Developing the Grid

Grids in art of the last century were mostly the Cartesian coordinate system kind, with orthogonal axes and rectangular cells. They were also usually module-based, having cells with common corner vertices, rather than coordinate-based. [1]. They didn't admit the possibility of overlapping and gapping of cells. Grids of many kinds – logarithmic, curvilinear, polar, geodesic, and unstructured grids like those used in surface modeling – are possible, but seldom used in visual arts.

A recent example of a grid-based image with a cubic function vertical axis.

Digital art is a recent exception. When artists have access to computer aided design systems they can use unstructured grids without having to calculate the data structure. Geometric primitives make up surfaces in the earliest stage of the graphics pipeline. Thanks to CAD, unstructured grids are commonly employed in architectural design, industrial design, and digital imaging. However, the fact that there is an underlying grid structure is usually lost in the final stages of rendering.

Victor Vasarely and Bridget Riley painted grids with quadrilateral cells. A few artists have been influenced by CAD imagery to create complex grid-like paintings without using a computer. Michael Knutson of Portland, Oregon, has painted grids that resemble polygon meshes or unstructured grids. They're based on the quasi regular rhombic tiling, but are mapped to freeform spirals that he lays out by hand. (See his paintings from 2007-2008 at the Blackfish Gallery in Portland, and this interview in Geoform.) The surfaces in the paintings of Vasarely, Riley, and Knutson undulate but never overlap or allow for gapping. They represent a development of the grid from flat surface to an optical bas-relief.

Simple programming, nothing as complex as a CAD system, can be used to generate grids other than unstructured and Cartesian grids. It's possible to create new designs using grids with nonorthogonal axes, and when coordinate-based, designs can include overlapping or gapping. Spirals can be used to map regularly spaced coordinates for grids resembling disc phyllotaxis. Coordinate or module-based grids can be generated from polynomial functions – sine waves, or cubic functions for example. This means that the grid no longer resists development. It's also neither flattened, nor antinatural. [2]

Grid scale also offers the possibility for development. Artists have just begun to use GPS systems. GPS systems use trilateration (not triangulation) to determine positions. If you think of the position of a receiver as a coordinate, the resolution or error in the system means there is a natural cell or area for each numerical position. Depending on the type of system, the accuracy may be from 10 meters to as little as a few centimeters. Appropriately larger cells of any size and shape could be calculated. The key concept though is that practical grids can now cover huge areas, such as an entire city. Alyssa Wright effectively used the entire city of Boston to map coordinates for "Cherry Blossoms". The coordinates make either an unstructured grid with significant coordinates as vertices, or a grid with all possible coordinates locating cells sized by the resolution of the system. For grid-like structures at the microscopic scale, look at diatoms and the drawings of Ernst Haeckel.

Another obvious area for development of grid-based art is in cell content. Until recently, grid cells were often modular, without gapping or overlapping, and filled with color (Ellsworth Kelly), texture (Nevelson), images (Warhol), or space (some Sol Lewitt structures). The natural world, from the atomic scale and up suggests that coordinate-based systems can position a variety of phenomena in grids. Disc phyllotaxis and social networks are two examples.

The more we consider algorithms, scale, or content (biological, social, political...) as means to develop the grid, the more we discover and invent underlying structures, the more possibilities the grid should offer artists.

Here are some graphic examples of new grids:

Cubic Functions
Sinusoidal (an almost sinusoidal grid in use, here)
Parabolic Spirals
Curvilinear (also here)
Curvilinear Triangles

References:
1. Jack H. Williamson. “The Grid: History, Use, and Meaning.” Design Issues, Vol. 3, No. 2. (Autumn, 1986), pp. 15-30.
2. Rosalind Krauss . "Grids." October, (Summer 1979), pp: 50-64. Reprinted in The Originality of the Avant-Garde and Other Modernist Myths. Cambridge, Massachusetts: MIT Press, 1985.