Architectonic Articulations

Symbolic Solids or: The Contract of the Architect

* This is the manuscript version of my forthcoming article for a book edited by José Aragüez: The Building (Lars Müller publishers, 2016).

“A computational approach enables architecture to be embedded with an extraordinary degree of information.“
(Michael Hansmeyer:

I would like to discuss three of Michael Hansmeyer’s recent experiments in computational architecture (Platonic Solids (2009), Subdivided Columns (2010), Digital Grotesque (together with Benjamin Dillenburger 2013)) in relation to what Peter Eisenman has recently foregrounded, in a conversation on the foundations of digital architecture: Architecture ought to find ways of coping with such miraculous deeds as “saving a void from the negative by making it positive,” or doubling and dividing volumes (cubes).[1] In this formulation, Eisenman was explaining how in his Biology Center in Frankfurt, Germany (1987) he was trying to rid the architectonics of this project from any figure/ground dialectics. His interest was to “use a computer as a modeling tool capable of drafting predefined forms in endless sequences based on logical statements in code”[2]. With regard to this, I want to suggest, Hansmeyer’s own experiments provide an inverse approach to the same interest (overcoming structuralist figure/ground dialectics as well as the poststructuralist infinitary rendering of it): Where Eisenman works by sequencing predefined forms (form as logical statement), through considering form as symbolically coated by code (symbolical in the mathematical sense of the word), Hansmeyer explores how we can consider form not as logical but as entropic (generic) statement, and, in its coded ‘coat’, as a means for algebraic ‘sourcing’ of ‘poly-tomic elementariness’ (poly-tomos, Greek for that which can be divided in many ways). His quest is one for form made up of symbolically coated atomicity or in-divisibility (a-tomos, Greek for that which cannot be divided), for which it indeed makes sense to say, as he does, that architecture “can be embedded with an enormous amount of information”[3]. This text elaborates on how we can think of ‘form’, with Hansmeyer’s experiments, as the ‘contractual form’ of ‘symbolic solids’. Like geometrical forms, they too regulate relationships rationally; but the can do so with larger or lesser capacity for dealing with the co-existence of ‘aspects that don’t seem to add up’ in unambiguous manner. I thereby suggest to refer to Eisenman’s ‘foundations’ as ‘architectonic plots’.[4]

Operations involving negativity, voidness, formality and elementariness are related to a set of classical problems in mathematics—among them doubling the cube, angle trisection, and squaring a circle— that has arguably propelled the development of mathematics as the ‘technics/art of thinking’ at large[5]: These apparently very specific geometrical problems are actually ‘genuinely’ abstract, or in other words ‘purely mathematical’, in the sense that they do not lend themselves handy for immediate application to a specific situation at hand. They make things appear more instead of less complicated and unsettled, at least initially; hence their treatment has often raised mistrust.[6] But on the other hand, the iterative and many-versed treatments have also driven the progressive movement towards mastering more and more abstract manners of counting and measuring, which is evidently indispensible for the invention of novel technical instruments and procedures.[7] These geometrical problems involve constructions that keep the study of form and its relation to number apart. They refuse to conflate exactitude (geometry) and rigor (rationality), they oblige us to treat meaning and measurement, attribution and assessment with regard to the constructed object, separately. How? By involving mechanical operations with symbolized ‘units’ whose intuitive ‘reference’ to graspable things escapes exact comprehension in principle: zero, negative entities (negative numbers), the root of a negative number (imaginary numbers), a bounded infinity (via the distinction between cardinality and ordinality) or the particle of a partitioned one (irrational numbers).[8] These symbols, hence, must count as algebraic, pure place-holders, notational substitute-characters one relies upon for reasons that cannot void themselves of their constitutive speculativeness,[9] because what these ‘superior’ symbols claim to stand in for counts, according to all established common sense and knowledge, at best as genuinely unsettled: mathematical symbols are in pact with the (mathematical) irrational.

