It is the first week of March, time for galleries from all over the world to display art at one of the half dozen large fairs in New York City. Since a lot of my own work involves paper, it makes sense that my first stop this year was the Art on Paper Fair. Here is just a quick overview of some of the work I thought had interesting mathematical connections.
Walking down Mulberry street I spotted this great sign in front of The Picture Room McNally Jackson Store.
The sign is the work of Benjamin Critton. It features a series of squares whose sides increase based on the Fibonacci Sequence. The first two squares are the same size. The third square has sides twice the side of the first. The fourth has sides three times as long as the first. This continues until the 7th square has sides 13 times longer than the first. They all are spiraled into a neat Fibonacci rectangle with sides in an 8:13 ratio.
Dikko Faust has been making prints using rectangular sections of grids and other geometric line patterns. By shifting the grids across the plane he has created a series of overlapping prints. Recently he has added a new twist to his process. Faust has invented a new printing tool that allows him to rotate the rectangle around a central axis point.
(A quick note about printers’ measurements: In the print studio distances are measured in picas and points. One inch is equivalent to 6 picas and 1 pica is equivalent to 12 points.)
To measure the rotation of the rectangle, Faust uses a straight edge to form a line from the bottom corner of the rectangle that is perpendicular to the horizontal bottom edge of his press, and then measures how far from the center point to the horizontal line. The initial measurement for a straight up and down rectangle would be 12 picas from the center (the rectangle is 4″x 6″ or 24 by 36 pica).
Faust has been experimenting with what happens to different patterns throughout the rotation process
To better explore the relationship between the grids,Faust has made series of two-color prints. He has selected only the prints that are the most visually interesting. Making consecutive prints with the number of ratio of pica differences to correlate with the Fibonacci Sequence is one technique.
The day I was in the studio, Dikko was working with a pattern he had created using airline (1/2 point) rules. He used parallel lines: there is 1 point of space between the first two lines, 2 points between the 2nd and 3rd line, then 3 points between the 3rd and 4th….. up to 6 points of space between the 6th and 7th line. Then the whole pattern repeats 12 times.
While I was at the printing studio Faust was making a single print with multiple rotational images. I took pictures throughout the process.
This is an early stage of the process: it has the original line print plus a 5 pt and 10 pt rotation clockwise and a 5pt and a 10pt rotation counter clockwise.
This is the finished print. There are 5pt, 10 pt, 15 pt, 20 pt, and 25 pt rotations in both the clockwise and counter clockwise directions. The process that Faust has developed to create these new prints is very algorithmic. It requires a commitment to experimentation trying different patterns and rotations. The outcomes are then judged on their aesthetic merit determining which prints are to be completed works of art.
The Cooper Hewitt, Smithsonian Design Museum in Manhattan was closed for renovation for three years before it reopened at the end of 2013. The current exhibition features an overview sample of their vast collection. I was very happy to discover that they have chosen to display quite a bit of work with direct Mathematical links. The debate over the critical delineations between Fine Art and Design is a hot button issue I am not going to address in this blog post. I have selected two pieces that have specific Mathematical themes.
“Prototype for an Environmental Screen, Fibonacci’s Mashrabiya”, 2009 is an architectural element designed by Neri Oxman at MIT Media Lab with Professor W. Craig Carter. It is was created using algorithms and digital processes but is based on traditional screens found in historic middle Eastern design.
The recursive Fibonacci Sequence was used to create the spiral pattern. Here is a detail of the center of the spiral.
Mathematician and artist Daina Taimina has been quite well known for her crocheted sculptures of Hyperbolic Geometry.
“Model of a Hyperbolic Space” 2011, is crocheted out of wool yarn. Working on these sculptures since 1997, Taimina has made major breakthrough on the modelling of figures in Hyperbolic space. Hyperbolic Geometry is a Non-Euclidean Geometry discovered by Janos Bolyai and Nicholay Lobatchevsky in the first half of the 19th century. In Hyperbolic Geometry each point has negative curvature and seems to curve away from itself.
At the Cooper Hewitt there were many more items that featured Mathematics as a design element. There was a very direct indication of the importance Mathematics plays in the field of both decorative and industrial design.
Summertime is a time to relax the rules. During most of the year my drawings require the use of grids and calculated templates. In the warmer months, when I am away from my studio, I continue to draw, but using a more organic approach. I have created two new types of small scale drawings based on the Fibonacci Sequence. These works are more about counted iterations then measuring. This allows the patterns to grow and develop more freely across the paper.
