Geometric Projections of Art, Mathematical Landscapes

By Özer Mumcu

"Exploring Mathematical Landscapes" delves into the rich tapestry of mathematical concepts that underpin artistic expression, from the mesmerizing patterns of fractals to the graceful symmetries of geometric shapes. Through insightful commentary, stunning visuals, and hands-on activities, readers will uncover the hidden mathematical structures that permeate the world of art, gaining a deeper appreciation for both disciplines in the process.

• Language: English
• Publisher: Özer Mumcu
• Release date: Apr 13, 2024
• ISBN: 9798224127559

Book preview

by Ozer Mumcu

ABSTRACT

The relationship between visual arts and mathematics has a long historical background. In the 4th century BC, the Greek sculptor Polykleitos mentioned the mathematical proportions of the ideal male body in his notes. Although there is no concrete evidence, the use of the golden ratio can clearly be observed in ancient art and architecture. During the Renaissance period, Luca Pacioli wrote an article titled The Golden Ratio (1509), referring to the works of Leonardo Da Vinci. Another Italian painter, Piera della Francesca, incorporated Euclid's ideas on perspective into his works. The famous engraver Albrecht Dürer made numerous references to mathematics in his works. In modern times, with the advancement of technology, computer science has become part of artistic practice. Computers are often used in artistic production involving fractals and in the algorithmic analysis of various artworks through X-ray fluorescence spectroscopy.

M. C. Escher, with significant assistance from Harold Coxeter, extensively used the regular division of the plane and hyperbolic geometry in his works. Additionally, mathematics directly influenced Escher's and many other artists' works through conceptual tools such as perspective types, symmetry analysis, Möbius Strip, recursion, and polyhedra. In Escher's works, besides Coxeter, the influences of certain scientists and artists can be observed periodically. Some of these individuals were personally acquainted with Escher, such as Roger Penrose and Albert Flocon, while others, such as Euclid, Jules Henri Poincare, Frederick Möbius, and Carl Friedrich Gauss, lived before Escher and left lasting impressions on the worlds of mathematics or art.

The study consists of three parts. In the introduction, information is provided about Escher's life, geometric topology, hyperbolic geometry, and curvilinear perspective. The first part examines Escher's division of the two-dimensional plane into regular subdivisions and how he subsequently brought these studies into the hyperbolic plane under the influence of Coxeter. The second part focuses on Escher's relationship with physicist Roger Penrose and how the two influenced each other's works. In the third part, Escher's knot works influenced by Albert Flocon are examined. Additionally, general information about knots, which have an extensive literature in mathematics and art, is provided. The conclusion is reached by mentioning Benjamin Sack, a contemporary artist influenced by Escher.

Keywords: M.C. Escher, Harold Coxeter, Roger Penrose, hyperbolic geometry, geometric topology, knot theory

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INDEX OF IMAGES

Image-1. M.C. Escher

Image-2. M.C. Escher, The Graphic Work, www.taschen.com

Image-3. M.C. Escher, The Graphic Work, www.taschen.com

Image-4. M.C. Escher, The Graphic Work, www.taschen.com

Image-5. Henri Poincare

Image-6. Königsberg Bridge, Geometric Perspective with Geometry, Assoc. Prof. Sadık Bayhan, Nazlı Tuğçe Baytaroğlu, Lakes Region Monthly Refereed Economic and Culture Journal, Volume 7, Number 76, July 2019

Image-7. Möbius Strip, Geometric Perspective with Geometry, Assoc. Prof. Sadık Bayhan, Nazlı Tuğçe Baytaroğlu, Lakes Region Monthly Refereed Economic and Culture Journal, Volume 7, Number 76, July 2019

Image-8. Möbius Strip, Geometric Perspective with Geometry, Assoc. Prof. Sadık Bayhan,