# order 2 rotational symmetry

# Sara VanDerBeek at Metro Pictures

“Pieced Quits, Wrapped Forms” is Sara VanDerBeek’s current solo exhibition at Metro Pictures Gallery. There is a variety of work on display, including large abstract photographs and monochrome sculptures, all with geometric themes based of quilts, Pre-Columbian patterning, and modern textile arts.

The totem pole-like structure “XXXVII” is a stacked tower of eight rectangular prisms or cuboids. By painting the entire structure white, the artist allows the viewer to focus on the patterns and shadows on each side of the prisms. Each rectangular side has a width to height ratio of 2:3. Bisecting the rectangle along the diagonal, order 2 rotational symmetry is achieved and two right triangles are formed. There is a series of consecutively smaller but similar triangles, that form a step-like pattern into the prism.

VanDerBeek has successfully taken the geometric basis from textile patterning and distilled it to its purest form, presenting these ideas in a contemporary visual dialog.

Susan Happersett

# Drawing Then at Dominique Lévy Gallery.

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.

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 The current show at Dominique Lévy gallery is true to this historical reference, focusing on work from the turbulent years from 1960-1969.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.

Susan Happersett

# Chelsea Hrynick Browne installation at Sprint Flatiron Prow Art Space

The iconic Flatiron building on 23rd street in Manhattan is home to the Sprint Flatiron Prow Art Space, a bright triangular room of windows, that can be viewed from the sidewalks of both Fifth avenue and Broadway. In coordination with Cheryl McGinnis Gallery, artists fill the space with interesting, temporary projects. This winter, Chelsea Hrynick Browne’s exhibition “Flakes” consists of vertical strings suspended with a multitude paper shapes.

Each two-sided shape is created by cutting and layering pieces origami paper. Browne’s intricate paper cutting relies on Mathematics to create the symmetrical patterns. These two examples each feature order-4 rotational symmetry. The use of contrasting colored papers affords an interesting expression of positive and negative space.

Each of the flakes is two-sided. Some flakes are circular and others are square, but my favorite flakes have 16 sides. These extra special flakes are formed by two back-to-back square paper cuts with a 45 degree rotation.

“Flakes” is a terrific installation for Winter in NYC. Each of the paper cut outs is different and they colorfully beckon to pedestrians navigating the snow and grey slush. I stumbled across this display and it made my whole day better. I am always amazed and happy to be able to find so many great examples of Mathematical art as I go about town.

Susan Happersett

# Gao Rong at Klein Sun Gallery

The current exhibition in the North gallery room of the Klein Sun Gallery is called “The Simple Line”. The show features the work of Beijing artist Gao Rong. Each of Gao Rong’s installation pieces is based on a circular hoop framework. Threads are stretched across the circle from evenly placed locations around the circumference. Although the basis of her subject matter is circular, through careful placement of the threads Gao Rong is able to create geometric arrangements featuring straight lines and angles.

This work contains a square grid of nine squares with the center square darkened with the overlapping of many lines of thread. this work at first appears to have two axis of reflection symmetry but this is only superficial. Upon closer inspection we see that some of the corners of the of the squares are much darker than others and taking that into consideration there is an order-2 rotational symmetry.

The next work is based on triangles. Starting at the bottom with single unit isosceles triangle, then moving up the structure, this single unit is overlapped by a triangle with a base twice as long. The next overlapping triangle has a base three times the length of the initial triangle Each subsequent triangle gets larger but also lighter in color. The shape seems to fade into the top of the circular frame.

There are two sets of theoretical juxtapositions in Gao Rong’s work. First and most is obviously the fact that the work illustrates linear structures within a curvilinear environment. Second, there is also the social statement of the use of colored thread, traditionally seen in women’s decorative needle work, to create very structured geometric diagrams that are heavily influenced by Mathematics.

Susan Happersett

# Frank Stella at the Whitney Museum

Happy New Year!

I decided to start 2016 with a big show and the Frank Stella exhibition at the Whitney Museum definitely qualifies as a really big exhibition. When the elevator door opens into the first gallery,the viewer is met by two very different canvases: a large, geometric, consecutive squares painting, and a huge abstract that is exuberant to the point of being Baroque. The dichotomy of these two works highlights the the range of styles and themes explored throughout the galleries. On display are the all black paintings from the late 1950’s, as well as the colorful geometric square-and-shape canvases from the 1960’s. Also included are the wall sculptures from the 1980’s and the more recent work created using 3-D printing.

For the purposes of this entry I decided to concentrate on Stella’s paintings from the 1960’s. These works are clearly about geometry. Some of the artist’s sketches and schematic diagrams are on display as a group. I highly recommend taking a close look at these plans, they really highlight the mathematical processes involved in the paintings.

The two canvases of “Jasper’s Dilemma” each have the same spiral geometric structure, but the left canvas features a system of the color spectrum, while the right canvas is composed of shades of gray. Stella has built these spirals within the squares by creating two sets of isosceles triangles. The set with vertical bases are slightly larger than the triangles with the horizontal bases. This results in only one diagonal line on each canvas and the four triangles do not all meet at the same point.

“Empress of India” is a monumental shaped canvas featuring a series of four V-shaped sections, each featuring a line of reflection symmetry and a 60 degree angle at the point of the “V”. There is also an interesting line of order-2 rotational symmetry running diagonally through the center section of the work.

Both “Jasper’s Dilemma” and “Empress of India” spotlight Frank Stella’s dedication to developing complex geometric structures in his work during the 1960’s.

Keep posted for many more observations on Mathematics and Art in 2016

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”:

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”:

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