Nonetheless, the underlying idea of transferring information from one grid to another has a long history in both mathematics and art. When the blank grid differs from the original grid, for example, a drawing can suffer intriguing distortions. In art, the result is sometimes called an anamorphic picture. Mathematically, you're looking at the results of a type of transformation or mapping.
To create one sort of anamorphic picture, you start with a piece of paper ruled into square cells and another ruled with the same number of trapezoids.
Draw your picture on the square grid. Then carefully copy the contents of each square of the original grid to the corresponding trapezoid of the other grid, stretching the lines of the drawing to make sure everything fits together. You end up with a distorted version of the original picture. Interestingly, if you now look at the final drawing at the proper angle from the edge, it appears undistorted.
You've probably experienced the same effect while riding in a car and encountering the word "STOP" painted on the roadway just before an intersection.
The white letters look normal from where you are sitting. If you were standing beside the sign instead of riding toward it, however, you would find that the letters are actually stretched out (see "Anamorphic Stop" and "Anamorph Bike").
Artist Douglas D. Peden has explored the use of grid transformations to create visually arresting paintings, in which each gracefully distorted abstraction ripples its own, subtly musical tale.
Transfiguration by Douglas Peden.
Bends and splashes of color seep and shift along narrow passageways, like dyes spreading in irregular patches up soaked strips of porous filter paper.
Peden started out in nuclear engineering, but he left the field during the 1960s to become an artist. In 1970 he moved to the Adirondack region of New York State.
"I found the combination of the natural beauty and solitude supportive of my creative needs," Peden said. "It also allowed me to pursue my own vision without the distraction and influence of the prevailing art fashions, styles, and market forces."
Peden's early paintings featured free-flowing amoebic forms radiating vibrant, contrasting colors. Later he turned to abstract landscapes—sky, water, land, trees, and structures presented symbolically and rhythmically in great, horizontal swaths. His visual patterns, with their carefully manipulated thematic variations, reminded you of musical forms such as the classical sonata.
Peden then began to formalize his approach, developing a geometric language and framework that he could use to create his abstract narratives. Instead of working with a Cartesian grid represented by horizontal and vertical straight lines, Peden focused on coordinate systems based on wavy lines. He called his scheme "wave space," or "gridfield," geometry.
In effect, instead of moving around the squares of a standard grid, he focused on changing the squares themselves into other shapes.
Peden would start a painting by penciling in a Cartesian grid, using a straight edge. After specifying the wavelength and amplitude, he would then draw freehand a set of wavy parallel lines oriented in the vertical direction. The corresponding set of lines in the horizontal direction was then chosen as a function of the vertical field. The combination produced what he called a "gridfield" configuration.
"If wave field amplitudes are reduced to zero, the gridfield becomes a Cartesian grid," Peden noted. "In other words, the commonly used Cartesian grid system is one of many grid/gridfield configurations."
One example of the sequence of steps that Douglas Peden may follow to draw a wavy gridfield to serve as the basis for one of his paintings.
Peden then would weave in another meandering wave field to produce additional distortions, making the final grid look like a strangely crumpled piece of fabric.
The choice of grid is up to the artist, as is the color of each of the grid's cells. "The shapes, themes, rhythms, and spatial textures are defined and influenced by the geometric configuration of the chosen space," Peden said.
A character drawn on a square grid can change shape drastically when transferred to different gridfields and to different positions within a given gridfield.
In the early development of his gridfield technique, Peden adopted a trial-and-error approach. "It was a matter of deciding on some wave field parameters, drawing them out, and using various colors, color rhythms, shapes, and textures to see what happens," Peden said.
"In other words, I had to experience the possibilities within different gridfield configurations to learn, to test—perhaps, in some sense, like Bach inventing pieces to see what the various organs were capable of doing," he added.
Detail from Douglas Peden's Serenade for a Lone Figure.
For Peden, gridfield geometry was not only an exciting spatial environment to experience and learn about but also a new language with its own structure and symbolic meaning.
"The gridfields themselves can be chosen to express rhythmic energy—in musical terms, allegro, andante, adagio, and so on," Peden remarked.
In his experience, Peden found that different grids contain the possibility of forming different types of shapes—ones that may be dynamic, passive, hard, soft, cutting, ugly, or lovely. The use of color then either enhanced or subdued the dynamics or mood suggested by the chosen grid.
Peden painted Transfiguration (above) as a memorial to his wife, Jan Peden. "It is basically a painting of hope and faith," Peden said. "The specific event in question is the pain and horror of his wife's cancer and the hope of a joyful conclusion, whether it be in the beauty of bodily healing or the painless union with God."
In creating a painting, Peden might start by simply picking a gridfield, penciling it on canvas or paper, then shading in an image.
"If the image expresses some sort of personal symbolism or has thematic appeal, I might tell a story," Peden said. "If the image is just a pleasing space, I might explore its variations and morphological possibilities, perhaps by adding or subtracting parts or elements, thereby creating other shapes—much in the same manner as a composer develops and explores a musical theme."
Come Together by Douglas Peden.
Eventually, the process evolves into an intimate conversation between artist and painting. At the same time, the panting style does not encourage trial and error. Perceived "mistakes" take a long time to correct.
"This is a slow painting process," Peden said, with much time spent visualizing and thinking before committing to paint.
Curiously, Peden did not use a computer to aid him in constructing his gridfields or to explore variations and plan his paintings.
"The possibility of computer-sketched studies is a fascinating thought and intriguing solution," Peden admitted. "It would certainly lead the way for quicker, more detailed, and more accurate studies and encourage the application of gridfield geometry to art and science."
"I enjoy the fact that my art includes and is inspired by such disciplines as music, literature, philosophy, mathematics, and physics," Peden noted. "I feel the most profound art is that which encompasses all life—at least as much as one can experience within one’s limited capacity for knowledge, understanding, and expression."
Peden's work helps shed light on the apparent tension between artistic freedom and mathematical constraint. Like many artists, Peden built a distinctive vision and style out of constraint—by exploring the seemingly unlimited, surprisingly rich possibilities offered by the set of rules that he developed (and freely chose) as the framework for his art.
See also "Scrambled Grids," "Pursuing Pursuit Curves," "Sliding Pi in Toronto," and "Binary Frieze."
Originally posted August 14, 2000
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