Piero della Francesca, Symmetry

In several previous blogs (Haeckel, Calder, Jess, Merian, and Baer) I referred to artists with technical backgrounds, and to the connections between math and art. I have also quoted artists and mathematicians who sought to separate math from art: Robert Mangold, — "Abstraction is an idea. Geometry is not."; Mel Bochner, — "Happily there seems to be little or no connection between art and mathematics (math deals with abstractions, art deals with tangibilities)."; and the mathematician, G. H. Hardy — "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

I received a Kindle for my birthday, and the first book I read was Ian Stewart's Why Beauty Is Truth: A History of Symmetry. This is a fascinating history of the idea of symmetry, how central it is to mathematics, and of the mathematicians that worked out what we currently know about the concept. (Much of the math is over my head.) In one short section Stewart explains how artists Filippo Brunelleschi and others formulated, Leonardo da Vinci applied, and Piero della Francesca perfected the mathematics of perspective, and how this fits into the continuous thread of discovery.

"In those days [during the Renaissance] mathematics and art were rather close; not just in architecture but in painting. The Renaissance artist discovered how to apply geometry to perspective. They found geometric rules for drawing images on paper that really looked like three-dimensional objects and scenes. In so doing, they invented a new and extremely beautiful kind of geometry."

Flagellation of Christ by Piero della Francesca
Galleria Nazionale delle Marche, Urbino, Italy

In the 17th century, Girard Desargues used this math to develop a new non-Euclidean, projective geometry. Desargues' contribution was a key step in the path that leads to the mathematics of higher dimensions.

So despite what Mangold, Bochner, and Hardy had to say, there is an important connection between math and art. That relation may have been strongest during the Renaissance, but continues in the math if not the art of today.

Here's one of my paintings from last year. It's one of several I did based on the idea that I could break the rules by combining multiple horizon lines in a single projective plane.

There is more than one way to do it.

"There is more than one way to do it" is an expression associated with Perl, a programming language developed by Larry Wall.

These two images are both modifications of grids mapped to a parabolic spiral. One line of Flash Actionscript code separates the two.



Here's the completed drawing, after several more modifications, but 95% of the code is the same as the above two:

Transfiguration of the Common Eye

Transfiguration of the Commonplace, by Arthur Danto, was published in 1981. As Danto explained in 2007, it is:

". . . a contribution to the ontology of art in which two necessary conditions emerge as essential to a real definition of the art work: that an artwork must (a) have meaning and (b) must embody its meaning."

The meaning embodied by this image of eyes in the pattern of Fermat's spiral escapes me. So, though not art, I offer it as design potential. It's utility would need to be determined before I knew if the colors are wrong, or if there are too many/few eyes, etc. Without meaning, and without utility it's neither art nor design.

There are 232 eyes, the spiral grid corresponds to some natural disc phyllotaxis, and each eye is content for a single cell in the coordinate based grid.Therefore, at least it's an example of content presented in a non-Cartesian, coordinate based grid.

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.