Ochoa 1 ​ Amy Ochoa Petersen Math 101: Tu/Th 29 March 2016 M.C Escher: Mathematics in Art “Are you really sure that a floor can't also be a ceiling?” Creating an alternate reality for ​ an artist is actually quite simple­the only requirement is creativity. But how did an architect choose art over his life’s study? Maurits Cornelis Escher is widely known for his distortion of the mundane through the artistic ability of creating alternate realities, this has been linked to his architectural background. Escher attended Haarlem’s School for Architecture and Decorative Arts, it was during this time that he decided to pursue graphic arts as his formal career after one of his professors/mentors encouraged his talent. Escher evolved into creating art that demonstrated his influences from mathematical theories he wanted to pursue. While his talents vary in forms of tapestry designer, muralist, book illustrator, and draftsman, his most impressionable pieces of art come from his prints. These were mostly carved out of wood or made with linoleum blocks during his early works. One of them is a portrait of his father, the great admiration he had for him helped Escher develop his interests in science and mathematics, which he implemented into his art. While in school, he slowly began to incorporate mathematical concepts into his wood prints, even though he did not completely understand the concepts himself. An example would be Eight Heads created in 1922, of which is the “first print ​ ​ to demonstrate the theory of the regular division of a plane” (M.C. Escher­ Life and Work, pg. 4). ​ ​ He then continued to implement this concept to create his most famous works, the art of Ochoa 2 ​ tessellations. These are geometric shapes that interlock with one another at appropriate angles to cover a surface or plane. This theory was tested by counting the number of times one print can be rotated to completely divide the plane into the available amount of sections provided by the print. Or better yet, described by the artist himself: "A plane, which should be considered limitless on all sides, can be filled with or divided into similar geometric figures that border each other on all sides without leaving any empty spaces. This can be carried on to infinity according to a limited number of systems"­M.C Escher (M.C. Escher­ Life and Work, pg. 14). ​ ​ The print of Eight Heads are Euclidean tessellations purposefully drawn on a flat plane with ​ ​ complementary angles. This type of geometry results in the idea that for every line there is a parallel line beside it, almost like a grid. Eight Heads, 1922, woodcut, block printed four times ​ Ochoa 3 ​ Shortly after he finished school, he married Jetta Umiker and decided to remain in Rome for the first part of their marriage. While in Italy, he developed an intrigue for painting landscapes, he would spend the better parts of his summers traveling and drawing the Italian countryside. When he would return to the studio, the prints he would make would enhance the actual architecture of the buildings and the roads would increase in complications. The vantage points would result in a dual perspective, and for the first time, one could look up and down at the same time. He not only changed the main focus of a picture through his creative architectural designs but he incorporated shadows and highlights that helped his tessellations become distorted and cohesive all at once. The use of these difficult and often “impossible” vantage points helped to establish him as one of the most creative illusionists. Escher’s paintings not only depicted impossible situations but they created a wondrous reality that could only exist through paper. Still Life and Street, 1937, woodcut Balcony, 1945, lithograph ​ ​ ​ Ochoa 4 ​ Escher later worked explore mathematical concepts that he had only ever read about but never applied them to numbers. Instead, “he focused on the geometry of space and the logic of space” and not only demonstrated but proved the theories through his art (Lybarger). For example, he was best able to prove the geometry of space through his collection of tessellations named Metamorphosis I­III. The series explores different stories of the different ways that ​ ​ shapes can morph into real animals. While starting with a shape on one end, the journey through the canvas can be of multiple animals distorting from the shapes of other animals. Through this series, his use of Euclidean geometry is best shown through color, as the parallel line work seems to continue through infinity. Cropped section of Metamorphosis II, 1940, woodcut in black, green and brown, printed from 20 blocks on 3 ​ ​ ​ combined sheets In 1941, while trying to improve his knowledge of geometric artwork, he wrote a now popular paper called “Regular Division of the Plane with Asymmetric Congruent Polygons.” ​ Which was a basic guide on how to add mathematical concepts to his art, and it was because of this paper that he was accredited among mathematicians. “In it, he studied color based division, and developed a system of categorizing combinations of shape, color, and symmetrical properties. By studying these areas, he explored an area that later mathematicians labeled crystallography, an area of mathematics” (M. C. Escher, Perception, Geometry, Thinking ​ Outside the Box). ​ Ochoa 5 ​ Studying crystallography is the science of determining the arrangement of an atomic structure, and by adopting this concept to his tessellations­he was able to further understand the correlations of space and geometry. Escher’s interest in topology also sparked multiple designs that deal with geometric shapes that are stretched and bent within a certain space. He learned from one of his fellow mathematicians, Roger Penrose, additional concepts that dealt with this bra­ Up and Down, 1947, lithograph in brown ​ ​ ­nch of math. This new knowledge was applied to Waterfall and Up and Down each ​ ​ ​ ​ of which express the ideas of impossible perspectives. Where is the water coming from? Is the floor also a ceiling? Waterfall, 1961, lithograph ​ Ochoa 6 ​ During his lifetime Escher was able to complete about 450 prints and over 2,000 drawings. The majority of which contained concepts of Euclidean and non­Euclidean geometry. However, he also expressed interest in creating art with the non­Euclidean branch of geometry. This type, specifically challenged the idea of a flat surface by introducing spherical models. Which are represented through, Lobachevskian geometry and Riemannian geometry. The first, Lobachevskian is shown through a drawing of a Poincaré disk, in which Escher was introduced to the idea that there existed more than one parallel line to any given line. A Poincaré disk actually defines “a line as the diameter of the disk, and a circular arc connecting two points on the boundary of the disk”(Johnson, Mowry, pg. 623). From there Escher was able to apply this concept to create the desired pattern on a sphere such as his Circle Limit IV. ​ ​ Circle Limit IV, 1960 Woodcut A Poincaré Plane ​ ​ Riemannian geometry brought Escher to experiment with the concept that a given line does not have a parallel line. He was then able to apply the Lobachevskian model to the Circle Limit IV and make it into a 3­Dimensional sphere without any parallel structures. Ochoa 7 ​ Riemannian Geometry model Circle Limit IV ​ Through art Escher was able to pursue his mathematical interests and create the alternate reality that all artists pursue in their works. Geometry is the basis of his tessellations and his architectural background helped him create these impossible shapes and realities. He explored the branches of Riemannian and Lobachevskian geometry as well as their counterpart, Euclidean geometry. This is why his shapes demonstrate parallelism, as well as stories that have dual­perspectives. His artworks helped influence many of today’s illusionists to create their own geometric shapes inspired by the same mathematical concepts that drove Escher to create these famous artworks. Ochoa 8 ​ Bibliography Johnson, David B., and Thomas A. Mowry. Mathematics: A Practical Odyssey. 7th ed. ​ ​ California: Charles Van Wagner, 2012. Print. Lybarger, Carey Eskridge. "M.C Escher." Escher. Web. 04 Apr. 2016. ​ ​ "M. C. Escher, Perception, Geometry, Thinking Outside the Box." M. C. Escher, Perception, ​ Geometry, Thinking Outside the Box. Web. 04 Apr. 2016. ​ "National Gallery of Art." M.C. Escher — Life and Work. Web. 04 Apr. 2016. ​ ​.