Self-Similar Vertices

I've been using the process described here to create tilings from a pentagon and triangle. These tilings, or tile patches, are one solution to a problem I encountered while generating tilings as structures for a series of vertex pattern diagrams. I needed a way to create greater density changes than I was getting with other tilings. Self-similar tile sets solve the problem.
My interest in tilings is more art than math. The tile set I’m using is closely related to, and can be derived from tilings described by Robert W. Fathauer (see Reference below). Fathauer found two families of self-similar tilings based on segments of regular polygons. One family includes an 18-18-144 triangle that is a segment of a regular decagon. The triangle is the s=10 prototile described by Fathauer. The tile set that I use includes a pentagon as well as the triangle. The pentagon is the shape that remains after removing five s=10 tiles from a regular decagon. Other hexagonal or square prototiles can be combined with triangles to make self-similar tile sets.

So, this method is based on an infinitely self-similar tile set consisting of pentagons and 18-18-144 triangle prototiles. Tilings should be edge-to-edge, with no overlaps. Gaps are inevitable, but they should allow lining with infinitely scaled tiles. Tilings will not fill the plane but should be infinitely scalable at the boundaries, as in a fractal. These are actually tile patches, not tessellations.

The initial pentagon and triangle prototiles are sized so the long side of the 18-18-144 triangle is equal to the pentagon side. Each subsequent pentagon-triangle pair is scaled so that the next pentagon side is equal to the short side of the previous triangle. Using this scheme, as the boundaries of the tiling grow outwards they form singularities, or gaps surrounded by tiles. The inside edges of these gaps can be continuously and infinitely lined with scaling pentagons and 18-18-144 triangles, or just 18-18-144 triangles.

An interesting feature of these prototiles is that the ratio of the areas of each pentagon to the next smaller pentagon (or triangle to triangle) is always 3.618... or 2 plus Phi.

It's possible to create numerous symmetrical tilings with these tiles, but I often choose to create asymmetrical diagrams. The processes, lattices, and patterns I use are not math. I'm influenced by structures in math, science, architecture, and design, but unconstrained by the rigorousness of math. These diagrams have no practical use or purpose other than art.

Reference:

Fathauer, Robert W. (2000). "Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons," presented at the Bridges Conference (July 28-30, 2000, Southwestern College, Winfield, Kansas).
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/Bridges2000.pdf
Also see:
http://www.mathartfun.com/shopsite_sc/store/html/Compendium/encyclopedia.html

Extrinsic Vertices from mixed n-gon tilings.

Plots of the extrinsic vertices (my term) for tilings from mixes of 4-, 5-, and 6-gon regular polygons with 3-gon tiles are generally more interesting than those with just triangles and one of the 4-, 5-, or 6-sided regular polygons. Less variegated plots come from tile sets that include only polygons with angles that are multiples of 18 degrees (pentagons and triangles that subdivide them), or multiples of 30 degrees (hexagons and their triangles), or multiples of 45 degrees (rectangles and their triangles).

These images show that it is not the complexity of the tiling, but the tile set itself that determines the patterning of the vertices and extrinsic vertices. The first image is from an animation. The second image is the tiling that is the underlying structure for the first and last images. The last image of the extrinsic vertices reveals its regularity. The tile set includes a pentagon and two triangles whose angles are multiples of 36 degrees. The tiling is highly irregular, including gaps that can't be filled with the tile set. It is an edge-to-edge tiling though.

Compare these images to those from my last post where the tile sets are generally mixed 4-, 5-, and 6-gon regular polygons with 3-gon tiles.

Extrinsic Vertices Animations

Here are the final frames of a few new animations. These are four of the most complex images from the series, "Extrinsic Vertices". These animations are controlled by tiling structures that preserve in spirit if not in fact preferences for tessellations from small tile sets that fill the plane, edge-to-edge, with no overlaps.

Tiling Generation

It’s difficult to design a tiling that fills the plane, is edge-to-edge, with no gaps and no overlaps, if you start mixing pentagons with squares and hexagons. This particular tiling is just VERY STRANGE in that it has two seemingly incompatible lattices in connection. So, from my point of view, it’s very jarring – even irritating in that it forces two systems into one, almost. It’s intentionally irksome.

Embodied Cognition

Compare this idea, that the body takes abstract thoughts literally (see Abstract Thoughts? The Body Takes Them Literally, by Natalie Angier) with this idea — that the integration of the human body into performance is a fundamental problem with electronic music (see "Human Bodies, Computer Music", by Bob Ostertag). In the first case, the field of embodied cognition shows us how the body reacts to abstract ideas. In the second case, the lack of body is seen as limiting to the art form. Perhaps the study of embodied cognition might hold some clues as to why electronic music and computer art in general are cold to our "senses". Though we can grasp the abstract idea of computer art we have no experience connecting the idea of the art to our bodies. We have all seen live performances of music in which the bowing of the violin, the strumming of the guitar, the blowing of a horn, or the beating of a drum were all accompanied by corresponding volume, pitch, intensity and the emotions of the performers. With a recorded performance, we still react in empathy with performers we have seen. The abstract idea of a song is felt with our bodies partly because we have been trained by performance artists to feel what they emote. Correspondingly, if we sense that the body was removed from the performance, with electronics, we react less with our bodies. You get into a hard rock guitar solo as though you were the guitarist, but computer code and electronics tamp down all those emotions. We can't imagine a programmer strutting, thrashing, and banging out code.

Code and Art

Programming is simultaneously the problem with and the source of digital art's potential. With culture the abundance of digital solutions is directly related to the fact that coding is so intensely creative. Those who can love to code, so they flood the world with every conceivable image, animation, architecture, or other art form built with code. Programming is a thoroughly rewarding creative process. It's unstoppable because it's so completely satisfying to produce. It sustains innovation. The problem for digital artists is that though we can feel the emotion that goes into a song, painting, film, or novel, we can't empathize with the act of programming. Apparently devoid of emotion, the digital visual product is cold. The coded digital image, at home on the Internet, appears out of place in museums and galleries.

Another problem is with the attribution of the art to the artist. Writers, musicians, painters, directors, and architects don't have this problem. Digital art is more difficult. You never know how much credit to give to the hardware, the compiler, the application, the Internet, and the thousands of engineers that contributed to making it possible. Even if you program you're not apt to know quite how much credit to give the artist for an interesting image.

Coding is incomprehensible to those who don't, so a programmed image blocks potential empathy for the real creativity behind the image. Looking at a digital print, we're no more interested in the creativity of the coding than we are in the coding behind our browser app. We're satisfied if our browser works as well as other applications, and we'd be satisfied if a digital image could hold up against all other images, digital or otherwise. Again, even if you program you're not apt to credit an artist for their code.

A solution that has been tried is a programmatically arranged massive aggregate of unitized elements. Fractal art is an example. I’ve tried this approach, repeatedly, and I think it’s insufficient. I’ve also tried the creative use of math. This works no better than code with little or widely known math. I doubt there’s anything less appreciated in the Euclidean arts than Euclidean geometry.

The solution might be to animate. I’m confident that this works. Given the immediacy of motion, images come alive with potential for feeling and emotion. I’m concerned that it’s acceptable because it’s film, and not considered programming.

There may be a solution other than animation, but boy do I not know what it is, yet. Digital art can be cold, but coding is anything but. We should reject cold art, but it's an error to deny the process. New media is begging for a solution to this problem. Writers, musicians, cinematographers, and photographers are adapting to digital technology without appearing to abandon their art form. Visual artists alone are stuck with a preference for the hand made, and a prejudice against the machined. MP3s, 3D CG, and Giclée photo prints are acceptable uses of technology. The programmed digital print is not.

See "Human Bodies, Computer Music", by Bob Ostertag.

Onward through the fog. . .

Robert Motherwell on Math and Abstraction

Continuing a thread on quotes about math and art, I found these from Robert Motherwell:

"I have often quoted Alfred North Whitehead in what I think is one of the crucial statements on abstraction, that 'the higher the degree of abstraction, the lower the degree of complexity.' In that sense, mathematical formulae are (ironically) by nature of a lower degree of complexity than a painted surface with three lines, even if it's an Einsteinian equation."

"Advanced mathematicians say that when there are two mathematical solutions to the same problem that are equally valid, mathematicians will often reject one of the two solutions as less beautiful than the other. Even in something seemingly as cool and remote as mathematics, there is an element of the aesthetic involved."

Both of these quotes are from a lecture Motherwell gave on 2/7/1970, at St. Paul’s School, Concord, New Hampshire. I picked them up from The Writings of Robert Motherwell, edited by Dore Ashton with Joan Banach, 2007, Berkeley, CA: University of California Press, p. 250 ; from the article, "On the Humanism of Abstraction, the Artist Speaks", 1970, Robert Motherwell at St. Paul's School, exhibition catalogue, and reprinted in Tracks: A Journal of Artist Writings, vol. 1, no. 1, 1974.

I purposely selected these quotes for their reference to math, but the article they come from is not much about math. The quotes are out of context, and I recommend reading the entire selection, if not the book. Nevertheless, I take exception to the idea that a math formula is less complex than a painted surface with three lines. Applied mathematics, being the language we use to describe natural concepts (as in e=mc²), is not necessarily abstract. The formulas of pure mathematics are often as not the language of a larger process or proof, so though abstract are still complex. Sometimes three lines are equally as complex &/or abstract as a formula, as in three lines making a right triangle and the formula, a²+b²=c².

Here's a gratuitous design, Samurai.