Chaos – The Movie

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;

More Chelsea Galleries – February

Robert Morris at Sonnabend Gallery

Sonnabend Gallery is exhibiting large wooden sculptures by Robert Morris. Morris is one of the most important American artists and preeminent practitioner of Minimalism. The twelve sculptures in this show are from his “Hardwood Series” and they are all recent reinterpretations of plywood constructions from the 1960’s. Craftsman Josh Finn facilitated the actual production of the work. I was particularly drawn to three totem-like sculptures that were each stacked columns of square planks. In “Serrated Column” (2012) each consecutive plank is rotated 90 degrees. Each square has diagonals that are parallel to the sides of the squares above and below.

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Morris – Serrated Column – 2012 – Wood

“Twisted Column” (2012) is a stack of  40 squares that rotate a total of 90 degrees. Each square is only rotated 2.25 degrees. This subtle rotation gives the illusion that it is a smooth surface instead of separate square planks.

Morris - Twisted Column - 2012 - Wood

Morris – Twisted Column – 2012 – Wood

In “Spiral Column” (2013) the squares are rotated around a corner instead of the center. One full turn of the spiral is formed by planks. This work is an engineering marvel. Standing in front of this sculpture in the gallery it seems like magic that it does not tip over. Morriss’ column sculptures illustrate the many visual possibilities that can be explored using the repetition of a single geometric element.

Morris - Spiral Column - 2012 - Wood

Morris – Spiral Column – 2012 – Wood

Beth Campbell at the Project Room at Josee Bienvenue Gallery

In the Project Room at Josee Bienvenu Gallery, Beth Campbell is exhibiting her drawings and mobiles in an exhibition titled “My Potential Futures”. The works on paper are handwritten text-based diagrammatic drawings. The wire mobiles are a 3-D extension of the drawings. The structure of the mobiles create a binary fractal pattern. Each mobile is attached to the ceiling by a single wire that then divides into two wires, then each of those wires split again into two wires each. The 4 wires split into two wires each (now 8 wires). This continues through 7 iterations. Start at the top and then there is a choice of two possible routes, a yes or no question or ones and zeros if you are thinking in binary code.

Campbell - Mobile

Campbell – Mobile

Chelsea Galleries – February

Paul Glablicki at Kim Foster Gallery

On my recent visit to the Chelsea gallery district in Manhattan I noticed a number of exhibitions featuring art with Mathematical influences. At the Kim Foster Gallery there is a show of exquisite drawings by Paul Glabicki based on Einstein’s Theory of Relativity. These works have layers of scientific data, charts, and mathematical formulae. In drawing RELATIVITY #8 Glabicki has drawn a series of Pascal’s Triangles in the mix of images.

Glabicki- RELATIVITY #3email

Glabicki- RELATIVITY #8

A Pascal’s Triangle is a triangular array made up of numbers. The number of terms in each row corresponds to the sequence number of the row. For example, the first row has one number (1) , the second row has two numbers (1,1), the third row has three (1,2,1). In Pascal’s Triangle, the first and last term of each row is 1. The middle terms are calculated by adding the two numbers directly above. Here is an example of a small Pascal’s Triangle.

              1
          1      1
      1       2     1
    1     3      3    1
  1     4     6     4   1
 1    5   10    10   5    1

In RELATIVITY #3 Glabicki has drawn Geometric studies of internally tangent circles. These circles share only one point and the smaller is inside the larger.

Glabicki-RELATIVITY #8email

Glabicki – RELATIVITY #3

What I think is fascinating about these drawings is the way the mathematical and scientific elements are used as small pieces of the total work. They are transposed from abstract ideas into aesthetic elements of a much larger complex picture:  the artist’s expression of the Theory of Relativity exploring the physics of time and space, through the arduous process of  intense layering of images. Paul Glabicki is well known for his experimental animated films that use hand drawings.They have appeared at many film festivals and exhibits, including at the Whitney Biennial and the Venice Biennale.

