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

References

[1] Abdalla G. M. Ahmed. Mathematical hints for parameter selection for aa patterns. In Reza Sarhangi and Carlo H. Séquin, editors, Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture, pages 271–278, Phoenix, Arizona (2011). Tessellations Publishing.
[2] Abdalla G. M. Ahmed. Modular duotone weaving design. In George Hart Gary Greenfield and Reza Sarhangi, editors, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, pages 27–34, Phoenix, Arizona (2014). Tessellations Publishing.
[3] Abdalla G. M. Ahmed and Oliver Deussen. Colored aa bitmaps. In Proceedings of the Conference on Computer Graphics and Visual Computing, CGVC ’17, page 77–80, Goslar, DEU (2017). Eurographics Association.
[4] Abdalla G.M. Ahmed and Oliver Deussen. Monochrome map weaving with truchet-like tiles. In Eve Torrence, Bruce Torrence, Carlo S´equin, and Kristóf Fenyvesi, editors, Proceedings of Bridges 2018: Mathematics, Art, Music, Architecture, Education, Culture, pages 45–52, Phoenix, Arizona (2018). Tessellations Publishing.
[5] Anni Albers. On weaving. In On Weaving. Princeton University Press (2017).
[6] Jay Bonner. Doing the jitterbug with islamic geometric patterns. Journal of Mathematics and the Arts, 12(2-3) : 128–143 (2018).
[7] J.H. Conway, H. Burgiel, and C. Goodman-Strauss. The Symmetries of Things. Ak Peters Series. Taylor & Francis (2008).
[8] Alexander Guerten. Kuhkubus-3d-escher-tessellations. Journal of Mathematics and the Arts, 14(1-2) : 69–71 (2020).
[9] Vi Hart and Henry Segerman. The quaternion group as a symmetry group. In Gary Greenfield, George Hart, and Reza Sarhangi, editors, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, pages 143–150, Phoenix, Arizona (2014). Tessellations Publishing.
[10] Damien Hirst. Damien Hirst. Tate (2012).
[11] Liora Segal Itzhaky, Amnon Yoeli, and Shai Gul. Deformations of sl blocks. Journal of Interdisciplinary Mathematics, 24(8) : 2367–2372 (2021).
[12] Janis Jefferies. On weaving expanded edition. TEXTILE, 17(1) : 38–43 (2019).
[13] Craig S. Kaplan and Robert Bosch. TSP art. In Bridges 2005: Mathematical Connections in Art, Music and Science, pages 301–308 (2005).
[14] Pankaj, Gunjan, and Manju Pruthi. Unit graphs and subgraphs of symmetric, quaternion and heisenberg groups. Journal of Information and Optimization Sciences, 38(1) : 207–218 (2017).
[15] Ian Parberry. The unexpected beauty of modular bivariate quadratic functions. Journal of Mathematics and the Arts, 12(4) : 197–206 (2018).
[16] Sabina Puc and Riste Škrekovski. Contour map patterns. Journal of Mathematics and the Arts, 5(3) : 129–140 (2011).
[17] Bridget Riley and Robert Kudielka. Bridget Riley. Tate Publishing (2003).
[18] L. Roween. Algebra, Groups, Rings and Fields. A K Peters/CRC Press (1994).
[19] Kokichi Sugihara. Computer-aided generation of escher-like sky and water tiling patterns. Journal of Mathematics and the Arts, 3(4) : 195–207 (2009).
[20] Maheswara Rao Valluri and Shailendra Vikash Narayan. Public key authentication scheme over quaternions. Journal of Discrete Mathematical Sciences and Cryptography, 24(1) : 169–181 (2021).
[21] C. A. Yackel. Wallpaper patterns admissible in itajime shibori. Journal of Mathematics and the Arts, 15(3-4) : 232–244 (2021).

Views: 324Downloads: 112Citations: 1