A resident of Leeuwarden, Holland, and Maurits Cornelis Escher was born in 1898. He became recognized because of his exclusive and captivating mathematical creations in the works of art that explore and demonstrate a broad array of mathematical thoughts. Against the will of his family to turn into an architect, he had the talent for drawing and design, which eventually led him to a career in the graphic arts. His work was not much known till mid 50’s when in 1956 his foremost important exhibition, was published in Time magazine, which attained a universal repute. Mathematicians were one of the leading admirers, who acknowledged his effort as an amazing vision of mathematical ethics in spite of the fact that Escher had no proper mathematics education except till secondary school [Platonic Realm].

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The more his works got matured, the more he got stimulated from the already held mathematical ideas while understanding the geometry of structures and eventually utilized the geometrical principles in his master pieces. He looks enthralled with impossibility of the figures, and produced some intriguing works of art. M.C. Escher possessed exceptional visualization and perceptions. A number of the Escher’s mathematical ideas are not found somewhere else, specifically the interlocking shapes of people, birds and fish and reptiles, which recur on an even surface with no space in between.

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### The Mathematical Art

When asked about his style by an interviewer, Escher had replied about his aspiration to fill the space and his desire to fill a restricted level of surface without gaps that stirred much of his work [Biographical Sketches of Mathematical Giants]. His desire was further enhanced when he visited the city of Granada in 1922 where he happened to visit Alhambra palace, which had been built in the fourteenth century as a house for the last of the Moorish courts in Spain. He was fascinated by the designs of the building specifically the walls and floors which were wrapped with complex designs that were simple, yet at the same time were very detailed. The regular divisions of the plane, called “tessellations,” are arrangements of closed shapes that completely cover the plane without overlapping and without leaving gaps. These are the polygons or similar regular shapes, such as the square tiles often used on floors.

Escher, however, was fascinated by every kind of tessellation, regular and asymmetrical, and took special delight in what he called “metamorphoses,” in which the shapes changed and interacted with each other, and sometimes even broke free of the plane itself. This had motivated him to fill a plane with repeated contiguous shapes having no recognizable human or animal form and other simple geometric shapes. Un aware of the mathematical principles behind his work he continued working till until1940 he began his work for depicting standard partition of a flat surface. The work of Escher primarily contained a type of two-dimensional crystallography, which at present has become a science dealing with the formation and structure of crystals and rocks etcetera.

Having read works of other artists i.e. Polya, he used some basic geometric motions to create regular division of a plane including rotation, translation, reflection and glide-reflection. He worked on the development of a total theory for the regular division of a plane with a similar pattern of shapes that were not convex polygons. Ultimately he could reach his theory of “quadrilateral systems” in 1941, which was published, in his book “Regular Division of the Plane with Asymmetric Congruent Polygons.” He also worked on the creation of images that reduce in size until they became considerably minute. To create these prints the geometric principle of similarity is utilized in which the exact same shape but decreased in size is obtained.

It stands proved that of all the regular polygons, only the triangle, square, and hexagon can be used for a tessellation. Escher used these basic patterns in his tessellations; applying what geometers would call reflections, glide reflections, translations, and rotations .He also made these patterns by “deforming ” the fundamental form to depict them into animals, birds, and other figures. Escher’s work also enclosed a diversity of themes all through his life. His initial descriptions, Roman and Italian landscapes and of nature, finally led to expected partition of the plane. Many of his mathematical works were obtained from extraordinary view creating mysterious spatial special effects. Over 150 vibrant and identifiable works confirm the Escher’s originality and curiosity in regular division of the plane. [Maurits Cornelius Escher]

### Conclusion

Escher throughout his life saw the beauty in structure and infinity and forced the idea of meaningful lines into the mathematical framework of regular plane division. He liked to challenge the logic of seeing. One can see the mathematical divisions in form of the white birds and regard the black as background. The black and white horses can be seen separately and the black fish can be regarded as the white as background.