Austin Thomas at the Hansel and Gretel Picture Garden

At the Hansel and Gretel Picture Garden there is a show of work by Austin Thomas. Exhibited are twelve drawings on paper, all of which have interesting proportions and geometric elements. The work that seems to really express mathematical principles is a sculpture the artist refers to as a “steel drawing”, made of two black and two white rectangular prisms. These are 3-D line drawings and by stacking them in this perpendicular fashion Thomas presents a nice study on squares and rectangles. Viewing the work from different angles and positions throughout the gallery offers many possible relationships between shapes, as well as the positive and negative spaces created by the open structures. Although it does not move, this is not a static sculpture. The prisms can be stacked in other ways offering Thomas a multitude of permutations to explore.

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Thomas – Steel Drawing

Fibonacci Circle Curves

“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.

EPSON MFP image

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.

EPSON MFP image

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.

EPSON MFP image

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.

EPSON MFP image

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.

Fibonacci Circle Curve Red

Fibonacci Circle Curve Red

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.

– FibonacciSusan

Symmetry in Antique Peruvian Textiles

Last week I visited the Metro Show in New York. This is an art and antiques fair where 35 dealers display a wide range of items including folk art, outsider art, and ethnic antiquities. I did not necessarily expect to find Mathematical Art in this venue. Much to my surprise the first thing that caught my eye, as I walked into the exhibition hall was a Peruvian textile in the William Siegal Gallery  area, made by weavers from the Nasca Culture (sometimes spelled “Nazka”) from the Southern coast of Peru. It was made somewhere between 200-600 AD from camelid wool and natural dyes. This Stepped Mantle has interesting symmetrical properties. If you only look at shapes and ignore the colors, this is a great example of order 2 rotational symmetry, also called a “point symmetry”. Rotating at any point where all four colors meet you can rotate the four rectangles 180 degrees and still have the same pattern (disregarding colors):

Nasca Culture (200-600 AD) - Camelid Wool and Natural Dyes

Nasca Culture (200-600 AD) – Stepped Mantle – Camelid Wool and Natural Dyes

On another wall in the booth of the William Siegal Gallery there was a Stepped Cushma (one piece dress) also Nasca 200-600 AD. This textile demonstrates reflection symmetry,  also referred to as “mirror symmetry”. There are 7 vertical lines of symmetry that can be drawn through this example. If you consider each on the four columns of V-shaped chevron patterns, they have lines of symmetry through the center. Then, each of the two pairs of adjacent columns have a line of symmetry between them. Finally, the complete textile has a line of symmetry down the middle:

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Nasca Culture – Stepped Cushma – Camelid Wool and Natural Dyes

The moral of this blog is to keep your eyes out for Mathematical Art everywhere. The connections between Mathematics and Art can be found in unexpected places.

– FibonacciSusan

Experiments in Art and Technology

E.A.T Experiments in Art and Technology 1960 – 2014 is the current exhibition on display at the Payne Gallery at Moravian College in Bethlehem, Pennsylvania. This small show documents the collaborations of artists with scientists and engineers from Bell Labs in NJ. Two Bell Labs engineers, Billy Kluver and Fred Waldhauer, started working with artists, providing them access to the newest technology. In 1966 they helped bring together 30 scientists and engineers with 11 artists to produce a cutting edge performance art series called 9 Evenings: Theater and Engineering in NYC. Through these partnerships, the engineers were trying to do two things. They wanted to address the effects of technology on society, and they were looking for new ways to explore this technology. Not all of the work was performance art, it also included  sculpture, drawing and architecture.

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The Brooklyn Museum – Experiments in Art and Technology – Poster – 1968

What does this have to with Math Art? If you look at the time line for these collaborations you see that in 1966 computers were the new technology. Some of the art work done in these experiments was based on Mathematical algorithms.

Robert Rauschenberg

Robert Rauschenberg was one of the artists closely involved with E.A.T. One of his projects was a series of six “Revolvers”. “Revolver II” from 1967 is on display in the center of the gallery. It consists of 5 plexiglass circles that have been printed with silk screen. They rotate independently when one of five buttons is pushed. Because the circles are transparent, the different rotations (1, 2, 3, 4, or 5 circles at a time) create interesting geometric patterns.

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Rauschenberg – Revolver II – Silk screen on plexiglass – 1967

Andy Warhol was also involved with these Art Technology Experiments. On the second floor of the gallery there are 3 prints of Mao from sequentially produced photocopies made in 1973. Warhol started with a photocopy of a drawing of Mao Tse-tung and made a photocopy of the photocopy, then a photocopy of the second photocopy, etc. The copying machines 40 years ago did not make true to size copies, each copy enlarged the image .01%. By the 300th iteration of his process only an enlarged abstract portion of Mao’s face was visible.

Also in the exhibition are photos and ephemera from the largest collaboration undertaken by the engineers and artists: The design for the Pepsi Pavilion at Expo’70 in Osaka Japan. The exterior of the structure was a white Geodesic dome shrouded in a Fujiko Nakaya cloud sculpture. The photos made the structure look ethereal, almost delicate, like a folded paper origami dome. The interior was lined with Mylar, and there were some great photos of  visitors in the spherical mirror room showing unusual perspectives.

Although there was no mention of Mathematics in the wall text of the gallery, it is clear that Mathematics, through algorithms and geometry, played an important role in the creation of the art work made through Experiments in Art and Technology.

More from the JMM exhibition

A few days ago, I discussed a few of the artists exhibiting at the art show that was part of the Joint Mathematics Meeting in Baltimore. Here are my other favorites from that show.

Robert Fathauer

I have been a fan of Robert Fathauer‘s sculptures for years, but I feel Three-Fold Development is one of his best works. This ceramic vessel has a top lip sculpted to depict the development of a fractal curve through five iterations. Starting with a circle, then a three-lobed curve, then a nine-lobed curve. In each subsequent iteration the number of lobes triples.The sculpture has a wonderful organic quality, while still maintaining an elegant complexity. Fathauer has skillfully kept the spacing quite even between the ribbons of clay creating a graceful relationship between the positive and negative space.

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Fathauer – Three-Fold Development – Ceramic

Margaret Kepner

Mathematics enthusiasts have been fascinated with Magic Squares for centuries. Magic Squares are grids. Each grid square contains a number. The grids are constructed so that the sum of the numbers in each column, row and diagonal of the square are equal. Margaret Kepner‘s Archival Inkjet print “Magic Square 8 Study: A Breeze over Gwalior” is a an intriguing representation of a Gwalior Square: an 8 by 8 magic square which contains the numbers 0 to 63. The sums of the rows, columns and diagonals all  add up to 252. Kepner has translated each of the numbers 0 to 63 into graphic patterns using her own system, and formatting the numbers in either base 2 or base 4. The resulting print has a great optical effect of patterned color block grids that are both horizontal, vertical and across the diagonal. It reminds me of a Modernist quilt or a contemporary twist on some of Al Jensen’s paintings that resemble game boards. Kemper refers to her artistic process as “visual expression of systems”. I think that this print goes beyond merely expressing the Gwalior Square it celebrates the Mathematics in a bold field of shape and color.

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Kepner – Magic Square 8 Study: A Breeze over Gwalior – Inkjet print

Petronio Bendito

At the Art Exhibition at the JMM conference quite a bit of the art was digital printing on paper. Petronio Bendito – in contrast – prints his work on canvas, giving the prints more of a painterly feel. Bendito has developed algorithms to define his color palette, but there is also an element artistic expression in establishing the final images.”Color Code, Algorithmic lines n.0078″ is so vibrant that it beckoned me from across the room. Bendiito’s use of color and line creates  a cacophony of bright straight and curved thin ribbons of paint. The use of  the black background makes the exuberant frenzy of color jump out to the viewer.

