The Computation of Space


‘Technological progress’ has generated many new materials, and incorporated computation into many levels of their fabrication. This has established the possibility to create a new diversity of spatial forms and to begin to identify experimental methods of construction that weave the foundational abstraction of computation into their fabric. Once the joints are in place however, or the welds are made, the process of creation stops. We have yet to experience a space that is fully computational. This text aims to suggest a small means of doing so.

Bernard Cache1 has described the Constructivist project as something that is not finished but that is present as an abundant resource, of the imagination and of the synthesis of spatial formations coupled with social dynamics and multiplicative powers of abstraction. Aside from its typological visual characteristics, what constructivism allows for is this coupling of the abstract and the social, making them both concrete, material that can be experimented with in the here and now, but also allowing the fullness of the dimensions that they exist upon to become palpable, thinkable and more importantly to be worked in.

Much of the contemporary work on materials in architecture, although it has the capacity for thinking things through from the level of the multiple interactions of many parts by virtue of their ‘non-standard’ interaction amputates the potential of its working by staying solely within the domain of what it characterises as material. This is the continuing folly of engineering or of architects who consider themselves only special high–expertise servants or savants whose role is clearly demarcated. Instead, we propose that form-finding goes on at many scales, including the social.2 Cache’s image of Constructivism allows not for some codified vocabulary of forms, but for a sense of working with plasticity and constraints, affordances at any level of reality that may be to hand or that might yet be invented.

Perhaps Modernism, in what might be called its official guises, not those that like Alfred Jarry or Mallarmé had the sense to wreck their own pretence to authority from the very beginning, came to an impasse because it generated a codified vocabulary of forms rather than creating a codification mechanism for the generation of vocabularies? In other words, perhaps that architecture that fitted into the administrative functions of the state so smoothly, did so because it was borne of a failure to take a few steps up the scale of abstraction.


The current intensification of the mathematisation of space and spatial processes no longer allows us this option. In order to understand the city, one has to understand it at multiple scales, not simply of the local and the global, or other wavings of the magnifying glass, but as something that emerges out of interacting waves of force and materials operating at scales involving the following:

  • The informational normalisation of space, such as traffic control, security or logistics systems3
  • The weaving of informational space into the fabric of the city, layering it in with typologies of form characteristic of earlier eras, the mediaeval or modern city
  • The increasing production of forms of matter ready to correspond to its informational modelling, what Kas Oosterhuis calls file to factory production4, but also at the ‘lower’ scale of nanotechnological fabrication and the ‘higher’ one of pre-emptive actuations of whole futures via computer simulations
  • The increasing incorporation of previously ‘expert’ knowledges, of computing for instance, into popular culture
  • The development of algorithmic culture as a compositional force in areas such as music
  • The development of architectural software based upon the interaction of parameters rather than (ideal) forms
  • The increasing generation of organisational forms, especially in institutions, that are designed to meet the needs of informational schema, such as quality audits, before other considerations

Within this context, we propose an experiment that may help tease out some further possibilities in thinking through the production of space at a relatively high level of abstraction and also as something immediately made and experienced. But alongside the way in which informationalisation has mutated and layered urban space, it is also useful to first think about how it has developed a particular sub-set of practices within that space – architecture.

In terms of recognizing software as a new material within architecture a number of directions of its usage within different architectural practices are identifiable:

Case 1: Software as ‘a tool’, the most general and widespread use of software. Architects use AutoCAD, 3D Max, Maya, Rhino and Adobe Photoshop instead of a pen. They use software for faster drawing and quicker production of their design. In projects such as Maxilla by Stealth Unlimited5 which parametrically generates large-scale urban layouts, additional software is developed to work inside AutoCAD, and a spreadsheet program. In such cases software becomes less of a tool and more of a machine.

Case 2: Software as ‘a material’ for sculpturing buildings. Some architects, such as Lars Spuybroek6 and Greg Lynn,7 use software as a means for sculpting the space they are working on. They do not use software solely as a tool for production but more as a means for creating shapes and forms that are not possible to produce or to present on a sheet of paper. Here software is used more intuitively in the process of creation.