The first type of drawing I am calling Fibonacci Fruit. This type of drawing features pod-like forms with internal structures based on the consecutive terms of the Fibonacci Sequence. Here are two examples using the numbers 5 and 8.
In the first drawing there are 13 pods each divided into 8 segments and each segment contains 5 seeds.
Another type of new drawing I am calling Fibonacci Branches. In these drawings one branch divides into two new branches. Those branches each divide into three branches, then those branches each get five branches, then each of those gets eight branches until finally each of these branches gets thirteen new branches.1, 2, 3, 5, 8, 13. This creates a treelike arrangement.
In the next example, five sets of branches are scattered across the page. Each branch formation starts with one branch and grow in a similar fashion to the other drawing but in this case the final branch count is eight.
There are still a multitude of possibilities for the continuation of these two drawing series. It will be exciting for me to see where the Fibonacci Sequence will take me next.
It is my personal mission as an artist to illuminate the intrinsic beauty of mathematics in a purely aesthetic realm. Translating mathematical subject matter to the picture plane of my drawings, I strive to enable viewers to appreciate this aesthetic, regardless of their mathematical background. I express the grace and beauty I find in mathematics through symmetries, patterns and proportions in my art. Many of my drawings are related to growth patterns such as the Fibonacci sequence and binary growth. I begin my work process by creating a plan or an algorithm. I make all of the decisions for the work beforehand and make a detailed plan in a large spiral drawing tablet that I refer to as my plan book. After I write out all of the specifications, I generate the actual drawing by hand using the rules from the plan. Through my drawings I hope to express both the aesthetics of my mathematical subject matter, as well as the aesthetics of the process of algorithmic generation.
In the past few years I have become interested in generating drawings using fractal forms based on the repetition of similar shapes. I begin with a largest instance of a shape and incorporate copies scaled by powers of ½. I developed a drawing based on the four quadrants of the Cartesian coordinate system. Each drawing begins with 8 spokes. The line segments fall on the coordinate axes and the lines y=x and y=-x. Once I have drawn the initial shape, each spoke becomes the starting point for a new 8-spoke shape in which the line segments are ½ as long as the original spokes. Then those 64 spokes become the starting point for 8-spoke figures with line segments ¼ the length of the first line segments. Next, the 512 spokes each become the bases for an 8-spoke shape with line segments 1/8 the length of the original spokes. This process creates a circular fractal network of lines. While producing these drawings, I have developed a type of mantra to remember where I am in the drawing. I need to keep count and this becomes quite complicated and rhythmic, especially when I reach the third iteration.
Mathematics and art both enable humans to better understand the world around them by uncovering patterns and structures. Chaos Theory is one of the topics in mathematics that, I feel, particularly throws light on the intricacies of the human condition. Chaos Theory shows that even within apparent disorder there can often be found both order and structure. My investigation took me to the earliest ideas on Chaos Theory. In 1961 Edward Lorenz inadvertently discovered the phenomenon of sensitive dependence on initial conditions by noticing the effect of rounding off decimals had in a computer-generated sequence of calculations for weather prediction. This event marked the (re-) discovery of what is now commonly known as Chaos Theory. I decided to visually interpret this phenomenon in my drawings, by using my basic 8-spoke pattern and continuing with multiple iterations using stencils with a small margin of error. The errors accumulate to create these cloud-like, chaos- derived drawings. If the viewer spends a few moments gazing into what at first appears to be a chaotic cloud they will begin to see the pattern of the fractals develop. There is a hidden structure to these drawings, as well as a sense of growth through time. This process of layering iteration on top of iteration takes weeks of work and through the process the drawings go through interesting changes and developments. I wanted a way to incorporate this sense of time and change into my art. It was time to make a movie.
I started with a fresh large black sheet of paper. Then I installed a digital camera over my drawing table. I began my drawing process, but after each line I took a still shot of the drawing. I continued this process over months. I wanted the movie to have an organic handmade feeling to it so I made a number of changes throughout the process. The frequency with which I photographed the drawing fluctuated. Sometimes I would take a picture after each line, sometimes I would complete a small cycle of lines before taking a picture. This change produced skips and jumps in the rhythm. Occasionally, I moved the camera closer to or farther away from the drawing. I also included myself in the photos as the generating mechanism: there are a few shots where you can see my hands. At a point where the drawing was getting quite complicated, I adjusted the camera so you could see my feet coming and going from view: the drawing was becoming a dance. Leaning over to draw and then pulling away to take a picture created a very physical element to this work and I wanted to express that physicality. Thousands of still digital photographs were taken during the drawing process. These photographs were put into consecutive order and then repeated in reverse to create the sense of both growth and decay. The edited product is a 6 minute video titled “Chaos Night”.
