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PERFECT COLOURINGS OF ISONEMAL FABRICS BY THICK STRIPING

Part of: Real and complex geometry Discrete geometry

Published online by Cambridge University Press:  12 December 2011

R. S. D. THOMAS
R. S. D. THOMAS*
Affiliation:
St John’s College and Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada (email: thomas@cc.umanitoba.ca)
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Abstract

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Perfect colouring of isonemal fabrics by thick striping of warp and weft and the closely related topic of isonemal prefabrics that fall apart are reconsidered and their relation further explored. The catalogue of isonemal prefabrics of genus V that fall apart is extended to order 20 with designs that can be used to weave cubes with colour symmetry as well as weaving symmetry.

Keywords

fabricisonemalperfect colouringprefabricweaving

MSC classification

Secondary: 52C20: Tilings in $2$ dimensions 51M20: Polyhedra and polytopes; regular figures, division of spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 2 , April 2012 , pp. 325 - 349
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Clapham, C. R. J., ‘The bipartite tournament associated with a fabric’, Discrete Math. 57 (1985), 195–197.CrossRefGoogle Scholar
[2]Grünbaum, B. and Shephard, G. C., ‘Satins and twills: An introduction to the geometry of fabrics’, Math. Mag. 53 (1980), 139–161.CrossRefGoogle Scholar
[3]Grünbaum, B. and Shephard, G. C., ‘A catalogue of isonemal fabrics’, in: Discrete Geometry and Convexity, Annals of the New York Academy of Sciences, 440 (eds. Goodman, J. E.et al.) (New York Academy of Sciences, New York, NY, 1985), pp. 279–298.Google Scholar
[4]Grünbaum, B. and Shephard, G. C., ‘An extension to the catalogue of isonemal fabrics’, Discrete Math. 60 (1986), 155–192.CrossRefGoogle Scholar
[5]Hoskins, W. D. and Thomas, R. S. D., ‘Conditions for isonemal arrays on a cartesian grid’, Linear Algebra Appl. 57 (1984), 87–103.CrossRefGoogle Scholar
[6]Hoskins, J. A. and Thomas, R. S. D., ‘The patterns of the isonemal two-colour two-way two-fold fabrics’, Bull. Aust. Math. Soc. 44 (1991), 33–43.CrossRefGoogle Scholar
[7]Pedersen, J., ‘Some isonemal fabrics on polyhedral surfaces’, in: The Geometric Vein, (eds. Davis, C., Grünbaum, B. and Sherk, F. A.) (Springer, New York, NY, 1981), pp. 99–122.CrossRefGoogle Scholar
[8]Roth, R. L., ‘The symmetry groups of periodic isonemal fabrics’, Geom. Dedicata 48 (1993), 191–210.CrossRefGoogle Scholar
[9]Roth, R. L., ‘Perfect colorings of isonemal fabrics using two colors’, Geom. Dedicata 56 (1995), 307–326.CrossRefGoogle Scholar
[10]Schattschneider, D., ‘The plane symmetry groups: Their recognition and notation’, Amer. Math. Monthly 85 (1978), 439–450.CrossRefGoogle Scholar
[11]Schattschneider, D., ‘In black and white: How to create perfectly colored symmetric patterns’, Comput. Math. Appl. 12B (1986), 673–695.CrossRefGoogle Scholar
[12]Thomas, R. S. D., ‘Isonemal prefabrics with only parallel axes of symmetry’, Discrete Math. 309 (2009), 2696–2711.CrossRefGoogle Scholar
[13]Thomas, R. S. D., ‘Isonemal prefabrics with perpendicular axes of symmetry’, Util. Math. 82 (2010), 33–70.Google Scholar
[14]Thomas, R. S. D., ‘Isonemal prefabrics with no axes of symmetry’, Discrete Math. 310 (2010), 1307–1324.CrossRefGoogle Scholar
[15]Thomas, R. S. D., ‘Perfect colourings of isonemal fabrics by thin striping’, Bull. Aust. Math. Soc. 83 (2011), 63–86.CrossRefGoogle Scholar