Can you feel the algebraic structure? Quaternion and finite rings in the service of patterns and weaving

Barkan, Refael; Gul, Shai; Hochman, Aviv; Matzri, Eliyahu

doi:10.47974/JIM-1639

Open Access Research Article

Can you feel the algebraic structure? Quaternion and finite rings in the service of patterns and weaving

Refael BarkanDepartment of MathematicsHolon Institute of TechnologyHolon, 5810201, IsraelView full profile → , ^*Shai GulCorresponding authorshaigo@hit.ac.ilDepartment of MathematicsHolon Institute of TechnologyHolon, 5810201, IsraelView full profile → , Aviv HochmanDepartment of MathematicsHolon Institute of TechnologyHolon, 5810201, IsraelView full profile → , Eliyahu MatzriDepartment of MathematicsBar-Ilan UniversityRamat-Gan, 529002, IsraelView full profile →

* Corresponding author · click or hover a name for details

pp. 1787–1802Vol. 26Issue 8December 2023DOI: 10.47974/JIM-1639XML

Received:: 06 Jul 2022
Accepted:: 22 Nov 2022
Published Online:: 29 Dec 2023
Article type:: Research Article
Language:: EN
Article no.:: JIM-1639
Pages:: 1787–1802

Abstract

Artists frequently create symmetrical patterns, but disregard the mathematical structure that allows their formation. We here seek to expose this underlying construct and afford new ways of using mathematical structures to produce surprising, artistic symmetrical patterns. We introduce different multi-color symmetrical patterns whose generation entails only a transformation on an initial location matrix. By using different algebraic structures, we were able to obtain a range of surprising symmetric patterns. We demonstrate how planar patterns can be generated by generalizing a result of [15] to RGB and CMYK color formats using a Quaternion group. In addition, we show how finite rings can produces a perception of symmetrical weaving patterns. Lastly, we use the Quaternion group to produce spherical patterns in the RGB format. We believe these algebraic structures contain natural beauty that a variety of audiences will find interesting and applicable: digital artists, mathematics aficionados, teachers and college students.

Keywords

Digital images Planar patterns Spherical patterns Rings Algebraic structure

Subject Classifications

(2010) 03G100A6605C25

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