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

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.