EscherMath Fall 2010

FSEM Escher Math What is Mathematical Art?

Throughout this semester we have looked at and studied a variety of forms of mathematical art.  Along the way we covered some areas of mathematics that may have been new to you – isometries and tessellations, spherical and hyperbolic geometries, iteration and fractals, and probability are a few of the topics we explored.  For this last blog assignment I want you to consider what you will be taking away with you from this class.  Do you have a new appreciation for different kinds of art?  Did you learn some new mathematics?  What are some links between mathematics and art?  If there are other things you learned please write about them.  Finally, if you were asked by a friend to describe this class, what would you tell them?  Your blog entry (as a comment to the last posting) is due Friday December 3, the last day of class.

In my project I explored the concept of random art and probability. This lead to many attempts at random art, without success. I decided to focus on the randomness of letters and frequency of letters in the English language. It was very fascinating to me to be able to get words out of using probability and the correct proportion of letters. Therefore, this is the main focus of my project.

To begin, I used the game, ‘scrabble,” as a basis for determining the correct amount of each letter I should use. There were 100 letters to choose from. Each letter had its own corresponding frequency value. The letters were drawn, one at a time, getting replaced each time as to not disrupt the value amounts. The letters were then recorded on a large piece of paper. The paper has an underlined graph and therefore, it was not a challenge to keep the letters somewhat in line with each other.

Following the recording, there were some letters that did end up spelling words. This was expected and relieving to have the expected outcome became a reality. The letters were first done in pencil and then positioned a second time to be more inline. The second time going over the letters was done in black sharpie. This was to create the simple, but randomness effect that was intended. The spelled out words were then enclosed in a box and coloured. The colouring scheme was completely random. The only constraint on the colouring was not to colour two words next to each other the same colour. The colouring was done with coloured pencil and not done too heavily.

The pentagram has many mathematical properties. The major one is the presence of the golden ratio that is roughly the Fibonacci sequence with some altercations.  The sequence allows the pentagon to have other strange properties, like dilation within its self and infinite dilation that in many cultures was thought to be magical. The dilation occurs in the center of the pentagon at a very strange dilation scale. There are many cultural meanings of this symbol from the holiest to the darkest most evil. The pentagon has many mathematical properties like the Fibonacci sequence and dilation it also has many meanings in different cultures because of the mathematical nature.

I made my picture of the pentagram by approximating the golden ratio with the Fibonacci sequence. I used sketch paper, a pencil, and colored pencils. I first made to upright lines to make the top point of the star. These lines were 21cm long because of the Fibonacci sequence {1, 1, 2, 3, 5, 8, 13, 21……..}. Then from there I made the rest of the lines making sure to have all of them to be 21cm long.  This process gave me the big outside pentagram then I just connected the points of the pentagon and made smaller pentagram. I continued this until it was hard to make any more stars.

A Penrose triangle (also called the tri-bar or impossible triangle) is quite simply a piece of mathmatical art that shouldn’t work. The Penrose Triangle appears to not have a front or a back, but three never-ending sides. The three dimensional representation of a Penrose Triangle when glanced at the correct angle appears to be a solid object. When closer examined however, it is clear that the triangle is simply a series of three bars. These bars are positioned so that when viewed at a certain angle an impossible triangle appears. The Tri-bar is an impossible object because its two dimensional picture cannot be represented accurately in three dimensions.
I first tried to construct my own Penrose triangle and failed to produce fruitful results. While doing further research online I stumbled across a printable template for a Penrose Triangle. My attempts had failed because I was trying to make each angle sixty degrees as they are in a two dimensional triangle. In three dimensions each side must be ninety degrees, further proving this illusion cannot exist in three dimensions with an angle sum of two hundred and seventy degrees. Here is the template website if you are interested in constructing your own: http://my.opera.com/zhouye.ah/blog/how-to-build-your-own-impossible-triangle-with-a-printable-cutout-template

