The Whitney Museum is currently presenting “Carmen Herrera: Lines of Sight”. This outstanding exhibition examines work from 1948-1978. Born in Cuba and educated in Havana, New York and Paris, Herrera developed a distinctive hard-edge geometric style. This is a large show and would require more than one blog post to discuss in fill. I have decided to limit this post to paintings Herrera created in NY after she returned from studying in Paris (1952-1965).
“Black and White”, 1962 Picture courtesy of the Whitney Museum
“Black and White” from 1962 is an excellent example from this time period. The shape of the actual canvas is an important element in the architecture of the work. By rotating the square there are no horizontal or vertical lines, this immediately disrupts the visual experience. Herrera limited her color pallet to two colors creating a dynamic tension of positive and negative space. In this work the thicker white strips are the same width as the thicker black strips but in the gallery there is an optical illusion where the white seems wider. The alternating of black and white parallel lines on the isosceles right triangles creates an order-2 rotational symmetry.
“Horizontal”, 1965 Picture courtesy of the Whitney Museum
“Horizontal” from 1965 also features two colors and a square. This painting again relies on the shape of the canvas to define its structure, but in this case a circular format. The thin horizontal wedges amplify the push and pull of the red and blue triangles and circle segments, formed by the edge of the canvas (arc) and the sides of the squares (chords).
“Lines of Sight” is a long overdue solo museum exhibition for Carmen Herrera It is a welcome opportunity to appreciate the artist’s exciting use of geometry.
Victor Vasarely was the founder of the Optic Art or Op Art movement. His studies at the Budapest location of Bauhaus education in the 1920’s influenced Vasarely style of geometric abstraction. The paintings in the “Analog” exhibition at Maxwell Davidson Gallery demonstrate Vasarely’s ability to visually bend and stretch the plane of the 2-D canvas into 3-D space. The images seem to bounce and vibrate off the canvas.
The acrylic painting “PHOBOS” from 1979 uses the distortion of squares to create what looks like a square-shaped hole in the center of the canvas that is angled at a 90-degree turn from the edges of the canvas. The four isosceles right triangles in the corners of the canvas feature a pattern made up of purple or green squares. The four isosceles trapezoids have been filled in with distorted representations of squares creating the perspective of falling inward or protruding outward. The central square contains a grid of squares that again flattens out the plane.
Vasarely was the master of painting exacting geometric formations, but the most impressive element to this work is the exciting sense of space and movement. This is the first gallery show of Victor Vasarely’s paintings in many years It was very exciting to see a great selection of these geometric masterpieces all in one location.
There are a number of Upper East Side galleries that display museum caliber exhibitions of historically significant art. The current show at the Dominique Lévy gallery “Drawing Then, Innovation and Influence in American drawings of the Sixties” is an excellent example. It features work by some of my favorite artists like Eva Hesse, Agnes Martin, and Cy Twombly. The list goes on and on, there is even a Sol Lewitt wall drawing.
There are two works on display that relate the most directly to Mathematics. Mel Bochner’s “3” from 1966, is an homage to a Sierpinski Triangle. An equilateral triangular grid formation has been strategically filled in with hand written number 3’s and words that begin with letters “Tri”. The positive and negative shapes created delineate the fractal construction of a Sierpinski Triangle.
The second drawing is Josef Albers’ “Reverse + Obverse” from 1962. This line drawing is a 2-D rendering of 3-D constructions.
Josef Albers -“Reverse+Obverse” – 1962 Picture courtesy of the gallery
Both the top and bottom pairs of the figures employ a 180 degree rotation, an order-2 rotational symmetry. This work is a geometric expression of a form turning through space.
This year is the 40th anniversary of the MOMA’s ground breaking 1976 exhibition, “Drawing Now”. The current show at Dominique Lévy gallery is true to this historical reference, focusing on work from the turbulent years from 1960-1969. There is a wide range of work on display from drawings with social commentary, to drawings exploring the aesthetics of minimalism and conceptual rule-based art.
