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.
The current exhibition at the Luring Augustine Gallery in Chelsea, “Empty House Casa Vazia”, features sculpture that is associated with the Neoconcretism movement in Brazil from 1959 until 1961. Neoconcretism was a reaction against the rationalism of Concretism. Although Neoconcretism continued the use geometry to create abstractions, they were not interested in pure form. Instead, they introduced a human element.
Lygia Clark was an important member of the Neoconcrete movement. She added a participatory element to her sculptures. The viewer was encouraged to manipulate her hinged metal sculptures. I have written an earlier blog post about the MOMA exhibition of Clark’s work, but I was not permitted to take any photos, so I was thrilled the gallery is allowing me to share a photo now.
“Bicho” consists of a series of sheet steel isosceles right triangles (isosceles triangles whose vertex angle is 90 degrees). They are hinged together to form a complete loop that can be arranged in many different positions.
Lygia Pape’s series of wooden wall sculptures titled “Livro da noite e dia” features a series of 6 1/4″ squares. Each square has at least one geometric shape removed from the edge or corner. Then those shapes, triangles, squares, trapezoid….are shifted and layered onto another part of the square resulting in interesting symmetries.
“Estrela”, a copper sculpture by Amilcar de Castro, is made up of three rectangles. Each rectangle has been bisected diagonally and folded and joined together to make a sculpture with all sorts of triangular possibilities.
These practitioners of Neoconcretism employed mathematics in their work, particularly Geometry. But their art was about something even deeper, it was about how humans interact with the geometry. This is achieved in a different way by each of the artists: In the case of Lygia Clark through tactile manipulation, Lygia Pace’s intriguing puzzle-like squares encourage the viewer to ponder the missing pieces, and De Castro’s sculpture invites the viewer to walk around the work, because it changes dramatically depending on the location and angle from which it is viewed. In some ways these sculptures reveal more about our relationship with Mathematics than many other artistic movements.
We have been having a hot and humid week in NYC so it was probably not the most well advised plan to go traipsing around the Lower East Side. I was thinking to myself what am I doing here in the midday sun walking from gallery to gallery and then… I saw this amazing sculpture that just seemed to scream Mathematics in the Summer time.
This inflated vinyl hanging form”Squirm” is the work of Doreen McCarthy and is part of the group show titled “Object’hood” at the Lesley Heller Workspace. This sculpture has the materiality of a classic tube used for floating around in a pool on a steamy afternoon. Topologically we think of the traditional pool toy as being a donut-like torus, but this baby blue version is a knot instead. It is a 3D interpretation of a trefoil knot, Which is a basic overhand knot with the ends joined together. I found “Squirm” to be a refreshing topographical Summer treat. Susan Happersett
The G19 Artisan Gallery is located in Rockport, Massachusetts. Rockport is a picturesque town known for its art community for almost one hundred years. Historically, Rockport artists are known for their seascapes, but the G19 gallery exhibits art in a wide range of materials, styles and themes. I was so happy when I discovered some amazing geometric metal sculptures.
Dodecahedron and Octahedron
I was able to meet with the – somewhat reclusive – artist who would like to be referred to as Dan H. Dan told me that he first became interested in Dodecahedrons when they were referenced in an episode of the TV series “Doctor Who”. He started out making paper models, but after learning how to weld he found metal a better choice of medium. He quickly figured out that the angles for the pentagons need to be fairly accurate or the shape would not fit together. G19 is currently exhibiting Dan’s Dodecahedrons, Octahedrons, and Icosahedrons. These three shapes are all Platonic Solids. The faces of Platonic solids are congruent regular polygons where the same number of faces meet at each vertex. There are only 5 possible Platonic solids and Mathematicians have been studying them for thousands of years. Dan now uses computer software and laser cutting techniques to cut the metal shapes. The finally fabrication, however, is all done by hand. This gives the sculptures a wonderful organic element. The welded edges have a nice texture and the sides retain the flame patterns of the torch. The juxtaposition of using technology for accuracy of the geometry, but then adding the mark of the artists hand makes these sculptures a great example of how artists can use high-tech tools while retaining control of the aesthetics of their work .
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 Art Gallery of Ontario in Toronto is currently exhibiting a large solo exhibition called “Stephen Andrews POV” in their Contemporary gallery. Stephen Andrews is known for his photographs, videos, and paintings that address difficult societal issues, using both representational and abstract formats. A recent (2014-2015) series of paintings titled the “Butterfly effect” is Andrews’ expression of the Chaos Theory. Using a defined set of restrictions the artist explores the multitude of outcomes.