My three-dimensional sculpture is of the majority of a hollow sphere. I decorated the sphere with a tessellation of fish. On the sphere I was able to fit two rows of complete fish. One row consists of red, yellow, and orange fish, while the other consisted of black, blue, and green fish. The spherical sculpture is made up of 630 little triangles that are all connected to form the lines of the sphere. The triangles were made by folding little rectangles a certain way so that the tips of one triangle would fit in to the openings on the bottom of another triangle, this is how the triangles were connected to each other. I didn’t use glue, tape, paint, or colored pencils to make my sculpture, the only materials that I used were construction paper, scissors, and a lot of time! The fish are formed out of three triangles for each tail, and five rows of two triangles to make the bodies. So each fish contained sixteen triangles, and there were twenty one rows of triangles in the finished sculpture, which meant that the sphere contained ninety five triangles of each of the six colors that make up the fish, and then sixty white triangles were used. My tessellation of fish on the sphere is translated, rotated one hundred and eighty degrees, and if the fundamental unit is half of one fish then it is also reflected. The fish tessellation is there fore from plane group pg. Besides just tessellating, my sculpture deals with the large amount of mathematics that goes into making origami. Origami has a very intricate mathematical background. Even though it seems like its only folding paper, there happens to be a lot of strategy behind it. For my project though I only really used geometry and tessellations. I am a very math and art oriented person so my inspiration for this project was origami, one of my favorite hobbies, and I was also inspired by hyperbolic tessellations. If my spherical tessellation is held at the right angle, one could see a hyperbolic tessellation, even if it is very small. In spite of all, my spherical tessellation is just one of many examples of how art and math converge.

For the final project, I chose to base mine off of the topic of platonic solids. We studied this in class, and I learned that there are five different polyhedra, or Platonic solids. These are tetrahedron, cubes, octahedron, dodecahedron, and icosadedron. For my project I decided to work with tetrahedrons. I created a 3-dimensional sculpture made out of thick artist’s paper. I made lots of tetrahedrons by cutting and folding paper triangles, and finally gluing the edges together so that they remained intact. After making the solid tetrahedrons, I used a sharp knife and scissors to neatly cut out triangles from all sides of each individual tetrahedron form. I left one tetrahedron solid, and used that as my base. I then glued on the rest of the tetrahedrons, stacking them in a pyramid-like design so that they would fit together easily. After gluing on all of the necessary platonic solids, I spray painted my sculpture. First with blue, and then with a light misting of silver, in order to give it a more attention-grabbing element. Other than the thick artist’s paper, scissors, a sharp knife, glue, a ruler, and spray paint, my time and imagination were the only other materials used.
Because platonic solids involve fairly simple mathematics, there wasn’t a large amount of math behind my project. However, out of the 5 possible platonic solids to use, I chose to work with the tetrahedron because I find it’s shape interesting to observe, yet simple to create at the same time.
My inspiration or motivation for this project was to create something to was pleasing and eye-catching to all who viewed it. I wished to inspire and reach people not only on a matematical level, but on an artistic one, as well.

For me to create this design, I used poster board, glue, scissors, and a coin. I split the poster board into 336 1 inch by 1 inch squares. For each individual square, I flipped a coin and if it came up heads, then I pasted a logo on the square. This was then repeated 335 more times and ultimately, I ended up having 170 “successes” or heads.
In random art, there are many ways that mathematics can be incorporated throughout the process. One example involves probability. Probability is a way of expressing knowledge or belief that an event will occur or has occurred. Probability is associated with a random phenomenon for which the observed outcome cannot be predicted. I used an example of this when I flipped a coin. I knew that the outcome will be either heads or tails, but I never knew which would come next. Probability values always range from within 0 to 1 and their sum assigned to an outcome must add up to one. The other use of mathematics includes the concept of a binomial distribution. In a binomial distribution, there are only two outcomes. For my project the two outcomes were heads or tails. All heads are described as “successes,” while tails were “failures.” The number of repetitions for the project can be shown as the variable “n.” The numbers of observed successes are translated into the variable “k.” For example, the probability that I received my results of 170 successes can be shown as;
P(170) =.628
Therefore, there was a 62.8% chance that I would have gotten those results.
My motivation was looking at all those simple, yet effective designs that Dr. Hydorn showed us in class. I thought that this type of artwork is extremely attractive because your brain automatically looks for patterns, but the design is actually completly random.