The use of repetitive geometric patterns is a prevalent theme in abstract art. Lori Ellison’s paintings and drawings celebrate the hand of the artist, featuring a lyrical, hand drawn quality. Through the use of basic geometric shapes Ellison created lively compositions that hum, buzz and pulsate. The current exhibition at the McKenzie Fine Art gallery include small scale paintings on wood panels and drawings on notebook paper. All of this ambitious work was completed the year or so before the artist’s death in 2015.
This gouache on wood panel from 2015 measures 14 x 11 inches. Its compact format holds a profusion of triangles. The almost parallel columns of almost isosceles triangles are packed tightly on the plane. Alternating the the red and pink shapes, all of the red triangles seem to point right and all pink ones point left. This forms an interesting dialogue between positive and negative space.
In this close up of the same panel we can see more clearly that this work is not about the accurate measurement of pure clean geometry. It is some ways more complicated, more human. This is definitely a painting about lines, triangles, positive and negative, but it is also about the artist. The personal scale makes the viewer stand close to the work and be drawn into the patterning. Art can be about mathematics with out having to use a ruler or striving for perfection.
The use of computer generated drawing processes and inkjet printers is a popular means of expression at the Bridges conference. Some of the more interesting examples on display were created by David Chappell. The artist builds a system of rules to generate graceful line drawings that are mathematically to related plant growth through space and time. The lines begin from a rooted position at the horizontal bottom of the picture plane and playful grow up into reaching tendrils. In order to achieve this lyrical organic quality (not an easy feat using mathematical algorithm computer generation) Chappell modifies the rules throughout the process. This extra attention allows the drawings to change and develop in a more free-form manner.
David Chappell -untitled – 2014 33 x 40 cm – Archival Inkjet Print Picture courtesy of the artist and the Bridges Conference
Another means of creating computer assisted art is the use of laser cutting. In his work “Islamic Fractal Starflower”, Pill Webster has cut a lace-like pattern into a clear light blue acrylic sheet. The mathematics behind this pattern is a combination of two geometric themes: the symmetry in Islamic patterns and the recursive properties of fractals. This combination requires some heavy weight mathematics, but Webster’s choice of materials transforms the complex theories into an ethereal presence. It has the appearance of being built from delicate and complex ice crystal. The juxtaposition between the serious mathematical generation and delicate physicality of the work create an interesting tension.
Phil Webster – Islamic Fractal Starflower – 2014 38 x 38 cm – Laser cut acrylic, light blue Picture courtesy of the artist and the Bridges Conference
Nathaniel Friedman is one of my favorite artists for two reasons. First, he creates wonderful sculptures and prints and second because he is a very supportive of other artists. As the founder of the organization ISAMA – The International Society of Art, Mathematics and Architecture, he contacted me years ago to speak at one of the first Math Art conferences. This was my introduction into a whole community of other artists and mathematicians devoted to the aesthetics of Mathematics. I will be eternally grateful to Nat.
Nathaniel Friedman – Triple Twist Mobius – 2014 29 x 29 x 7 cm – Aluminum Picture courtesy of the artist and the Bridges Conference
But back to the sculpture…. “Triple Twist Mobius” consists of three equal-sized aluminum bars each with a single twist. They are joined to form a triangle shape. The clean lines and the simplicity of the form are deceiving, this is a powerful shape. The 2-D photo does not do it justice. In the gallery each vantage point offers a different geometry, it seems to change depending on where your stand. This act of looking at something from different perspectives is referred to as hyperseeing (a concept Friedman taught me, Thank You!)
Every Summer the Bridges organization holds a conference devoted to Mathematics and the Arts. Bridges is an international organization whose sole mission is to foster and explore these interdisciplinary connections. This year the meeting was held in Baltimore Maryland in the beautiful University of Baltimore Law building. Each year the Art exhibition is one of the highlights of the gathering. This year was a particularly impressive display of work in a light and open space over three floors. Here are two photos of the gallery.