Case 3: Software participating in the design process. This practice questions the position of the architect as an ultimate designer. Bernard Cache took part in the creation of the software “TopSolid” that, among other features has the facility to work with an implementation of parametric design. This suite of applications was originally made for the “drafting, design and manufacture of products”8 but architects took on the exploration of the possibilities of its implementation in their practice. Parametric design is determined by the constraints on the structure to be made. These are fed into the software and a correspondingly narrow or broad spectrum of possible results can be derived and tweaked. The composition or structure produced by the architect in this way is a numerically determined geometry. A great advantage is that the architectural design produced in this way can be immediately manufactured in the factory or workshop because all the elements of the construction have their precise specifications.

Case 4: Software as an ‘interactive media’ or augmentation of space with software. Architects and designers, such as Attila Kovacs,9 Mark Goulthrope (dECOi)10, Daan Roosegaarde11 and others, use these in direct application to architectural elements in space, such as façade walls. In this way they intend to create a playful and receptive architecture. Here software and hardware are additions to the space and not its essential elements.

When making an architectural design for a building, an architect develops a programmatic model of how a building should be used. He or she creates a sort of ‘programming table’ for the planned life of the inhabitants inside the building. Even the most farsighted architects of Modernism, excited by every new machine that appeared, cars, grain processors, aeroplanes, machines used in factories, didn’t see beyond the functionality of the machines and the aesthetics that followed this functionality. They did not imagine the possibility of a machine as an abstraction, the algorithmic logic by which a certain machine works, in order to make such a logic applicable to their architectural concepts. Instead they saw building ‘as a machine’ in a purely functional sense: a house that works as perfectly as a machine. What, however, would a constructivism of the abstract machine imply?


Whilst computation offers the mechanism of fixing, of identity, and the schematic formatting of realities it is also a manifestation of the possibility of synthesis, and of the generation and development of mathematico-material drives in which permutation becomes the compositional principle. In his text, ‘A. and Pangeometry’ of 1925 El Lizzitsky states that the realm of irrational numbers allows that, “We now enter a realm that cannot be directly registered by the senses, that cannot be demonstrated, that follows from a purely logical construction and therefore represents an elementary crystallisation of human thought.”12 It is the vastly productive alienness of mathematics from ordinary experience of the world that provides the power released when its scalar level is brought into composition with others.

A question one might have with the principle of identity to be found in the logical preposition ‘A is A’, is about the conditions in which it is uttered. If it is a computation specifying, say, the transition of an email from one point to another, that is one thing: a guarantee of an adequate transmission. If it is a statement made by an entity placed in a position to determine one or other, or both, of the instances of A in a case where A is, from the perspective of another scale, equal to more than A, or something other than A alone, one is left with neither certainty nor accuracy. Such a statement might also equate to a violation.

Asking such questions means that logical structures, even those as simple as ‘this equals that’ need to be understood in terms of the grammars they weave between entities in the world. It is not possible to adequately describe an entity such as a city in logical terms, nor by a mathematical description that would not itself be larger than the city. Neveretheless, the violence of logical reason, the way it opens itself to ridicule, but also its power, is generated in part in its capacity to map relations between entities. But this is often a process in which the act of mapping itself is excluded. Perhaps it is possible to think this through with reference to minimalist sculpture and thus onto space. If we are to take a square of magnesium or of slate, as in the work of Carl Andre, and identify each of the entities as corresponding to the logical statement ‘A’, each additional statement of ‘A’ corresponds to each ‘A’ in the sculpture. However, what Andre’s work allows us to understand and literally to feel, especially in the case of the magnesium, in terms of a heat differential between the viewers’ feet and the metal, is that each enunciation of ‘A’ takes place displaced in time and space from other elements but also in arrangement with them.13 Each square of slate or magnesium has minor differences in quality, which might be derived from manufacturing and shipping processes, their handling whilst entering the category of ‘art’, or from the long timescale of the material’s geological gestation. Each square, like a timeslice of cells in a cellular automata, is placed in varying relation to the other entities in the composition. Thus there is a grammar of differentiation inherent to the repetition of what is ostensibly a ‘same’. This minor differentiation alters the meaning of ‘A’ without however necessarily altering it at the scalar level of logic or of the statement which describes the work at the level of matter-of-fact linguistic articulation. And it is perhaps that they are brought together by a description of sameness at a certain scale, which also produces the capacity for the sensing of their differentiation. Finding the means to articulate the traction or compositional force that logical statements have on other scales of reality, allows us to understand them also as materially productive.


