Math at the E/AB Fair

This week there are numerous art fairs in NYC that emphasize prints and artist’s books. I am participating in the E/AB fair with the letterpress publisher Purgatory Pie Press. We are exhibiting the first of series of three prints based on my Fibonacci Spiral drawings.

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“Fibonacci Spiral 1” – 2015

Using an algorithmic process of folding and tearing double-sided prints, we have made an edition of a book called “Galactic Collision, Fibonacci Spiral”. This book breaks up the spiral patterns into small segments of the curves.

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“Galactic Collision” – 2015

Bernard Chauveau Editeur brought some very interesting work from Paris including “Mineral Skin”,  a limited edition cut and folded paper sculpture by Arik Levy.

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Mineral Skin – 2013

“Mineral Skin” is a single sheet of paper that has been cut and folded to create a surface of pentagons and hexagons.

At the Wingatestudio booth Sebastian Black’s large scale accordion books are on display .

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“Period Piece, Simple Sequence” 2014-2015

“Period Piece, Simple Sequence” is a series of two sets of counting books. The first starts with one randomly placed black square on the first page. Each subsequent page has one more square, up to ten squares. The second set begins at eleven square marks and continues up to twenty.

There is a very diverse collection of work at the E/AB fair and I was quite pleased to find some work with mathematical themes.

Susan Happersett

Math meets Art at the EA/B Fair in NYC

This weekend at the Editions/Artist’s Books Fair Purgatory Pie Press will be exhibiting limited edition letterpress artist’s books featuring my mathematical drawings. I have been collaborating with Purgatory Pie Press for fifteen years and we have published numerous Mathematically themed artworks.

“Box of Growth” is a set of five small accordion books. Each features a series of my counted marking drawings based on different growth patterns created using the Fibonacci Sequence.

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Another topic we have explored is Cantor Set. “Infinity Remove” has two sides; one with self-similar gridded marking drawings, the reverse had famous quotes about Infinity.

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“Fibonacci Flower” shows the development of a Mathematically generated flower using the Fibonacci Sequence.

61-3Our most recent project is “Box of Chaos” is a series of four paper sculptures with my fractal chaos drawings.

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The EA/B Fair is free and open to the public this Friday (November 7, 2014) to Sunday at 540 West 21st Street NYC.

— FibonacciSusan

Math Unmeasured

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.

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The second drawing has 21 pods and again each pod has 8 segments with 5 seeds each.

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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.

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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.

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I am always interested in the negative space in my drawings. A good way to explore this is to make a white on black drawing.

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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.

 

Susan

Off the Wall in Chelsea

I discovered a very interesting trend at the Chelsea galleries this week. I found three different exhibitions where an artist presented drawings, paintings, or sculptures, but also built an installation work that protrudes off of a gallery wall.

Robert Curry at Bryce Wolkowitz Gallery

Bryce Wolkowitz Gallery  had a collection of Robert Currie’s perspex cases with monofilament line 3-D drawings.

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9,772 inches of Black and Red Monofilament (2013)
Picture courtesy of the artist and the gallery

In the sculpture “9,772 Inches of Black and Red Monofilament”,  Currie uses a series of threads hand-strung in grids to form angled wedges of red and black that intersect at the center, forming an area of at what – at first – looks like disorder. Upon closer inspection the consistency of the patterns becomes clear. This work has a number of mathematical connections: The careful measurement of the monofilament is a defining factor in the title for this work. Currie uses a series of grid patterns to thread the work. There are intricate geometric shapes created within the cases. The finally mathematical connection is his allusion to Chaos Theory, where there is underlying order in what at first appears to be disorder.

At the entrance and in the hall of the gallery, Currie has installed a site-specific thread drawing based on the architecture of the room “12 miles 1647 yards of Black Filament”. This work explores the gallery space using repetitive straight lines.

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12 miles 1647 yards of Black Filament
Picture courtesy of the artist and the gallery

Mark Hagen at Marlborough Chelsey Gallery

At the Marlborough Chelsea Gallery, Mark Hagen has created an aluminum and stainless steel space frame installation named “To Be Titled Ramada Chelsea #3”,  that climbs in front of his “To Be Titled Gradient Painting #35”. This geometric construction features cube formations meeting at star formations formed by 12 line segments radiating out from a central point.

