These from [Coplien on Nature of Order] - Brad Appleton's notes on Jim Coplien's presentation
Through repeated experiments, Alexander has correlated this objective sense of beauty to the presence of fifteen fundamental properties which recur throughout human cognition and are apparently rooted deep within the psyche. The number of these properties (15) is not at all "magical" in any way. It is completely arbitrary, and merely the byproduct of correlating the results from Alexander's experiments. These 15 fundamental structural properties are:
Within or around any given center, there will exist smaller centers
which are one level of scale lesser in size. The difference in scale
should be somewhere between a factor of 4 and a factor of 10.
[[cf. Salingaros in "The Laws of Architecture from a Physicist's
Perspective" where he states the three laws of architecture as:
Cope reads from the NoO draft: "any perturbation of irregularity which develops near [center] 'A' causes a latent center 'B'. As this latent center gets stronger, this will then cause an aggregation or nucleation near the first center. It must be a jump in scale or else it will not preserve the structure."
Whenever a center is formed, successive structure preserving transformations will call out smaller centers around and within it, which, by virtue of their spatial positioning, are arranged in just such a way that it strengthens the first center. As the architecture unfolds and more structure preserving transformations are applied, this will happen with all the centers in the system.
Alternating repetition involves "latent centers" (centers which are not yet completely formed) in the adjacent spaces between the repetition of centers. When the latent centers become more fully formed, they will become a repeating sequence of centers all on their own; one which alternates in between the sequence of centers from which it emerged. If the spaces between repeating centers are at all similar, this will eventually occur between all centers in the first system of centers and the second. After a number of these transformations, alternating repetition will appear.
Each center gives rise to one or more (fuzzy) boundaries which enclose the center itself, and also which enclose the adjacent centers around it. As the centers are intensified and more centers are added (using structure preserving transformations) the boundaries will intensify and form bigger boundaries which intensify the centers within them. As this process repeats itself under more structure preserving transformations, these boundaries will occur repeatedly throughout space.
As centers become reinforced and intensified, latent centers become definite centers and take on "good shape" which in turn are reshaped as the design unfolds. [How is this different from "Boundaries"?] In this manner, vague centers are reshaped and strengthened with more definite centers. The overall shape of the centers is intensified by these transformations and itself becomes a center possessing "good shape."
[[NOTE: at this point Cope is spending less and less time talking about each of these properties and is picking up speed, zipping through them. Im not sure how much more description Ill be able to provide for the rest of them.]]
The pattern "City-Country-Fingers" from APL is an example of deep interlock and ambiguity. [Hmmn - I wonder if many of the classic drawings by M. C. Escher would also be good examples of this.] Along any edge between two centers, random perturbations will form disturbances whish exist as latent centers. As these seemingly random latent centers are intensified, the proceed into one zone, or the other, or along the edge. When centers along the edge are strengthened, they penetrate more deeply into the two zones (centers) on either side; thus centers are established which penetrate both of the large centers and new centers along the edge will appear to belong equally to both sides.
One effect of structure preserving transformations is that they give each center more distinctness by differentiating space between them and within them. This differentiation from the surrounding space is progressively intensified to achieve "contrast".
In order to ensure that smaller centers work together to form larger ones, smaller centers are often irregular in shape ("syncopated in shape and arrangement") to create a smooth fit of the smaller centers into the larger centers. This kind of "imperfect similarity" gives rise to "roughness", and is an outward sign of deeper order as larger centers are being perfected.
Echoes are repeated structural similarities. They are structures which repeat with slight variations from one another, as opposed to being precisely identical [cf. variations of a theme in classical music]. Cope (reading again): "In the weatherbeaten face of an old man, the lines and angles all over his face, make a similar pattern, centers are organized with a similar morphology; there are echoes from point to point throughout his face."
These similarities of process create structural similarities (or "echoes") in different parts of the system: systems of similar angles and shapes which bear a familial resemblance to each other in the different centers where they occur.
"The Void" results from periodic "cleaning out" or "self-organizing". Centers, or groups of centers, will sometimes become too abundant and/or too intense. The resulting conglomeration begins to look a little chaotic or "too busy" (perhaps even confusing). "The Void" is a kind of protective-response [cf. basic animal instincts of self-preservation, and preservation of the species] to preserve the structure of the system by purging some of the overly-intense centers, replacing them with a "homogeneous emptiness" which differentiates and clarifies some of the smaller, less visible centers (perhaps even heretofore unnoticed [latent?]).
In general, to protect structure and the multitude of smaller and less important centers and overly differentiated space gets purged or cleaned out -- replaced by a "homogeneous emptiness." Some of the smaller centers are then strengthened by this resulting emptiness and homogeneity. So "the Void" takes on the purpose of strengthening smaller centers that were previously unnoticed.
A kind of "intensification by simplification" of centers which removes unnecessary elements and distills the resulting centers to their more basic essence. As the cleaning out of irrelevant structures continues, and centers are further intensified by simplification, a state slowly appears in which nothing unnecessary remains present. All irrelevant and confusing centers that irritate the structure or reduce the value/importance of other centers are swept away until a simple, inner-state arises naturally as part of the structure-preserving process. [[Hmmn - this one is starting to sound a lot like "self-organizing systems", even more so than the void IMHO.].
Cope reading again: "as the structure develops through its uncompleted forms, the pressure to unify continues. Each part becomes wedded more firmly to the others. Exaggerated differences are eliminated.
Not-separateness is a kind of ambiguous unity (e.g.: where does the pond end and the shore begin? or where does the shoreline and and the sky begin?) [How does this differ from "Deep Interlock and Ambiguity"?] As the last centers are placed to create an almost sheet-like unity, each part begins to appear inseparable from the others.
[[These last three properties seem to be about taking things out, rather than adding things into the mix -- cleaning house if you will.]]
Each "wholeness" is made entirely from centers which consist of these elements in appropriate combination, and the resulting latent structural boundaries (cf. "Golden" rectangles). As the wholeness of the system unfolds, these 15 structure-preserving properties recur more and more frequently at all levels of scale (they are both comprehensive and dense in their pervasiveness). While this is happening, latent centers in the system become progressively more intense and more differentiated. Supposedly, this reflects precisely how the 15 properties recur in nature and natural processes. [[Why does all of this stuff keep making me think more and more of fractals?]]