Method of Construction of Decagonal Self-Similar Patterns

Because of its unequal beauty and mathematical sophistication, Islamic art has received a great attention from several scientists. Hence, several works have been done to investigate its mathematical structure, and to discover its principle of construction. Up to now, no method of constructing new self-similar patterns were proposed. In this paper, we will present a method for constructing new self-similar patterns. The proposed method is based on successive subdivisions of the golden mean triangles.

[1]  A. K. Dewdney,et al.  The Tinkertoy computer and other machinations , 1993 .

[2]  J. Hogendijk.  Mathematics and geometric ornamentation in the medieval Islamic world , 2012 .

[3]  Craig S. Kaplan,et al.  Islamic star patterns in absolute geometry , 2004, TOGS.

[4]  R. Ajlouni.  Octagon-Based Quasicrystalline Formations in Islamic Architecture , 2013 .

[5]  Branko Grünbaum,et al.  Interlace Patterns in Islamic and Moorish Art , 1993 .

[6]  Aziz Khamjane,et al.  Generating Islamic Quasi-Periodic Patterns , 2018, ACM Journal on Computing and Cultural Heritage.

[7]  Mohamed Ould Djibril,et al.  Islamic geometrical patterns indexing and classification using discrete symmetry groups , 2008, JOCCH.

[8]  Peter R. Cromwell,et al.  Islamic geometric designs from the Topkapı Scroll I: unusual arrangements of stars , 2010 .

[9]  Issam El-Said,et al.  Geometric Concepts in Islamic Art , 1976 .

[10]  W. K. Chorbachi,et al.  In the Tower of Babel: Beyond symmetry in islamic design , 1989 .

[11]  Peter W. Saltzman.  Quasi-Periodicity in Islamic Geometric Design , 2015 .

[12]  Peter R. Cromwell,et al.  The Search for Quasi-Periodicity in Islamic 5-fold Ornament , 2008 .

[13]  S. J. Abas,et al.  Geometric and Group‐theoretic Methods for Computer Graphic Studies of Islamic Symmetric Patterns , 1992, Comput. Graph. Forum.

[14]  Craig S. Kaplan,et al.  Islamic star patterns from polygons in contact , 2005, Graphics Interface.

[15]  Peter R. Cromwell,et al.  Cognitive Bias and Claims of Quasiperiodicity in Traditional Islamic Patterns , 2015 .

[16]  Jean-Marc Castéra,et al.  Arabesques: Decorative Art in Morocco , 2009 .

[17]  B. Wichmann,et al.  Islamic Design: A Mathematical Approach , 2018 .

[18]  Aziz Khamjane,et al.  A computerized method for generating Islamic star patterns , 2018, Comput. Aided Des..

[19]  E. Hanbury Hankin.  Some Difficult Saracenic Designs. III. A Pattern Containing Fifteen-Rayed Stars , 1936 .

[20]  Peter R. Cromwell,et al.  A modular design system based on the Star and Cross pattern , 2012 .

[21]  E. Hanbury Hankin.  Examples of Methods of Drawing Geometrical Arabesque Patterns , 1925 .

[22]  John Rigby,et al.  Creating Penrose-type Islamic Interlacing Patterns , 2006 .

[23]  Jay Bonner,et al.  Islamic Geometric Patterns , 2017 .

[24]  Natasa Bulatovic,et al.  The CENDARI Infrastructure , 2016, ACM Journal on Computing and Cultural Heritage.

[25]  R. Ajlouni.  The global long-range order of quasi-periodic patterns in Islamic architecture , 2012 .

[26]  Oleg Grabar.  The Mediation of Ornament , 1992 .

[27]  Victor Ostromoukhov,et al.  Mathematical Tools for Computer-Generated Ornamental Patterns , 1998, EP.

[28]  Alpay Özdural,et al.  Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World , 2000 .

[29]  E. Makovicky,et al.  The first find of dodecagonal quasiperiodic tiling in historical Islamic architecture , 2011 .

[30]  P. F. Hach-Alí,et al.  Mirador de Lindaraja: Islamic ornamental patterns based on quasi-periodic octagonal lattices in Alhambra, Granada, and Alcazar, Sevilla, Spain. (El mirador de Lindaraja: dibujos islámicos ornamentales de la Alhambra de Granada y del Alcázar de Sevilla (Es , 1996 .

[31]  R. Penrose.  Pentaplexity A Class of Non-Periodic Tilings of the Plane , 1979 .

[32]  Jay Bonner,et al.  Three Traditions of Self-Similarity in Fourteenth and Fifteenth Century Islamic Geometric Ornament , 2003 .