. Soc. 16 (1984) 209–240. 776 D. Frettlöh M at he m at ic al / Th eo re ti ca lA sp

colors wherein two points of Λ 1 are assigned the same color if and only if they belong to the same coset of Λ 2 . In this case, the set of colors C can be identified with the quotient group Λ 1 /Λ 2 so that the color mapping c is simply the canonical projection of Λ 1 onto Λ 1 /Λ 2 whose kernel is Λ 2 . Denote by G the symmetry group of Λ 1 and fix a coloring c of Λ 1 . A symmetry h in G is said to be a color symmetry of c if it permutes the colors in the coloring, that is, all and only those elements of Λ 1 having the same color are mapped by h

.: On the power subgroups of the extended modular group. Turk J. Math 28 (2004) 143–151. [21] Senechal, M.: Color symmetry. Computational Math. Applica- tion 16 No. 5–8 (1988) 545–553. [22] Schwarzenberger, R. L. E.: Colour symmetry. Bulletin London Math. Society 16 (1984) 209–240. Enumeration of index 3 and 4 subgroups of hyperbolic triangle symmetry groups 551

. Acta Cryst. A41/5 (1985) 484–490. [17] Roth, R. L.: Coloring non-characteristic crystallographic orbits. Z. Kristallogr. 183/1–4 (1988) 233–244. [18] Schwarzenberger, R. L. E.: Colour symmetry. Bull. London Math. Soc. 16/3 (1984) 209–240. [19] Senechal, M.: Color Symmetry. Comput. Math. Appl. 16/5–8 (1988) 545–553. Enumerating and identifying semiperfect colorings of symmetrical patterns 491 Mathematical and Theoretical Crystallography Edited by Hans Grimmer The articles correspond to lectures or posters presented at the workshop “Crystallography at the start of the

On subgroups of hyperbolic tetrahedral Coxeter groups Ma. Louise Antonette N. De Las Peñas*, I, Rene P. FelixII and Glenn R. LaigoI I Mathematics Department, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines II Institute of Mathematics, University of the Philippines – Diliman, Diliman, Quezon City, Philippines Received August 24, 2009; accepted May 4, 2010 Coxeter tetrahedra / Tetrahedral Coxeter groups / Tetrahedral Kleinian groups / Subgroup enumeration / Color symmetry Abstract. In this work we address the problem on the determination of