This post is a short excerpt from a paper recently published in Architectural Theory Review 15(2) - a special issue on architecture and geometry with lots of good (Australian) stuff. My paper (pdf) is a critical look at space-filling geometry in generative design. It touches on several things already blogged - the Water Cube and ideal foams, and some generative projects that use self-limiting growth. This excerpt looks at the Voronoi diagram as a space-filling process.

The Voronoi diagram has become a ubiquitous motif in recent generative architecture and design. It, too, can be usefully read as a space-filling model. In formal terms, a Voronoi diagram is a way of dividing up space into regions so that, for a given set of sites within that space, each region contains all points in the space that are closer to one site than any other. The result is also foam-like, but as a model the Voronoi diagram has attributes quite different to the ideal Kelvin or Weaire Phelan foams.

Firstly, while the formal model is again based on a strict set of conditions (in this case proximity) it works with an arbitrary input — the given sites —rather than defining a regular structure. The Voronoi is thus a procedural geometric structure in a way that the ideal foams are not: its structure emerges through the application of a specific process or algorithm to a given set of inputs. In this way, the specific spatial relations between neighbouring cells depend on, and emerge locally from, the given spatial relations of the specified sites. This trait also gives the Voronoi model a kind of malleability; sites can be added, removed, or moved, and the spatial structure readily adapts

Again we can read off the attributes of the Voronoi as a model in this way. It is multiplicitous, but in a different way to the grid-like uniformity of the foam models. In this case, the multiplicity can, in fact, be irregular: the sites can be positioned anywhere within a given space. However, this does not amount to much, in terms of heterogeneity: while the sites can be positioned arbitrarily, the procedure, and the relation between sites that it encodes, is entirely uniform. Each site, taken as a formal entity, is identical to every other; this is a kind of uniform diversity. Like the foam models, the Voronoi diagram treats space as indefinite and extensive: it can go on forever; its only practical limit being the computational resources required to calculate the diagram. The model itself has no way of defining an edge or bound. Finally, the variability of the Voronoi can be phrased another way, as arbitrariness; in other words, that there is no inherent reason for a given site to be where it is. There is nothing internal to the model that can generate that differentiation.

In Marc Newson's

*Voronoi Shelf,*for example (above), we see a characteristically organic variety: a range of cell sizes and shapes, different wall thicknesses, all in an agreeable state of harmony. The form gives an impression of inherent logic. It is as if the harmony of the relationships between the cell sites assures us that there must be a reason for them to be as they are. This is unsurprising, given our familiarity with, and aesthetic attunement to, naturally occurring structures that resemble these cells. The visual signature carries an association of organic logic: but in formal fact the cell sites are arbitrary, that is to say, designed. There is no necessary relation of one to another, only (we can but assume) a designer's choice, which is concealed by an appearance, much as the surface of the Water Cube conceals the regularity of its foam model.

Conversely, some designers directly address the arbitrary input to the Voronoi diagram, treating it as an opportunity and exploiting the malleability of the model. As Dimitris Gourdoukis writes, "the problem of deciding on the initial set of points is, I think, one of the most interesting in relation to voronoi diagrams." In Gourdoukis'

*Algorithmic Body*project (above), the locations of the Voronoi sites are specified by a second generative system, a cellular automaton; here the Voronoi acts as a geometric filter, interpreting and interpolating one set of spatial data into another. In Marc Fornes'

*POLYTOP*, the designer proposes a mass-customised product in which customers can design the point cloud that drives the Voronoi geometry; here a problem of arbitrary choice is turned into a feature, towards uniqueness and specificity.