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“To Be Titled Ramada Chelsea #3″in front of “To Be Titled Gradient Painting #35”
Picture courtesy of the artist and the gallery

Ryan Roa at Robert Miller Gallery

The Robert Miller Gallery is presenting a group show titled “Six Features”.  One of the artists, Ryan Roa, is exhibiting drawings that relate to fractions and geometry. In the same room he has created site-responsive installation that create a sense of movement within the space.

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Site-specific installation
Picture courtesy of the artist and the gallery

In his drawing “12X12 series #01”,  Roa has drawn a multitude of equal line segments radiating out from two opposite corners of the square, creating two equal quarter circles that overlap along the diagonal.

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12X12 series #01
Picture courtesy of the artist and the gallery

In “12X12 series #02”, the artist uses the same technique of drawing equal line segments, but in this case they radiate out from the two left corners of the squares. The circles overlap to form a pointed dome shape.  The right square is not completely filled in with lines: it  retains the curves of the circle segments.

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12X12 series #02
Picture courtesy of the artist and the gallery

It is fascinating to me how Roa has been able to create two drawings with such different proportion shapes and energy using basically the same technique by only changing one parameter.

It is amazing that within the course of an afternoon walking only a few blocks I was able visit three installations of Mathematical constructions by artists with very different practices and techniques. By expanding their formats off the gallery walls, each artist has created an exciting space to engage with the measures, proportions, and geometries that make up their work.

Susan Happersett

Dikko Faust – Tesselations

Dikko Faust is the printer and co-owner of Purgatory Pie Press, a letter press publishing company in Tribeca, Manhattan that he runs together with Esther K. Smith. Faust also teaches a course on Non-Western Art History at the City University of NY. It was his experience in looking at Non-Western patterning that has lead to his recent series of prints called Tesselations. The prints are made by hand setting bits of lead to create the pattern, using only red and black ink. Each patterned print has its own set of distinct symmetries. Today, I will discuss two prints from the series.

The first one is “Tesselation 4 -Nessonis 1: Pyrassos”. Printed on the back of the card is the following descriptive text: “A serving suggestion for a Middle Neolithic stamp seal design found in three sites in Northern Greece”:

Dikko Faust - Nessonis 1: Pyrassos - Hand set block print - 2012

Dikko Faust – Nessonis 1: Pyrassos – Hand set block print – 2012

I see this print as a fragment of a wallpaper symmetry, because the repetition in the pattern is based on the symmetries between the shapes. The white figures with the black outlines that resemble a $ or an S and the 8 red squares around them have order 2 rotational symmetry. If you rotate the figure 180 degrees, you have the same figure again. Each of the $ or s shapes has glide reflection symmetry with the upside down $ or S in the rows above and beneath it. In a glide reflection symmetry we see the mirror image of the original shape, but then it is glided or moved along the plane (in this case, along the paper).

The second print is “Tesselation 6- Magnified Basketweave”.  The text on the back of the print states “aka Monk’s Cloth or Roman Square Quilt As seen on NYC sewer covers”:

Dikko Faust - Magnified Basketweave - Hand set block print - 2013

Dikko Faust – Magnified Basketweave – Hand set block print – 2013

This print is a great example of reflection symmetry. It has two lines of symmetry: one horizontal though the center, and one vertical through the center. Another interesting mathematical feature of this print is the similarity between the larger sets of black or red bars and the smaller sets. Two figures are similar if they have the same shape and are only different in size. Both the large set of bars and the small set of bars form two sides of a square:  all squares are similar. The inner rectangle of larger bars measure 5 sets by 7 sets. It requires a rectangle of 11 sets by 15 sets of the smaller squares of bars to frame the large rectangle. There is a border with the width of one small square, so after subtracting 1 set from each dimension, we have the inner rectangle of 5 by 7 surrounded by a 10 by 14 rectangle of smaller sets of bars. The ratio of the dimensions of the larger to the smaller is 2:1.

Faust has made a whole series of these striking Tesselation prints. He has been inspired by what he has encountered teaching  art history, and what he sees all around him looking at art, and in the case of Tesselation 6, the streets of New York City. The mathematics in these prints go beyond the patterns themselves and connect the viewer with distant times and cultures, and links us all in a visual aesthetic.

– Susan Happersett

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;

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.

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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.

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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.

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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.

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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

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