The Dutchman’s Pipe: Fifteen Geometric Properties of Wholeness


Recently, Narendra Dengle, one of our staff members here at Building Beauty, noticed a flower in the yard next door to his home. He became fascinated by it because it so clearly evidences the 15 geometric properties of wholeness that Christopher Alexander describes in The Nature of Order. After a bit of investigating, Narendra found this is an Aristolochia, or Dutchman’s Pipe. He drew some sketches to study the structure of the flower.

When considered from a Euclidean point of view, the markings are chaotic or irregular. Yet, there is profound wholeness, a natural beauty and order to it. The extent to which a configuration has life, or unity within itself, is caused by the geometrical connections among its elements, and by its relation to its immediate surroundings. These properties generate wholeness.

The fifteen properties are found everywhere in nature. They can certainly be observed in the examples of the flower below.

  1. Strong Center. A center is a distinct physical system that has special marked coherence. Centers can be strengthened by any combination of the other properties.
  2. Thick Boundary. A thick boundary around, or partly around, the zone occupied by a center, helps to make that center more coherent.
  3. Levels of Scale. Centers can be embellished with smaller centers typically one-half to one-third their diameter, created within the original center, or in the space adjacent to it.
  4. Alternating Repetition. The repetition of centers form a local array. This may happen in one, two, or three dimensions.
  5. Local Symmetries. One or more local symmetries – smaller, symmetrical centers within the whole – strengthen the center.
  6. Positive Space. Empty spaces within the center can enhance the center. The “positiveness” of the empty space comes from a combination of good shape, local symmetries, boundedness, and above all, the appropriateness of the space for human purposes.
  7. Roughness. In the course of unfolding, as the different properties push and shove to make various things happen, very often something does not quite fit neatly. Adaptation is more important than Cartesian regularity.
  8. Gradient. Gradients point toward or away from a given center – gradients of size, of contrast, spacing, or orientation.
  9. Contrast. The coherence of a center is enhanced by contrast of color, of material, gradient, or density, which enables it to stand out more strongly from the field.
  10. Deep Interlock and Ambiguity.This is an interface between two adjacent centers, creating a zone, usually an ambiguous zone, that forms a third center between the original two.
  11. Echoes. Predominant angles, curves, ratios, or proportions in the shapes that have been created, give the whole system of centers a family resemblance shared by many of the centers.
  12. Good Shape. If some rough outline of a shape has been generated, examine the overall convex pieces of the shape, and try to strengthen or emphasize those pieces to make the overall shape more distinct, more recognizable.
  13. Inner Calm. A clean-up tool, working along the lines of Occam’s razor, it simplifies the configuration.
  14. The Void. At the core of any center, there is always some undisturbed and perfectly peaceful area.
  15. Not Separateness. The purpose is to overcome any separation that is caused between the configuration and its environment, or between any individual center and its immediate environment.

Think about these for a while, then take a walk in a park, or a beautiful landscape, you will see the properties everywhere. When first thinking about working with them, it is tempting to see these qualities as elements, or ingredients to be assembled and combined in a work. But of course, this raises the crucial distinction between mechanism and organism. A flower is not assembled. It grows. If we are to achieve beauty on the level of a simple flower, assembly is an obstacle rather than an aid, and this runs counter to many deeply engrained preconceptions.

The most daunting, yet exciting prospect that these observations present, is that, in order to use these qualities to their full potential, we step beyond the mechanistic worldview that has been so pervasive for the past 500 years and seek to understand their significance in our potential to make wholeness.

Dan Klyn