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:

1. Levels of Scale
[cf. aggregation, hierarchical (de)composition?] 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:

Order on the smallest scale is established by paired contrasting elements, existing in a balanced visual tension. Large-scale order occurs when every element relates to every other element at a distance in a way that reduces entropy. The small scale is connected to the large scale through a linked hierarchy of intermediate scales with scaling factor approximately equal to e = 2.718. I find it particularly interesting that the "level of scale" suggested here happens to be the root of the so called "natural" logarithm]] 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."

2. Strong Centers
Centers may overlap but they should not completely lose their identities. Strong centers are centers that have been reinforced by other centers and intensified by structure preserving transformations. Cope reads from NoO again: "In the growth of a flower, there is a field effect taking place, caused by chemical gradients in the sap. As one center forms -- the position of the flower heads -- leaves, stems, and other parts of the flower then rearrange themselves to support the flower with sap, and in the process create a field effect which actually intensifies the center." 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.

3. Alternating Repetition
These are recurring/repeating structures throughout the design. Cope reads (yet again): "Atoms, waves, leaves, grains of sand, cirrus clouds, all have the repetition of some given type of center, many times spread through a portion of space." 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.

4. Boundaries
[cf. encapsulation?] 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.

5. Positive Space
Positive space is the region of the design filled with "latent centers". Centers which have not yet emerged, but which, when they do emerge, will fill the empty space in a "positive" way, with centers that reinforce it. [I asked the question of whether or not this was at all like the way Michelangelo claimed he could "see" the human form in the empty block of stone before he sculpted it.] Repeated structure preserving transformations will take such latent centers make it more center-like, resulting in center that are more "solid". As this empty space of latent centers is filled, pushed, pulled, connected, each piece of it becomes a center and hence more "positive".

6. Good Shape
[cf. abstraction???] 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."

7. Local Symmetries
Recurring, and often co-located, structural similarities which strengthen and reinforce centers, and which are themselves strengthened, reinforced, and replicated. [Sounds like fractals to me.]. Local symmetry will almost always intensify any center. Each local symmetry intensifies some local center. Gradually, the density of such local symmetries will increase.

8. Deep Interlock and Ambiguity
[cf. cohesion?] 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.

9. Contrast
Is the zebra white with black stripes or black with white stripes? 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".

10. Graded Variation
As centers are added and intensified, the spaces around them form step-by-step gradients pointing toward the center and strengthening it [the "field effect" again?] Slowly graded variation will appear throughout space, around many of the centers at both large and small levels of scale. The gradients supplement and reinforce existing centers.

11. Roughness
As the system evolves, centers (particularly the ones at larger layers of granularity) connect in interesting ways to form "rough" boundaries, edges, layers, and other shapes (sometimes unusual). Roughness should occur down to 1/60 of an inch [is that another level of scale?]. One example is a system of growing crystals. 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.

12. Echoes
[cf. specialization inheritance] 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.

13. The Void
[cf. class hierarchy refactoring] "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.

14. Simplicity and Inner Calm
[cf. elegance? abstraction?] 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.].

15. Not-Separateness
[cf. coupling? connectedness?] 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?]]

Extract from: “The Chicago Patterns Group presents: James O. Coplien on Christopher Alexander's latest work entitled: The Nature of Order Tuesday, 5 August 1997 [notes taken by Brad Appleton]“