I knew from the beginning of the process that I would add music into the final production. I contacted composer Max Schreier, and discussed the structure and mathematics I wanted incorporated into the music. I wanted to make sure the number 8 played a major role in the structure of the music to mirror the 8 spokes of the drawing. Max agreed to write and perform a 6 minute composition based on these specifications. Influenced by Arnold Schoenberg, he based the music on a series of 8 sequential notes. While the bottom voice of the organ plays a drawn out rhythm associated with the first iteration of the drawing, the violin accelerates with the increased speed of the smaller iterations. The right hand of the organ creates small disturbances, each catalyzed by the random insertions of hands, feet and rulers in the video.
– Susan Happersett
Originally presented at Bridges Art Exhibition – Banff, Canada – July 2009;
“How does an artist take inspiration from a Mathematical concept and transform it into a work of art?”
This is a question people have asked me many times. Each artist follows her own path, but translating the aesthetic elements of a mathematical topic into the visual realm of Art is my personal journey. I will discuss the process I developed to to create my most recent series of drawings, which I refer to as “Fibonacci Circle Curves”. I will map the artistic process from my selecting a Mathematical theme, through the many steps it takes to complete a drawing. This is a process that took 18 months to develop.
Through the years I have made many drawings exploring the Fibonacci Sequence. The recursive nature of the sequence makes it an interesting subject for abstract drawing. My new series of drawings investigates the visual qualities of intersecting circles whose area measurements are in proportions related to the Fibonacci Sequence. This experiment is a different way to look at the ratios of consecutive Fibonacci numbers.
The measurement of the area of the first circle in the sequence determines the area of each subsequent circle.The measurement of the area of the second circle is the same as that of the first circle. The measurement of the area of the third circle is twice the first. The measurement of the area of the fourth circle is three times the first. The measurement of the area of the fifth circle is five times the first, etc. This series of circles illustrates the Fibonacci Sequence: 1,1,2,3,5,8…, though the measurements of their areas.
I made templates for the first eight circles in the series and started to experiment. I started off by drawing the circles in a straight line. I drew the first circle and marked its center point. then I began the second circle at that center point. Then each subsequent circle started at the center point of its predecessor. In this format it is possible to draw a straight line connecting the center points of each of the circles. I immediately noticed there were some aesthetically interesting shapes created by the intersecting circles, but I was not satisfied. I decided to continue to manipulate the circles. I broke up the straight line connecting the center points into angled line segments. Instead of having the center points of the circles line up, the line segments connecting the center points should create angles less than 180 degrees. After some time it became clear that the best angle to use was the Golden Angle. The golden angle has a measurement of approximately 137.51 degrees. It is the smaller of the two angles formed by two radii that divide the circumference of a circle into two arcs so that the ratio of the measurement of the large arc to the small arc is equal to the ratio of the measurement of the total circumference to the measurement of the larger arc.
After curving the series of circles, the space created between the arcs started to look much more interesting. I was still not satisfied with the image, however. I began a process of using this curve as my basic building block. I made a number of curves on transparent paper and I began to superimpose and shift the images. I did not want the drawing to look static but wanted the image to have a sense of movement. I came up with a method of drawing using the line segments created by connecting the center points of adjacent circles. Using these line segments as a guide, I dragged the template of the first circle, so that the center point stayed on the guideline. Then I drew multiple circles until the first circle was completely inside the second circle, sharing one circumference point. I repeated this with each of the circle templates. The finished product was finally an image with potential.
This elegant structural unit is the starting point for all of this new work. I have made numerous drawings using multiple Fibonacci circle curves. either shifted or rotated or, and superimposed on top of each other, creating some surprising interactions. I continue to explore the shapes produced through this process. I have made work emphasizing the negative spaces, painstakingly filling in between the lines. By cutting up the drawings and rearranging the sections I have made collages and Artist’s books allowing the viewer to focus on small sections of the curve.
I hope this detailed explanation of my artistic practice offers an interesting behind-the-scenes tour of my process, beginning with my thinking about Fibonacci ratios and circles, and progressing through experiments leading to new drawings.