A formalization of the field of computation was achieved in 1936 by Alan Turing. His concept was very simple. He imagined a machine that would be capable of carrying out any achievable calculation by means of an algorithmic process. The genius of his idea consists in its position as a meta-algorithmic statement, that is it produces an algorithm by means of which, as far as it is currently known, all other algorithms or ‘effective procedures’ for the purposes of computation may be written. 14

The standard model of a Turing Machine consists of an infinite tape divided into cells of equal dimensions, positioned one next to the other.15 Before the machine starts working the cells are blank. When the machine starts working, each cell may be filled with a symbol from some finite alphabet. This alphabet contains a special symbol to identify a blank cell. The Turing Machine also contains a scanner or a head, that scans the cells reading what is written in the cell, carrying out any instructions given in the ‘programming table’, recording the results of instructions and writing another symbol according to the ‘programming table’ set up. Instructions might include functions such as copying a string of symbols, deleting a symbol, moving the head a specified number of positions in one direction, writing a specified symbol in the current cell carrying out a concatenation or addition. Each computer language differentiates itself by making certain kinds of such activities, including far more complex ones, more or less built in and accessible by more or less condensed rather than convoluted commands. What is crucial about the Turing Machine is that anything that can be fully formally described can be described within the terms of this abstract machine.

Whilst it forms the crucial logical substrate of all current computers, the Turing Machine is, by and large, an imaginary machine. That is to say, that it is rare for anyone to actually set out to make it manifest both at the level of concept and as material practice, with pencil, paper, and the human computer originally imaged by Turing. Whilst much of what is revealed by such an exercise is a) the mundanity of computation carried out as a task16 and b) the normally imperceptible speed with which electronic computers carry out such work, it is possible, and perhaps fun as an exercise, to set it out. Equally, it is possible to construct a basic set of symbols corresponding not only to actions within the Turing Machine, but also to the manipulation of spatial elements ‘outside’ of it. It would also be possible to feed the results of the actions carried on at the level of work with materials back into the Turing Machine as the results of computations, or to carry out part of the computation in the handling and behaviours of the materials.


What we would like to propose is a logical continuation of the thought that already exists, and given above in the list of four tendencies in architecture. But instead of augmentation, as the last step in the architectural-computational evolution we suggest merging of the logic of the machine, in our case Turing Machine, and the logic of the architectural space with all of its connotations, social, cultural, formal, mathematical, etc. What we suggest is creation of the space that will function in algorithmic manner. We suggest three ways in which this might happen: a) creating an algorithm that will deal with specific number of basic forms (i.e. as a simple sub-set of all numerically determinable shapes, the Platonic solids: sphere, cube and cone; or using another set such as fogs, slimes and blobs) which will when executed create a form and, depending on the algorithm(s) used, might even create a different form with the same elements on every execution. The accent is on shape; b) creating an algorithm that will deal with only one element (i.e. sticks or tiles). A result of the algorithm is the structure composed by specific mathematical rules. The accent is on structure; c) create an algorithm that when executed creates a specific kind of state of the space and when as machine/space passes different steps that are previously programmed in the algorithm newly created space can be: 1) A space that can go through different states within itself – metamorphosis of the space. 2) A space that can provoke or elicit different states within the observers – metamorphosis of the inhabitants and constructors of the space.

1) The metamorphosis of the space would imply that a space changes its numerically describable characteristics, such as size, volume, colour, shape, sound, texture, etc. There is a possibility of having a database of different materials this space can use. The states of this space are changed with the input of the inhabitant but related to a previously programmed algorithm. A more illustrative example could be in having parametric design software running real-time, not only inside the computer, but in working space. The software would run a space so that an inhabitant could change all the elements of a space, could programme what shape, colour, size, texture one wants to have and then live in its own experiment till the time one wants to make another experiment. This might be like living inside a sintering machine17, or a space equipped with swarms of ‘always-on’ Computer Numerical Control (CNC) drills18. Equally, one can imagine using a magnetic fluid that can reciprocate in creating shapes with programmable forces, or, over a larger area, banks of particles interacting with positional sound waves. This would be an ever-changing, perhaps interestingly nightmarish, space.

2) The metamorphosis of the inhabitant of the space would be concerned with the experience one has whilst inside the space. One might argue that the modernist grid itself proposes a universal machine corresponding, when reduced to one dimension, to an infinite stretch of cells. Are we not simply proposing a greater degree of formalisation? To inhabit and co-construct such a space would be a significant way of both coupling and uncoupling sensual and cognitive experience in a vivid interplay generated by the momentary degree of emphasis in working with the space. We imagine the spatial entities and processes thrown up by such procedures would allow people to more fully experience the logical and computational scales in which they exist. As an extension of this argument, and in resonance with the continuing potential of constructivism, by opening up a mechanism for the generation of spatial entities, such a project might tend to induce access to a conceptual feel for the relations of dimensionality whose multiply interlocking scales compose life.


In the exercise we have produced for the Calculation Space project at Transmedia we will use a mechanism of the Turing Machine to compose and shape an architectural space. The goal is to write a ‘programming table’ composed out of a set of algorithms that will when executed create a shape. Instead of using a computer for the calculations we return to human computers, instead of pencils and paper, simple construction materials. The structure for this exercise is:

  • Decide whether to operate via a predefined matrix to which materials are aligned or altenately by using the form-finding capacities of the materials. (Turing’s ‘infinite tape’ may be used to produce translated an infinite 3D matrix, since architectural space in computer terms has often been defined with x, y and z axes19). If using a matrix, define and create it.
  • Identifying the material (i.e. wooden rods, balloons, paper strips, string, etc)
  • Creating an algorithm for shaping an individual material element. For instance by trimming, or filling with water, sand or air. These elements will make a composition when processed through further algorithms – ‘what’ is composed
  • Creating an algorithm for positioning of created elements – ‘where’ they are composed, whether in the matrix or in reletionship to other elements
  • Creating an algorithm for composition of created elements – ‘how’ they are composed
  • Creating an algorithmic routine for the observation and adjustment of the shaping, positioning and composition of elements.

This is the first in a line of experiments we are hoping to produce related to this subject, staging new encounters between abstract machines and the experiential and social working of space. A number of questions naturally arise from such work, including discussion of: the generation of adequate notation and scripting symbols; how far the Turing Machine as logical substrate or pretext remains adequate when different compositional forces, such as other data structures or even imagination, take over; and, necessarily, the relationship of the experience of boredom to that of formalisation


1 Bernard Cache, Earth Moves, the furnishing of territories, trans. Anne Boyman, ed. Michael Speaks, MIT Press, Cambridge, 1995

2 For a useful set of arguments see, Michael Hensel and Achim Mendes, ‘Morpho-Ecologies, towards and inclusive discourse on heterogeneous architectures’ in, Michael Hensel and Achim Mendes, eds. Morpho-Ecologies, Architecture Association, London, 2006, pp.16-59

3 see, Steven Graham and Brian Marvin, Splintering Urbanism, Routledge, London, 2001

4 Kas Oosterhuis, ‘A New Kind of Building’, in, Arie Graafland and Leslie Jaye Kavanagh, Crossover, architecture, urbanism, technology, 010 Publishers, Rotterdam, 2006, pp.240-269

5 “The input comes from layouts of urban areas in an AutoCAD drawing and parameters saved in a Microsoft Excel spreadsheet. Maxilla generates the streets, envelopes and buildings from scratch.” See,

6 See, Nox Architecture,

7 See, Greg Lynn Form,


9 See, Attila Kovacs, (1969-1970) “Electronic-cybernetic scultpture and wall”

10 In work such as the ‘Aegis Hyposurface’. 2004

11 See, Studio Roosegaarde,

12 El Lizzitsky, ‘A. and Pangeometry’, in, Charles Harrison and Paul Wood, eds. Art in Theory 1900-2000, an anthology of changing ideas, (2nd. Ed.) Blackwell, Oxford, 2003, pp.317-321

13 i.e. Carl Andre, 144 Magnesium Square, 1969, Tate collection.

14 Alan Turing, On Computable Numbers, with an Application to the Entscheidungsproblem (1936), in, B. Jack Copeland, ed., The Essential Turing, Clarendon Press, Oxford 2004, pp.58-90

15 In some models of the Turing Machine the ‘scanner’ moves and the tape is motionless. Others contain multiple tapes, although ultimately these distinctions are inessential.

16 David Alan Grier, When Computers Were Human, Princeton University Press, 2005

17 A sintering machine is often used in rapid prototyping. Plastic granules are melted, layer up layer, using computer-controlled lasers, in order to build up three-dimensional entities.

18 Computer Numerical Control is one means of directly manufacturing from file. Such drills, often with several heads working in different dimensions, carve directly into blocks of material.

19 For a reading of a computational alternative to Cartesian co-ordinates see, Adrian Mackenzie, ‘Losing Time at the PlayStation: real time and the ‘whatever’ body’, from, Transductions, bodies and machines at speed, Continuum, London, 2002, pp.146-171