The Andrea Rosen gallery in NYC is exhibiting the work of Matt Keegan. I found two of the powder coated steel wall sculptures of particular interest. These structures originate as folded paper cut-outs that are then fabricated in steel. The type of fold that is used to make the paper forms is called a French fold. To make a French fold you take a sheet of paper and fold it in half. Then without opening the paper you fold it in half again perpendicularly to the first fold. When you unfold the paper you have two types of folds: valley folds, which are concave, and hills folds that are convex.
In the sculptures Untitled (Navy) and Untitled (Neon) the French fold technique creates horizontal valley folds running through the centers. The top portion of each sculpture shows a vertical hill fold through the center, and the bottom half has a vertical valley fold through the center. Disregarding the fold directions both sculptures have two lines of reflection symmetry, vertical and horizontal.
Keegan celebrates the simplicity of the folded and cut paper by transforming the patterns into substantial steel structures .
The Howard Scott Gallery in Chelsea NYC is currently exhibiting a selection of Charles Thomas O’Neil’s recent abstract paintings.
Untitled 2740, 2013 Picture courtesy of the artist and the gallery
The painting “Untitled 2740” (2013) has a vertical line of reflection symmetry running through the center of the canvas. The top section of the features a rust colored bridge-like shape enclosing a white rectangle. The bottom section of the painting has a variation of the bridge shape in dark grey.
Untitled 2741,2013 Picture courtesy of the artist and the gallery
The oil painting on panel “Untitled 2741” (2013) is a 2-D rendering of what appears to be a 3-D impossible object. It looks like a rectangular bar with square ends positioned so both ends are visible to the viewer. This work has 180 degree rotational symmetry.
O’Neil’s geometric designs are enhanced by his use of saturated colors that immediately draws in the eye of the viewer. I also appreciate his use of visible painterly strokes which keep the work from looking flat and static.
Mitra Khorasheh has curated a fascinating exhibition of the paintings, sculptures, videos and performance art of Rachel Garrard title “VESSEL” at Gasser Grunert. All the work in the show is about geometry, a very personal geometry, based on the physical measurements of the artist’s body. In the press release from the show Garrard is quoted as saying: “I see the human body as a microcosm, a seed encompassing all the geometric and geodesic measures of the cosmos, as a container for something infinite”.
One of the geometric forms used by Garrard is the isosceles triangle.
The work “Convergence 2004” (quartz dust on linen) features layers of transparent isosceles triangles, 4 with the bottom of the canvas as the base and three with the top of the canvas as their base. The vertex angles are lines up on a vertical reflection line of symmetry that runs through the center of the canvas. This expresses the symmetric nature of the human form, with a vertical line of symmetry, but also the non-symmetrical nature, i.e. the absence of a horizontal line of symmetry.
The geometry for “Blue II” (Ink on canvas, 2004) is takn diretcly from the outline of the artist’s body. Garrard uses various rectangles to create a structure that relates the proportions of her body and again displays a verical line of reflective symmetry.
Garrard has also created videos and performance works that are based on her techniques of dividing up her body into a sort of grid of points. The artist then connects these points with either tape lines, directly on her body, or paint lines on a clear panel.
The sculpture “Geometric Void” (paint on perspex) is the result of an 8-hour performance from 2010. Rachel Garrard has created a new way to express geometry based on the proportions of her body. Although the nature of this work is very personal, the essence of these symmetries and proportions reveal universal truths.
Benigna Chilla has incorporated mathematics into her art practice throughout her career. Her recent, large scale canvasses on display at Tibet House are inspired by her stay in Bhutan in 2011.
Overview of the exhibition Picture courtesy of the artist and the gallery
Chilla has included small segments of cultural pattern and textiles into the texture of these paintings. This enhances the connections between the bold symmetries and traditional Tibetan Art. In the painting “Two black Triangles” there is the obvious reflection symmetry of the black triangles, but there are also subtle almost-reflective symmetries. Near the bottom of the canvas there two added sculptural elements, but the right one is higher than the left. On the right hand side of the bottom border there are two red triangles with grey circles on top. On the left hand side, the triangles re grey, but the circles are red.
Two Black Triangles – Mixed Media – 8′ x 6′ – 2012 Picture courtesy of the artist
The painting “Full Moonstone” features a large central Mandala with 8-fold rotational symmetry.
Full Moonstone – Mixed Media on Canvas – 8′ x 6′ – 2013 Picture courtesy of the artist
In the press release for this exhibition, Chilla discusses the importance of both the meditative and physical processes involved in the creation of these works. There are not many artists who can discuss creating mathematical symmetries and meditation, and I personally find that combination very inspiring.
Using symmetry to create a work of art is one way that Mathematics can influence an artist. But what happens when an artists uses symmetry and then makes one small change to upset that symmetry? The resulting work can express very different and exciting dynamics. I saw an exhibition of work by Julije Knifer (1924-2004) at the Mitchell-Innes & Nash Gallery in New York. Looking at Knifers drawings and paintings it is obvious that using reflection and rotational symmetries were a major aspect of his work.
APXL – Julije Knifer – 2003-04 Courtesy of Mitchell-Innes & Nash Gallery
In APXL made in 2003-2004 there is a vertical line of reflection symmetry, but what is also interesting is how each of the “S” like figures would have an order 2 rotational symmetry if the artist had not truncated the outer columns.
Knifer used some type of symmetry in a lot of the work on display at this exhibition, but in much of the work the perfect symmetries were in some way altered.
MK 69-43 – Julije Knifer – 1969 Courtesy of Mitchell-Innes & Nash Gallery
The figure in the painting MK 69-43 from 1969 has an unblemished order 2 rotational symmetry but the figure is not centered on the canvas. There is an interesting sense of tension in this canvas because of the imbalance.
MS 09 – Julije Knifer – 1962 Courtesy of Mitchell-Innes & Nash Gallery
In the painting MS 09 from 1962 there is a horizontal reflection line of symmetry running through the center of the figure except for two small lines. There is a line connecting the center columns at the top and a line connecting the two right columns at the bottom. These act like bridges connecting the columns.
What I find so interesting about these works is how they make me think about symmetry. At first I was not going to write about this exhibition because only one painting had a complete and clear symmetry. But I kept thinking about the paintings and drawings and after a while I realized that by creating these imperfect symmetries Knifer has given us a different but inspired way to look at symmetry. There is enough of a framework provided so the viewer is looking for order in the symmetries, but is thrown off balance. Maybe this uncomfortable imbalance is the perfection of this work.
Julije Knifer is is one of the most important Croatian artists of the 20th century. His work is in many museum collections including MOMA in NYC and has had exhibitions at The Centre Pompidou in Paris and the Museum of Contemporary Art in Sydney Australia. In 2001 he was selected to represent Croatia at the Venice Biennale.
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
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
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.
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):
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: