Quasiperiodic Crystal Pattern

http://tilings.math.uni-bielefeld.de/tilings/substitution_rules/square_triangle

Quasicrystals are aperiodic structures that produce Bragg diffraction, i.e. they show pure point spectra. Thus they share a defining property with crystals, but differ from them by lacking a regular repeating structure. They were considered to be mathematical artefacts, known as aperiodic tiling, but physical experiments gave evidence of their material existence. Within the field of crystallography and solid state physics the discovery has produced a paradigm shift which is indeed a minor scientific revolution. - from Wikipedia

Gestalt Principles

It might very well be that the Gestalt Principles of Perception act as the Geometry of Cognition. Geometry is a way to abstractly represent and describe the relationships between points, lines, planes, areas, surfaces and volumes in physical space -- a way to represent and understand physical matter -- the arrangement of objects in space and their relationship to other objects.

What if the Gestalt Principles of Perceptions are to mental space what Geometric Principles are to physical space? What if the Gestalt Principles provide the rules that give rise to the shape of cognition, rather than merely just applying to the principles of visual perception? What if the Gestalt Principles extend further - and are more widely applicable - than we thought? While this may seem a very obvious question to ask, it isn't really, if you imagine these principles as quite literally determining parameters for how information is perceived, processed and eventually filed in human memory.

Lists of the Gestalt Principles seem highly variable in their content but here is a list that seems quite complete:

- Proximity (objects close together in space are thought to belong together; are grouped together)
- Similarity (similar objects are grouped together)
- Continuity (we strive for good continuity; a break or disruption can be overlooked and perceived as continuous)
- Closure (we strive to close or complete an open shape)
- Smallness (smaller things are perceived to be foreground objects first, rather than objects that are farther away)

http://www.artinarch.com/vp12.html


From Wikipedia:
A notable weakness with the Gestalt laws of Prägnanz is that they are descriptive not explanatory.

My thoughts on this: Of course they are descriptive. They rely on external input from the world around us. Our perceptions are judgements that we make on the world around us based on fuzzy information that involve thresholds and contextual information. We perceive things based on what's around, based on the context. It's an assessment of the figure/ground relationship that is cited as a fundamental component of the Gestalt Principles.

On another topic: Someone should take a look at the Reiser file system and see how this could represent information in human memory. The Reiser file system is a very interesting model.

Geometry is the Glue

Geometry is the way we describe the glue that binds together our world. Geometry is the language of physical structure. Geometry encompasses the principles that define relationships between points in space (or objects in space, or locations in space).

The image above is a Pennsylvania Dutch hex sign.
From Wikipedia:

Hex signs are a form of folk art, thought by some to be of a talismanic nature^{[citation needed]}, found in the Fancy Dutch tradition in Pennsylvania Dutch Country.

Artfully painted hexagonal star-like patterns are a well-known sight on Pennsylvania Dutch barns in and around central Pennsylvania, especially Berks County. However, the decoration of barns is a late development in Pennsylvania Dutch folk art. Prior to the 1830s, the cost of paint meant that most barns were unpainted. As paint became affordable, the Pennsylvania Dutch began to decorate their barns much like they decorated items in their homes. Barn decorating reached its peak in the early 20th century, at which time there were many artists who specialized in barn decorating. Drawn from a large repertoire of folk designs, barn painters combined many elements in their decorations. The geometric patterns of quilts can easily be seen in the patterns of many hex signs. Hearts and tulips seen on barns are commonly found on elaborately lettered and decorated birth, baptism and marriage certificates known as Fraktur. In recent years, they have been increasingly used by non-Pennsylvania Dutch persons as talismans for folk magic rather than as items of decoration. Some scholars argue that they have never had any connection with superstition or magic^{[citation needed]}. They are viewed as decorative symbols of ethnic identification, possibly originating in reaction to 19th Century attempts made by the government to suppress the Pennsylvania German language. In the 20th Century, mobile signs were produced as commodities. These signs could be bought and then mounted onto barns.

Paul Klee - in Search of Primitives

"The monument of Klee's obsession with this metaphysics was a singular book, The Thinking Eye, written during his teaching years at the Bauhaus - one of the most detailed manuals on the "science" of design ever written, conceived in terms of an all embracing theory of visual "equivalents" for spiritual states which, in its knotty elaboration, rivalled Kandinsky's. Klee tended to see the world as a model, a kind of orrery run up by the cosmic clockmaker - a Swiss God - to demonstrate spiritual truth. This helps account for the toylike character of his fantasies; if the world had no final reality, it could be represented with the freest, most schematic wit, and this Klee set out to do. Hence his reputation as a petit-maître.

"Like Kandinsky, Klee valued the "primitive," and especially the art of children. He envied their polymorphous freedom to create signs, and respected their innocence and directness. 'Do not laugh, reader! Children also have artistic ability, and there is wisdom in their having it! The more helpless they are, the more instructive are the examples they furnish us ....' In his desire to paint 'as though newborn, knowing absolutely nothing about Europe,' Klee was a complete European. His work ferreted around in innumerable crannies of culture, bringing back small trophies and emblems from botany, astronomy, physics, and psychology. Music had a special influence on him. He believed that eighteenth-century counterpoint (his favourite form) could be translated quite directly into gradations of colour and value, repetitions and changes of motif; his compositions of stacked forms, fanned out like decks of cards or colour swatches, are attempts to freeze time in a static composition, to give visual motifs the "unfolding" quality of aural ones - and this sense of rhythmic disclosure, repetition, and blossoming transferred itself, quite naturally, to Klee's images of plants and flowers. He was the compleat Romantic, hearing the Weltgeist in every puff of wind, reverent before nature but careful to stylize it. Klee's assumptions were unabashedly transcendentalist. 'Formerly we used to represent things visible on earth,' he wrote in 1920, 'things we either liked to look at or would have liked to see. Today we reveal the reality that is behind visible things, thus expressing the belief that the visible world is merely an isolated case in relation to the universe and that there are many more other, latent realities ...'
"Klee's career was a search for the symbols and metaphors that would make this belief visible. More than any other painter outside the Surrealist movement (with which his work had many affinities - its interest in dreams, in primitive art, in myth, and cultural incongruity), he refused to draw hard distinctions between art and writing.

http://www.artchive.com/artchive/K/klee.html

Here's my question: Why are Klee paintings celebrated as masterpieces in the Guggenheim museum, when they look like quilts? People have been generating pattern fields like the ones seen in Klee's paintings for thousands of years. If Klee was searching to represent ‘the reality behind visible things’, then why does so much of his work look like the "primitive" arts and crafts of women and men throughout history? Rugs, pottery, patchwork quilts, and paintings of the aboriginals from Australia...were cultures in the past also trying to represent the reality behind visible things? The cognitive reality that all humans share?

Perhaps Klee's work was found particularly novel, especially within a 20th century context -- when people had lost sight of fundamental patterns, and needed to be reminded of them.

image reference:

http://www.metmuseum.org/special/se_event.asp?OccurrenceId={D88B5218-2B74-46D0-B36A-4ADC7B4D6A3A}

Patternmakers as Mirrors of Nature

Today, I started searching for patterns in nature, thinking that pattern makers are trying to mirror nature. We see these remarkable patterns in the world around us, and we copy them and reproduce them as objects. Patterns of honeycombs, and Komodo dragon scales, and spirals found in cactus. And then I searched for the oldest life on earth and found a one-celled creature. But I thought this level was still too high, too ‘gross’, so I searched for molecular structures and saw some beautiful patterns. And then I searched for atomic structures, and found a beautiful hexagonal matrix pattern for silicon that is visible at the atomic scale in an article: The World According to Nanometers. Again, I have seen this pattern in quilts.

The image (above is) of a silicon surface, captured by Taisuke Ohta, Fumio Ohuchi, and Marjorie Olmstead using a scanning tunneling microscope, patterns are visible at the atomic scale. Bright spots are individual atoms.

http://images.google.ca/imgres?imgurl=http://www.artsci.washington.edu/news/WinterSpring04/Photos/nano_image.jpg&imgrefurl=http://www.artsci.washington.edu/news/WinterSpring04/Nanotech.htm&h=300&w=300&sz=182&hl=en&start=50&tbnid=WS3-ztUHH3tunM:&tbnh=116&tbnw=116&prev=/images%3Fq%3Datomic%2Bpatterns%26start%3D40%26ndsp%3D20%26svnum%3D10%26hl%3Den%26hs%3DOTK%26lr%3D%26client%3Dfirefox-a%26rls%3Dorg.mozilla:en-US:official%26sa%3DN

In the late 1920s, an American quilt revival brought in a new color palette of pastel prints. Quilts made from these fabrics are sometimes referred to as Depression Quilts, since the styles and fabrics continued through the Great Depression. Mamie ordered her fabrics from a catalog, either Montgomery Ward or Sears, Roebuck and Co.

http://images.google.ca/imgres?imgurl=http://www.sohe.wisc.edu/depts/hlatc/quiltsExhibit/images/Med_sized_pix/GrandmaFlower_D1_MED.jpg&imgrefurl=http://www.sohe.wisc.edu/depts/hlatc/quiltsExhibit/flowerGarden.html&h=216&w=288&sz=25&hl=en&start=2&tbnid=ezPzIjSvW6jkFM:&tbnh=86&tbnw=115&prev=/images%3Fq%3Dgrandma%2527s%2Bflower%2Bgarden%2Bquilt%26svnum%3D10%26hl%3Den%26lr%3D%26client%3Dfirefox-a%26rls%3Dorg.mozilla:en-US:official%26sa%3DN

Now, the chances are quite likely that ‘Grandma’ never used an atomic microscope. And yet how did she know to replicate the atomic structure of silicon in her quilt pattern? This is the mystery that I’m trying to solve. I want to know how these two things are connected.

Norah Gaughan has written a new book ‘Knitting Nature’ April 2006 and she has based her patterns on natural patterns

http://www.berroco.com/knitbits/html/KB137_interview.html

Again, I’m thinking, ‘So what? People imitate nature? That is something we’ve all known about for a long time, right?” True, but what we haven’t discovered is how these patterns may be used in our own minds to organize information.

What type of information might be organized in a hexagonal pattern? We commonly organize and understand information that is organized in a graph, as a matrix. This is a pretty standard form of information organization that has been used for several centuries.

There is something quite circular with patterns, and our understanding of them.

We see pattern structures in nature at all levels of scale, from the atomic on up to plants and trees and shells and landscape. We ‘learn’ these patterns through living in our environment and we record these patterns in our memories. Then we attach information onto these structures at a low level, learn how to intellectually abstract from real, concrete (physical) instances into an abstract mental structure, and we see the same structure again in a virtual fashion… that we then describe using mathematics. This is considered ‘higher thinking’.

So, we internalize the patterned structures that we see in our environment, load these structures up with information until we can achieve a mental level of abstraction, so that we can create or replicate using our own minds and hands what we see out in the real world. Weird. We internalize and then we externalize. We are not just pattern recognizers, but we then also learn to become pattern generators. This is a uniquely human capability. This is what creativity is all about. Seeing patterns, internalizing patterns, abstracting patterns and making more patterns.

We are pattern recognizers, pattern recorders and pattern makers. Truly.

Patterns are an efficient way for physical matter to be organized. Patterns are rather social, when you think about them. Individual items are held together in relationship by an ‘agreed’ structure of organization, just like a society of humans. There are roles and rules.

As below, so above.

The Sound of Simple Shapes

A rather bizarre question popped into my head today. ...What does a spiral symbol sound like? If a spiral is a path, and the center of the spiral is the start point and the spiral is 'unwound' and transformed onto a line that represents time, then the sound would start out high-pitched (high frequency) and soft (low amplitude), and get lower in pitch (low frequency) and louder (higher amplitude) over time.

The sound of a circle? A short (brief) continuous tone of equal amplitude and frequency.

The sound of a square? A rising continuous tone, then held for a brief duration, then falling and held for a brief duration....remarkably similar to the prosodic structure for approval.

This is the general idea. If phosphene patterns are commonly found in prehistoric art (and carry on throughout the ages) in cultures with an assumed verbal language (but no written language), then is there a link between SOUND and these simple shapes/patterns? Perhaps the sounds of these shapes correspond with sounds heard in the natural environment, or perhaps the sounds correspond with chanting utterances of sacred rituals?

Keep in mind that the sounds I'm talking about here are not sounds arbitrarily assigned by humans to the phosphene symbol. The sounds here are achieved when the actual path of the symbol is 'unravelled' and transformed into a sound wave. Also rather strange is that the sound waves derived from these symbols look very much like the acoustic mappings of bird calls.

In prehistoric pottery, border patterns are prolific. If a border contains several discrete symbols, the resulting sound map would reflect simultaneity, achieving a harmonic set.

Phosphene Lexicon

Some qualified speculation about a Phosphene Lexicon....

"But what turned out to be truly amazing," said Cardenas-Arroyo, "was that many of the patterns made by the Tukanos were nearly identical to the phosphene patterns described 40 years earlier by Knoll." Moreover, he showed how phosphene-inspired patterns used in contemporary indigenous cultures are very similar to designs on ancient pottery.

"I believe that many of these design patterns were like an unspoken language that held symbolic meaning through their iconography, which transcended linguistic and cultural barriers," he said.

"Archaeologists should make a greater effort to decode the symbolism beyond the art."

- Felipe Cardenas-Arroyo
Columbian Archaeologist

http://www.iadb.org/idbamerica/Archive/stories/1998/eng/e998k.htm