Uniform Diversity: Space-Filling and the Voronoi diagram

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.

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Links, Tangents, Grids and Foam (Olympic Arrays II)

The previous post brought in a wave of traffic (hello!) largely due to the connective clout of City of Sound (thanks Dan). More importantly the ideas I sketched there have been drawn in to various other discussions, including some fascinating anthropological thoughts on sport, the (transhuman) individual and the collective. Interestingly some of the more thoughtful responses came via the newly-enlarged notes field in delicious. Rodcorp wonders if the opening ceremony will come to be seen as "the defining image of peak energy" - I'm not sure; I think the massive deployment of LEDs makes a sort of cake-and-eat-it statement about abundance and the spectacular. In other words, this is "efficient" spectacle. Does anyone have the numbers on the power actually drawn by the whole show? Sevensixfive takes up the "human pixels" as a harbinger of Deleuze's society of control - though for me Deleuze's notion of control as "modulation" resonates more (and almost uncannily) with the architecture:

[T]he different control mechanisms are inseparable variations, forming a system of variable geometry the language of which is numerical (which doesn't necessarily mean binary). Enclosures are molds, distinct castings, but controls are a modulation, like a self-deforming cast that will continuously change from one moment to the other, or like a sieve whose mesh will transmute from point to point.

Which brings us neatly back to the aesthetics and cultural valency of computational architecture. I suggested that the organic multiplicity of the Birds' Nest and the Water Cube can be thought of as a post-industrial grid, where the regularity or (via Deleuze) control has in a sense moved inwards, to the level of the generative system. What we have is a kind of managed heterogeneity, where each element in the structure can be unique because computation ensures that it all works out the same and guarantees against failure. It's interesting, tangentially, that in the organic "models" for these structures - a nest of twigs, a blob of foam - the morphogenetic forces have more to do with local adaptation (to failure) than any global summation


This reading - managed heterogeneity - seems to work for the Birds' Nest, though on closer inspection the Water Cube is actually a far more literal grid. As Dan Hill points out, its bubbles are based on the Weaire-Phelan structure, a formal solution to an optimisation problem posed by Kelvin: how to divide space into equal sized volumes, with minimal surface area. One by-product of this minimal surface quality is that the structure resembles a lattice of bubbles - but these are highly idealised bubbles: uniform in volume. Moreover the Weaire-Phelan structure is spatially regular - as outlined here, "two irregular pentagonal dodecahedra (12-sided) and six tetrakaidecahedra (14-sided) form a translation unit with a lattice periodicity which is simple cubic." In other words, it's a cubic grid, though the modular unit is built from two different, irregular solids. Although I'd never noticed it before, the regularity of the grid is clearly visible on the building's surface - notice the recurring pattern in the image above (by Chris Bosse, via archidose).

As this Science News article reports, the design masks the grid structure of the foam by slicing it obliquely: "by cutting at an angle of about 111 degrees, [Carfrae] found a pattern that looked entirely natural. In fact, the pattern actually repeated in ways that were very hard for the eye to detect." At the same time, "That repetition was key, because it meant the building would be far easier to construct." So this is a formal grid, that's trying to "look natural" - and as such it's exactly like the terracotta warriors, whose "simulacrum of diversity" is constructed from permutations of mass-produced modules (see City of Sound quoting Craig Clunes, earlier in this conversation).

I'm not sure what this amounts to - perhaps it simply shows how metaphors (foam) and categories (post-industrial) are never quite what they seem, especially as they meet the pragmatics of real architecture and engineering. The Water Cube's hypermodern image rests on an aesthetic of "natural" variety and multiplicity - and after a week of the Games it's unquestionable that these structures are fundamentally machines for producing images (televisually, and literally, in the case of the Cube's illumination). But that aesthetic rests in turn on the modern industrial logics of the grid: modularity and regularity.

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Array Aesthetics (Olympic Edition)

The opening act of the Olympics was routinely spectacular in one sense, but it can also be read as a more interesting cultural, geo-political and aesthetic sign of the times. I'm talking about the massive array of (2008) drummers, who kicked off the show with a synchronised sound and light barrage. The traditional Fou drums they played were "augmented" with pressure-activated arrays of white LEDs set into the top, creating a modern variation on the opening ceremony staple of choreographed human/graphic arrays (image from boston.com).


This not-very-subtle interweaving of old and new was a major theme of the ceremony, of course, but the drummers resembled nothing so much as a giant United Visual Artists show: the crisp light of the LEDs, the transitions from flickering chaos to global order, the articulation of individuality and global coordination (individual as pixel) and the intrinsically tight audiovisual sync. Intentionally or not, artistic director Zhang Yimou hit on major motifs in recent media art (blogged earlier). As well as UVA, see Monolake's Atom, Rafael Lozano-Hemmer, and many more.

The opening ceremony is sending the same message, then, as the Games architecture: cultural and technological leapfrog. The Water Cube and the Birds Nest don't simply display China's modernity, they claim a jump into a digital, sustainable, mega-scaled future. The computational aesthetics of multiplicity that mark these structures are, again like the opening ceremony, a powerful cultural narrative: coherence, strength and beauty made of countless tiny pieces. Like the flickering grid of the drummers, the ordered diversity of these structures is important too, in that it's not total uniformity, a simple (modernist) grid. In fact these buildings contain a kind of post-industrial grid, where the uniformity or regularity is not literal or material, but procedural or computational - the computer's ability to resolve complex distributions of force is what enables the "organic" multiplicity here. Of course this post-industrial process hits the ground, on site, in the form of human labour, and that's where the social narrative begins to unravel (image by theojones).


The other none-too-subtle message of the opening ceremony was about technology, and specifically LEDs. LEDs are post-industrial lighting - semiconductors instead of mechanical-era glass and metal - not to mention efficient, bright, flexible, ubiquitous. They are already a feature of the olympic landscape in GreenPIX, a "zero energy media wall" that is also the world's largest LED display. They were everywhere in the opening ceremony, in the drummers sticks as well as the drums, the massive scroll display surface, and in the olympic rings. There was another old/new interplay here around lighting technology, with (digitally deployed) fireworks in the "old" role, but LEDs held centre stage in the stadium. Again this is a technology that is nascent in the West, and being taken up by the cogniscenti in art, design and architecture. And again, here China makes a show of trumping the West in a display of cultural and technological advancement and literally massive industrial clout. Here, above all, more is more.


p.s. Dan Hill's latest post offers an interesting angle on this, quoting Craig Clunas on the "modular mass production" of the Terracotta Army. Another array, of course.

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