Extrinsic Vertices

Extrinsic Vertices are lattices and patterns developed from tilings of polygons. I create edge-to-edge tilings or tile patches, and then plot the extrinsic vertices of potential tiles, creating a lattice structure. From these lattices I germinate patterns. The pattern lines extend out from the vertices of tiles and potential tiles.

The processes, lattices, and patterns in this project are not math. I’m influenced by structures in math, science, architecture, and design, but unconstrained by the rigorousness of math. The patterns have no practical use or purpose beyond this project.

Extrinsic vertices and potential tiles are terms that describe objects unique to this project. They are elements I designed to create lattices and patterns, but are not recognized elsewhere. I begin with tile sets of two or more regular polygons, rectangles, isosceles triangles, isosceles trapezoids, or rhombuses. The selection and arrangement of tiles is not necessarily predetermined as in a periodic or symmetrical pattern. At any step in building a tiling I can make multiple selections from the tile set, each with potential vertices. The vertices of potential tiles are extrinsic to the tiling. These vertices lie in characteristic dot patterns or lattices depending on the angular properties of the tile set.

I almost always borrow from math a preference for edge-to-edge tilings, simple polygons, and filling the plane with no gaps or overlaps. In contrast to typically symmetrical tilings like those in Islamic architecture, I often opt to create nonperiodic tilings, to select asymmetry. Ultimately I obscure the tilings with overlapping patterns.

These designs begin with tile patches, not in fact tilings or tessellations. They are tile patches – a finite number of tiles from some tiling. All of them could be extended, and most if not all might fill the plane if extended. However, it’s not necessarily clear how they would be extended or what a tiling extending any patch might look like.

My tilings based on a pentagon tend to include polygons with angles that are multiples of 18 degrees. Those based on a hexagon tend to include polygons with angles that are multiples of 30 degrees. Tilings based on a square tend to include polygons with angles that are multiples of 45 degrees. Tile sets with just one of these three angle groups have vertices and extrinsic vertices that lie in characteristic patterns depending on the angles used. Other mixed tile sets have their own characteristic vertex lattices. As tilings become complex with mixed tile sets the lattices reveal new patterns characteristic of the tile set, often repeating rosettes.

For reference, I refer you to the dot patterns described in Tilings and Patterns, by Grunbaum and Shephard (p. 238-246). Some of my simplest lattices correspond to Bravais lattices in 2 dimensions. The more complex lattices are possibly overlain combinations of dot patterns or lattices. I emphasize that the images and techniques in this project are not math.

I advocate using complex technology to create art that mimics natural beauty. If you know the technology and use it assertively, resolutely, and creatively you can produce something new and interesting. If it’s a bit innovative it might also be instructive, at least for other interested artists. Maybe it opens a path with potential. You can work with the natural beauty of math, though no one is likely see it in your work. You can also mimic or parallel nature. I do this kind of art because I think I’m particularly good at it: connecting, organizing, coding, elaborating, extending systems, following narrow paths that haven’t been pursued.

Tiling, Bravais Lattice, Squiggles

This is the first successful image in a new project that grew out of my tilings project. I created the image in three steps. First, I made a non-periodic tiling. Then I generated a lattice of vertices including all potential vertices around each tile. Then I plotted a shape or lines (in this case, a squiggle) at each lattice point. The plot of vertices is somewhat like a Brazais lattice, though I'm extending this math concept for my own purposes.

Tilings of the plane using tile sets of regular polygons, rectangles, isosceles triangles, trapezoids, and parallelograms, have vertices that lie in characteristic patterns or lattices depending on the properties of the polygons — its angles and sides. The Bravias lattice system categorizes these patterns. The five Bravais lattices for two dimensions approximate the arrangement of vertices of simple periodic tilings. I'm applying this concept to complex, non-periodic tilings with large tile sets.

Storm, Sepia Approach (Hexamerism Lost)

This is titled "Storm, Sepia Approach". The tile set could have made a hexameric tiling starting with the center hexagon. I forced the off-balance tri-radial asymmetry.

Gratuitous link: I suggest you look at the paintings of Janice Biala.

Pentamerism

This is the only non-periodic tiling I've done that is pentaradial. It has five polygons, but the square and rectangle (which has an aspect ratio of 1:1.618033989) can only be used along the five rows extending from the center pentagon. If I attempt to use them anywhere else, a gap is created.

Camouflage

In 2007, the Portland Art Museum exhibited Camouflage, an exhibition of eight paintings that explored artists’ use of pattern. The exhibit included a large camouflage pattern painting from a series that Andy Warhol did around 1986.

In his 1940 essay, A Mathematician's Apology, G. H. Hardy (full text here) discusses, among other things, pure versus applied mathematics with one difference being the application of mathematics to war. He stated that "a mathematician was a maker of patterns of ideas, and that beauty and seriousness were the criteria by which his patterns should be judged".

In Arthur Danto's The Transfiguration of the Commonplace: A Philosophy of Art, and 25 years later in his 2008 reply to his critics in CA online, Danto uses Warhol's Brillo box to discuss what makes an object an art work. (See Ontology, Criticism, and the Riddle of Art Versus Non-Art in The Transfiguration of the Commonplace. Contemporary Aesthetics, Volume 6, 2008.)

The Warhol camouflage series is interesting in that it turns a pattern designed for war back into art. Hardy said "A painting may embody [an] 'idea', but the idea is usually commonplace and unimportant." I wonder what he would have thought about the Warhol painting's ability to wrap all these concepts — beauty, seriousness, the ontology of art — in one art work.

Here's an unrelated, gratuitous Bezier curve image from a series:

Here are a few close-up pictures of a painting (a polyptych) just under way. It will resemble the digital print, Slip, but will undoubtedly have a completely different color scheme. I shot these as close as I could to show what I look at all day while I'm laying down these outlines in white on white oil paint. All the curves are paint applied with a small brush along pencil lines. The pencil lines are plotted/interpolated Bezier curves. That is, I plot about ten points per curve, and interpolate with ships curves or other French curves. The plots follow a spreadsheet of grid points that I exported from the original digital print program data.

Erosion

First test drawing of new program about erosion and alluvial fans: