Research Question: What would Maurits Cornelis Escher’s Regular Division of the Plane with Birds look like on the torus?
Maurits Cornelis Escher was born in Leeuwarden, Holland in 1898. He showed an interest in design and drawing, and this led him to a career in graphic art. His work was not given much recognition until 1956 when he had his first important exhibition which led him to worldwide fame. He was inspired by the math he read about and his work related to those mathematical principles. This is interesting because he only had formal mathematical training through secondary school. He worked with non-Euclidean geometry and “impossible” figures. His work covered two main areas: geometry of space and logic of space. They included tessellations, polyhedras, and images relating to the shape of space, the logic of space, science, and artificial intelligence (Smith, B. Sidney). Although Escher worked with a wide variety of art, the main focus of this paper will be tessellations. This brings me to my research question: how does Maurits Cornelis Escher’s Regular Division of the Plane with Birds relate to the tiling view of the torus?
Tessellations and the torus are related to mathematics in the areas of geometry, topology, and the geometry of space. “A regular tiling of polygons (in two dimensions), polyhedras (three dimensions), or polytopes (n dimensions) is called a tessellation.” (Weisstein, Eric W.). Tessellations, or regular divisions of the plane, cover the entire plane without leaving any gaps or overlapping (http://www.mathacademy.com/pr/minitext/escher/). The word “tessellate” comes from the Greek word “tesseres” which means four in English. This relates to tessellations
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While the studies at Governor’s School are noticeably more advanced and require more effort than at regular public schools, I see this rigor as the key to my academic success. For me, the classes I take that constantly introduce new thoughts that test my capability to “think outside the box”, are the ones that capture all my attention and interest. For example, while working with the Sierpinski Triangle at the Johns Hopkins Center for Talented Youth geometry camp, I was struck with a strong determination to figure out the secret to the pattern. According to the Oxford Dictionary, the Sierpinski Triangle is “a fractal based on a triangle with four equal triangles inscribed in it. The central triangle is removed and each of the other three treated as the original was, and so on, creating an infinite regression in a finite space.” By constructing a table with the number black and white triangles in each figure, I realized that it was easier to see the relations between the numbers. At Governor’s School, I expect to be provided with stimulating concepts in order to challenge my exceptional thinking.
“Relief of a Winged Genius.” Museum of Fine Arts Boston. N.p., n.d. Web. 9 Nov. 2013.
Throughout the vast history of visual art, new movements and revolutions have been born as a result of breaking past conventions. This idea of moving past traditional styles was done by many artists in the 1950s and 1960s, including those artists who participated in the many different abstract movements. These artists decided to abandon old-fashioned techniques and ideas such as those of classical Renaissance, Baroque, or even Impressionist art. One of these new conventions, as discussed by art historian Leo Steinberg in his essay, “The Flatbed Picture Plane,” is the concept of a flat and horizontal type of plane in a work that does not have a typical fore, middle, or background like that of the traditional art from classical periods previously mentioned. The flatbed picture plane that Steinberg refers to is similar to that of a table in which items can be placed on top of, yet they are merely objects and do not represent any space. In his article, Steinberg explains that the opposite of this flatbed plane is the
In the science-fiction short story “And He Built a Crooked House” by Robert A. Heinlein, a mathematically inclined architect named Quintus Teal constructs a house based on the unfolded net of a tesseract in order to save on real estate costs. However, to Teal’s dismay, an earthquake occurs the night before he shows a friend the house, and the house had fallen through a section of space and seemingly had been shaken into an actual tesseract. Despite its mathematical basis, “And He Built a Crooked House” is a quality example of science-fiction.
(Misturelli, F. and Hefferman, C., 2008). I wrote this paper in a way that challenges you to put
Visual demonstration of the Delian Problem." Aesthetics No. 13 (2009): 179-194. Japanese Society for Aesthetics . Web. 1 May 2014.
This paper will discuss three specific instances: Le Sacrifice, Psappha, and Metastasis. The first principle that I will discuss is the Golden Section. The Golden Section can be found in art and architecture dating as far back as the Parthenon, as well as different places in nature, such as the nautilus shell. The Golden Section is essentially a proportion that is established by taking a single line and dividing that line into two separate sections of unequal lengths, one quite longer than the other.
Nevertheless, that day followed me, and I tried to understand more about fractals through the resources I already had at my disposal-- through courses I was taking. Sophomore year, through my European History and Architecture courses, I learned about many ancient architectural feats-- Stonehenge, the Pyramids of Giza, the Parthenon, many Gothic Cathedrals, and the Taj Mahal-- and that they all somehow involved the use of the golden ratio. I will come back to how this relates to fractals later in the article, but for now know that each of these buildings use different aspects of their design to form the golden ratio. I was intrigued by the fact that fractals, what seemed to be something only formed by the forces of nature, were being constructed by human hands. Although I wanted badly to find out more, I waited until that summer, when I discovered a YouTube account by the name of Vihart. Vihart’s videos are not tutorials on how to do math, however Vihart’s ramblings about the nature and the concepts of the mathematical world have a lot of educational value, especially on topics that are more complicated to understand then to compute. Her videos on fractal math and their comparability to nature, helped to show me that...
Josef Albers was a well-known and influential artist of the twentieth century. He was known for his use of vivid colors and interesting and abstract shapes. He was instrumental in ushering in the Modernist movement as he was a teacher to many of the great artists of the 1950s and 1960s. In 1963, Josef Albers released a book surrounding a series of paintings he did, The Interaction of Color. This book was crucial when it came to art education and various applications in his and his student’s works. His final series was his Homage to a Square that only used squares and rectangles with varying colors to demonstrate spatial relationships between the shapes and the colors. Albers use of shape and color, particularly in his Homage to the Square
"The Foundations of Geometry: From Thales to Euclid." Science and Its Times. Ed. Neil Schlager and Josh Lauer. Vol. 1. Detroit: Gale, 2001. Gale Power Search. Web. 20 Dec. 2013.
... middle of paper ... ... Berk, L. (2007). The 'Standard'.
Schattschneider, Doris. “The Fascination of Tiling.” The Visual Mind: Art and Mathematics. Ed. Michele Emmer. Cambridge: MIT Press. 157-164.
Fractal Geometry The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it provides an answer, a comprehension, to nature, the world, and the universe.
The abstractions can be anything from strings of numbers to geometric figures to sets of equations. In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world. A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced. Mathematicians are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest richness or complexity but on the contrary, in finding the greatest economy and simplicity of representation and proof.