It has been very difficult for me to just single out a few art works to write about, for a complete overview I suggest checking out the Bridges website. Today I will focus on two works by two different artists that struck me particularly.
Taneli Luotoniemi – “The Hyper Cube” – 2015 Pencil on paper – 42 x 40 cm Image courtesy of the artist and Bridges
I will start with a pencil drawings by Taneli Luotoniemi. I have a real affinity for hand drawing and I feel Luotoniemi is able to achieve a remarkable subtly of line form and grey scale using only a pencil. “The Hypercube” Is a 2-D representation of a 3-D depiction of a 4-D cube. There have been many example of two dimensional art referencing hyper cubes but this is definitely a a more organic representation then most. This is achieved by the use of thick curved lines that meet at crossings of more solid shapes, instead of small points. By adjusting the grey scale of the pencil mark Luotoniemi gives the lines the appearance of weaving over and under each other. This is one of the most graceful visual interpretations I have seen.
David H Press – “Three ¾ Great Circles in Orange” – 2015 Laminated wood and cotton thread – 40 x 40 x 40cm Picture courtesy of the artist and Bridges
David H. Press builds elegant hanging sculptures that are a type of 3-D line drawings. The support structures are curved shapes but the wires within these frameworks are straight lines that form what appear to be curved surfaces. Symmetry plays a major role in Press’ work. In “Three Great ¾ Circles in Orange” the use of three circles would have created a sphere, but the ¾ circles create an asymmetrical frame work. Within the wire line work, however, there are some smaller areas with symmetrical properties. We are used to seeing complicated symmetries in Mathematical sculpture, but the use of the ¾ circles rips open the sphere, granting the viewer a fresh look.
There were so much interesting work on display this year it is hard to discuss it all in one blog post, I will write more next week.
In between observations on math art in Manhattan galleries and beyond, a quick shout-out for my own art work. Two of my collaborations with Purgatory Pie Press are now for sale at the new “Paper Project” gallery at the Metropolitan Museum of Art (back of the lobby on the left side when you come in through the main entrance).
Box of Chaos is a series of 4 paper sculptures based on Chaos Theory.
The “Happersett Accordion” is a modified, folded Moebius Strip
This is another big week for Art Fairs in New York. I will be at the Select Fair in Chelsea, New York with Purgatory Pie Press. It will run from Wednesday night through Sunday. If you are in the area stop by for a visit and see some new work. I will be showing some of my Mathematical Marking Drawings, Fibonacci Flowers, Spirals and Trees. Dikko Faust’s Tessellation prints will be on display, as well as his newest work (the ink is still wet) with Mathematical Moiré patterns. This is an exciting new process Faust has developed using rotating grids. It should be an exciting week!
Judith Lauand is referred to as “Dama do concretismo” or “The First Lady of Concretism”. She is an important figure in 20th century Brazilian Art. Concretism (called “Arte Concreta” in Brazil) is an international post WWII artistic Movement that included the use of a networks of mathematical geometry to build precise abstract systems of pattern.
The exhibition at Driscoll/Babcock is Lauand’s first solo show in NYC. Dr Aliza Edelman has curated “Judith Lauand: Brazilian Modernism 1950s-2000s”. This collection of paintings and drawings demonstrates Lauands significant geometric vocabulary. Her paintings feature bright flat hard edge figures.
Concerto 66 – 1957
“Concerto 66” is a circular panel with four lightening bolt shapes radiating from the center, creating a four fold rotation symmetry.
“Concerto 178” is tempera on canvas and is more of a line drawing. Two rhombi are surrounded by a host of triangles building a tiling type of pattern with 2 fold rotational symmetry.
Lauand’s work is a great example of the emphasis on mathematics in important post-war abstract artistic practices.
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.
The second drawing has 21 pods and again each pod has 8 segments with 5 seeds each.
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.
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.
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.