Each work in the series consists of a white canvas with six rectangles that are identical except for color. Two of the rectangles are red two, are blue, and two are yellow. Each of the primary colors has equal representation. The oil paint has been applied using mylar sheets to ensure the integrity of the rectangles. It is the placement of the rectangles that changes with each painting. There are an infinite amount of possible outcomes. Andrews is interested in the accumulation of colors that eventually make black. The areas where all three colors overlap become black rectangles. These black rectangles appear in different locations on each canvas. At first these painting look random but upon closer inspection you realize they all share the same geometric elements, there is consistency in the chaos.
Stephen Andrews has expressed his ideas about the chaos he sees in our troubled world by using the ideas of the mathematical Chaos Theory to create abstract geometric paintings.
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
Whenever I have the opportunity to travel I make a point of visiting the local museums and galleries and I am always on the hunt for MathArt.
In Toronto, I visited the Distillery District, which is a collection of brick historic buildings that was once a whiskey distillery, but is now filled with galleries, restaurants and artist’s studios. I was in the Thompson Landry gallery when I spotted mathematical formulas that seemed to come directly out of my old Integral Calculus text book. I became immediately interested in the work of Étienne Gélinas. He uses a variety of techniques to create multi-media work: scratching geometric drawings into a thick base coat of paint or medium, collaging with paper blueprints, floor plans and garment patterns, and carefully painted shapes and formulas. The artist also adds a random accidental quality to each work by including an expressive element of abstract splatter drip and mark painting.
Étienne Gélinas – “Composition 365” – Mixed media on wood Picture courtesy of the artist and the gallery
In the work “Composition 365” Gelinas has used a circle as his underlying geometry. There is a series of larger concentric circles which have been segmented into 8 equal sections with smaller series of concentric circles within each segment. Around this circle pattern there are mathematical formulae, specifically integral formulae. This formulae painted in white on the black background are quite beautiful. Dividing the work horizontally, the artist has placed layers of vintage patterns for making clothing. On top of the collaged element Gélinas has painted a free form abstract painting. There is a lot going on in this work and that is what I like about it. The seemingly disparate techniques yield complex work with a great textuality. To me the work addresses the layers of mathematics in society. There is the obvious association of the calculus formulae, and the geometric implications of the drawn diagrams relate to the geometry used in making the paper garment patterns. Finally, the wild abstraction of the gestural painting adds a level of spontaneity and emotion to the visual dialog. It is not often I find single work of art with so levels mathematical aesthetic.
The current exhibition at Pace Gallery takes its name from the Edgar Allan Poe poem from 1848. The press release contains a quote from the poem: “I design to speak of the Physical, Metaphysical and Mathematical-of the Material and Spiritual Universe: of its Essence, its Origin, its Creation, its Present Condition and its Destiny…….”
This group show features work form the 1840’s to 2010 that builds a links between science and Mathematics and the artistic spirit. In one of the first galleries there is a copy of Edwin Abbot’s 1884 book “Flatland: A Romance of Many Dimensions”. In this novel Abbott creates a two dimensional society and introduces a three dimensional character with interesting results and exciting prospects about further dimensional expansion. Abbott’s art allows his readers to imagine the possibility of a fourth dimension, a Mathematical idea that was very new at the time.
Installed in the largest room of the gallery is Tim Hawkinson’s large rotating sculpture “Gimbled Klein Basket” a wonderful homage to the “Klein Bottle”. A Klein Bottle is an impossible form first introduced by mathematician Felix Klein in 1882. Like a Moebius strip it has only one side, but a Klein Bottle has no boundaries, whereas a moebius strip has boundaries at its edges. Compare to, for instance, a sphere, which has no boundaries either.
The basket structure of Hawkinson’s “Gimbled Klein Basket” creates an interesting grid pattern on the shape, adding another visual element to the form. The hand crafted quality of the object makes it seem as if this shape is actually possible in 3-D. By rotating the sculpture the viewer has a chance to examine the form from all angles.
Installation view of the exhibition, Eureka, Pace Gallery, 508 West 25th Street, New York, May 2–June 27, 2015. From left: Hawkinson, Gimbled Klein Basket, 2007; Siena, Battery, 1997; Jenson, Physical Optics, 1975. Photograph by Tom Barratt, courtesy Pace Gallery.