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Bendito – Color Code, Algorithmic lines n.0078 – Digital print on canvas

Lilian Boloney

Lilian Boloney is a textile artist who uses crocheting to explore the geometry of hyperbolic figures.There is an elegant simplicity to the off-white cotton thread she used to crochet the sculpture “Boy’s surface”. This allows the viewer to explore the complex topology of the figure with out the distraction of patterns or color. Boloney not only has a clear understanding of her Mathematical subject, but she transposes their beauty into graceful objects. Instead of models of Hyperbolic figures I see them crocheted portraits.

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Boloney – Boy’s surface – Crocheted cotton

I hope you enjoyed the samples of work from the JMM exhibition as much as I did. The Art Exhibition at the JMM conference was organized by the Bridges Organization, an international organization that promotes the relationship between Art and Mathematics. Each year they have a conference where Mathematicians, Artists and educators meet to discuss, explore and learn about Math Art.

This year’s Bridges 2014 conference will take place in August in Seoul, South Korea. This is the first Bridges conference to take place in Asia. It is a wonderful opportunity. I encourage all artists who are interested in Mathematics to attend and participate at this conference. The deadlines for paper and art submissions are fast approaching all info is on the Bridges website.

-FibonacciSusan

Mathematical Art

In this blog, I will be sharing my observations on Mathematical Art that I see in galleries museums exhibitions and art fairs. What is Mathematical Art? I will choose work that meets at least one of the following three criteria: The art

  1. is based on a Mathematical phenomenon, or
  2. it is generated by a Mathematical process, or
  3. it is a personal response to Mathematics by the artist.

JMM – Baltimore 2014

Each year in January, thousands of Mathematicians gather at the Joint Mathematics Meeting (JMM) to discuss current issues in their field.  For the past 11 years, an exhibition of Mathematical Art has been part of the event. This year the Joint Meeting was held in Baltimore at the convention center. The art exhibition was held at one side of the general exhibition hall.

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Joint Mathematical Meeting – 2014 Art Exhibition

I have participated in the exhibition five times in the past six years and over that time the exhibition has matured, both in the range of work exhibited, and in the quantity of interesting – or even exciting – work.

Exhibitions like this are really a mixed bag of prints, drawings, paintings and sculpture of all types. You can find full catalogs of the shows online. here I will discuss just a few of my favorites from this year’s show.

Shanti Chadrasekar

Kolam-93X93 is a painting on canvas based on the fractal patterns of Kolam drawings. Shanthi Chadrasekar has incorporated the rules of Indian Kolam drawings into her artistic practice. Kolam drawings are traditionally drawn by women, each day, at the entrance of their homes. In this painting, Chandrasekar has created an elaborate 93 by 93 dot grid with a single thread-like line that gracefully winds around each dot, completely enclosing the dots in a web. I find the intricacy of this painting mesmerizing. Spending a few moments with this work, the viewer feels as though they too could be encircled by this unbroken thread. The patterning on this painting is so dense that a small image of the entire piece will not do justice to the work so I am providing just a close up of a small section.

Chandrasekar - Kolan 93X93 - Paint on Canvas - 24" x 24" (detail)

Chandrasekar – Kolan 93X93 – Paint on Canvas – 24″ x 24″ (detail)

Karl Kattchee

Karl Kattchee has developed a unique process to use Mathematics to create his digital prints. His work starts with hand drawn abstract drawings that are then multiplied and manipulated using  a camera, a computer and a printer. He creates reflections, translations, etc. until the image appears to have fallen into chaos. Kattchee then builds patterns using these chaotic elements. What I find very interesting about these prints is that the whole process begins with what  Kattchee refers to as” abstract automatic drawings”. The freedom of this stream-of-consciousness type of drawing lends a whimsical quality to the initial pictures. After they have been subjected to all of the technical process, they retain a playful quality: the drawings dance across the page.

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Karl Kattchee

More about the art exhibition at JMM in Baltimore next time.

– FibonacciSusan