M.C. Escher
One great artist that has embodied art, mathematics, and the mystical is Maurits Cornelis Escher. He lived from 1898 to 1972. He was a graphic artist who revolutionized a new kind of art form that tickled the imagination of people. He has created around 2,000 drawings, and around 448 lithographs and wood art (M.C. Escher Foundation). He went to the School for Architecture and Decorative Arts in Haarlem, Netherlands to study architecture. However, an instructor of his convinced him to focus on graphical art instead of architecture upon seeing samples of his graphic art. Particularly, what inspired him in his later works was his visit to the Alhambra, a 14th century Moorish Castle in Granada, Spain in 1922. From what he learned at Alhambra, he has accomplished a total of 137 drawings which he named the Regular Division of the Plane series.
Works of Art
Noteworthy to mention are these four art pieces: (a) Relativity, (b) Day and Night, (c) Print Gallery, and (c) Circle Limit IV. First to be discussed is the Relativity (See Figure 1). It is a lithograph made by Escher in 1953. It depicts a complex three-dimensional world where the laws of gravity do not apply. It shows seven bulb-head people, and seven staircases. It is mind-boggling since there is the interplay of illusion and reality. It is science fiction, but fascinating. It is Escher’s masterpiece that has amazed the human mind. Second is Day and Night which is part of the Regular Division of Plane series (See Figure 2). Escher researched on F. Haag and G. Polya’s work on congruent convex polygons, periodic planar tiling, and symmetry groups. Looking at the work Night and Day, one can see the contrast of dark and white as well the opposing flow of the white doves from the black ones. In this particular work, he has focused on reflection. In latter works, the concepts of translation and rotation are emphasized more. Third is the Print Gallery made in 1956 (See Figure 3). This artwork is quite unique in the sense that he introduced the concept of the distorted grid. This created the imagery of circular expansion and transformation. With research and practice, Escher has grown his talent and skill. By 1960, Escher has extended his concepts to woodwork through the four-part Circle Limit series. Shown in Figure 4 is the Circle Limit IV. In this art piece, Escher even further developed from reflection, and used translation and rotation. Through the use of the ruler and a compass, Escher has made hyperbolic patterns in his work. The Circle Limit series were made in wooden beech spheres. The concepts Escher presented in his drawings had prompted contemporary researchers and mathematicians to study and to explore them in the hopes of discovering new knowledge.
Figure 1: Relativity by M.C. Escher (1953)
Figure 2: Day and Night by M.C. Escher (1938)
Figure 3: Print Gallery by M.C. Escher (1956)
Figure 4: Circle Limit IV by M.C Escher (1960)
Mathematical Concepts
For Escher’s work, two main characteristics can be observed: impossible structures and the regular division of the plane (de Smit & Lenstra 457). For the making of impossible structures similar to the Print Gallery, Escher used the theory of cyclic expansion. This can be expressed using straight lines or curve lines, but curve lines offer a distorted view of the space. As the grid line moves counterclockwise around the center, the corresponding squares grow by a factor of 44= 256. In 2003, two students, B. de Smit and H.W. Lenstra made a mathematical model to reconstruct the original Print Gallery into a straightened one. Moreover, they were successful in recreating a copy similar to the original Print Gallery using the enhanced straightened version (their previous output). To do this, they have used a computer program by J. Batenburg (Robinson 3). The only difference was that the blank center was the disappearance of the blank center. It is an amazing feat in mathematical modeling to recreate a specific work of art made by a great artist such as Escher.
The regular division of the plane has been developed further by Escher through reflection, translation, and rotation of his tessellations (combinations of irregular shapes that interlock to completely cover a surface). His research on Haag and Polya’s work even convinced him to correspond with Polya to ask question regarding shapes of his tessellations. He even devised his own rotation such that S refer to 180° rotation, L refer to 90° rotation, = refer to related by translation and || refer to glide-reflection (Schattschneider 709). Another mathematician, H.S.M. Coxeter influenced Escher particularly in the latter three Circle Limit series in 1954 onwards. The hyperbolic tiling in Circle Limit II, III and IV was successfully explained mathematically by Coxeter.
Reflection
Escher, as cited by Schattschneider (2010), explains “although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists.” This is basically what Escher understood of his works. As a graphic artist, he mixed art and mathematics in a very special way only grasped by mathematicians. This is where the mystical comes in. This reflects the impossible in his structures, the beyond in his curved lines, and the without beginning and end in his cyclic expansions. He is one of the greatest Dutch graphic artists. The assistance of mathematical software is needed to analyze his art pieces. His tessellations have been the inspiration for fractal art which in a nutshell could explain basic designs and patterns in nature. With the power of human mind to create comes the inspiration to pursue the truth and search for answers to current problems.
References
de Smit, B. & Lenstra, H.W. The Mathematical Structure of Escher’s Print Gallery. Notices of the AMS 50:4. pp. 446-457. Retrieved from http://www.ams.org/notices/200304/fea-escher.pdf (December 14, 2014).
M.C. Escher Foundation (2014). Biography of M.C. Escher. Retrieved from http://www.mcescher.com/about/biography/ (December 14, 2014).
Robinson, S. (2002). M.C. Escher: More Mathematics Than Meets the Eye. SIAM News 3:8. Retrieved from http://www.msri.org/people/members/sara/articles/siamescher.pdf (December 14, 2014).
Schattschneider, D. (2010). The Mathematical Side of M.C. Escher. Notices of the AMS 57:6. pp. 706-718. Retrieved from http://www.ams.org/notices/201006/rtx100600706p.pdf (December 14, 2014).