We can see better now the philosophical relevance of what is at stake with the ‘foundations’ of digital architecture, or, respectively, its ‘architectonic plots’. One may ask, however, what is supposed to distinguish the works of architecture from those of the mechanical arts[10] —which were indeed referred to as ‘arts’ for precisely the reason that they could deal with the infinite in indefinite manner. In any case, such ‘foundations’ touch upon the nature of the sacred, its relation not only with art but also with the problem of violence and the practices of sacrifice and exclusion, and through that with the anthropologically invariant role of myth for the constitution of a collective ‘we’.[11] This is indeed why I want to speak of Eisenman’s ‘foundations’ for architecture in the sense Aristotle began to use the term ‘plot’ in his poetics as the dramatically versed “arrangement of events that happen in a story”, which I read essentially as a theory of how to dramatize myth in a symbolical as-if domain of the quasi, where the irrationality (infinitaryness) of real deeds are symbolized by substitute characters (e.g. acts, actors, characters, performance).[12] This distinction is also what relates the topos of the ‘work’ to that of ‘art’, and with George Bataille to the idea of a general economy of the sacred as the infinite work, with Maurice Blanchot to the idea that literature, by speaking about what cannot be said, is the work of death, and with Jean-Luc Nancy to a decoupling of communication from work, as the im-possible constitution of the in-operative community.[13]

As a manner of decoupling this architectonic plot from all tentative immediacy, I suggest to view ‘the building’ as ‘the plot of dramatizing architectonic articulation’, and the arrangement of events it is to accommodate as the dramatic prism through which we can regard a building as an actual and active form of thinking. A prism in optics is a diligently cut and polished solid figure, which breaks the rays of light that shine through it according to particular patterns—hence a prism affords perspective and method, yet in a manner in which precision and rigor are kept apart. A building then is a form of thinking because it resolves a particular ‘insight’ as a spectrum—hence a mediated ‘in-sight’. Of such a mediated insight, we can think of as a ‘theoretical promise’: theoretical because what is at stake in such a promise becomes fixed only as the promise is being referred to as a building’s singularly dramatized, yet generic, plot. The architect does not properly speaking realize such a promise, I would suggest, she dramatizes it. The building’s plot, in that sense, becomes positively in-definite rather than positively in-finite (as mythical divinization or theological revelation would suggest). Being a ‘natural’ myth (rather than the mythological story of a tradition), in that it embraces means of ‘superior’ mathematics, such a ‘building as theoretical promise’ is inevitably in pact with an anonymous, impersonal agency. Yet it is not the agency of a tool (like the computer, a ruler or a compass), it is the agency at work in mathematics as the art of learning. A building’s plot engenders a presence (is a ‘natural’ myth), and hence is actual, precisely because its rational accounts cannot exhaustively legitimate what it promises to manifest.

Architectural disposition in planning (ichnographia, scenographia, orthographia)[14] then depends not alone on intention and design (on the side of the architect as cognitive subject) nor on objective requirement (on the side of a building as object of cognition), but also on a kind of mastership that belongs to the object, that relates to what I called ‘the impersonal agency’ that is at work in architectonically active form (form of thinking as the theoretical promise that articulates formulations, indefinitely so, of what can be learnt). It is this mastership of objectivities with which an architect ‘partners’ when she formulates the promise a building is going to manifest; the architectonic qualities can no longer refer to an essentialist reference of ‘the beautiful’, ‘the adequate’, or ‘the useful’ immediately, but only in a manner that contracts the abstract identity of such quality symbolically.

This has two important aspects:

  • With regard to itself, such contractual quality is always perfect, fulfilled, yet in a speculative manner void of reason because algebraic symbols are self-referential in the sense that they refer to one and the same cipher (the total of a cipher’s notational symbols must always make up zero (‘cipher’ is a word for nothing, zero)). Like all speculation, algebraic symbols proceed circuitously: they build on what is assumed to be given, then they support explanations for the givenness of what has been assumed to be there in the first place.
  • With regard to the real stakes that belong to the abstract identity of such quality, that is being contracted in the formulation of a promise, such quality is never perfect or fulfilled—because it is the very character of a promise to be overdetermined with possible meaning, to build upon polytomic elements, and, we might add to highlight the role of sophistication (and sophistry): to sparkle in its formulations with more or less objective brilliance or mastership.

So how can we look within this context at Michael Hansmeyer’s experiments in computational architecture?[15] His interest is “to embed an extraordinary degree of information,” as he puts it[16]. As long as ‘form’ is taken as equivalent to ‘pre-determined logical statement’, such an interest appears vain, and the involved computation reduces to sophistry. What I would like to suggest is that all three of Hansmeyer’s works explore how an architect can ‘partner’ with this ‘impersonal agency’ that is at work in active forms of thinking. Furthermore, all three works explore and seek getting used to the formality of symbolic self-referentiality that is not only ‘tauto-logical’ (and reproduces variations of the same), but rather invariantly ‘tauto-nomical’ because it regards the elementariness of form in a quasi-atomist manner; ‘quasi’ because it explores the symbolic polytomy of the atoms in relation to the code in which they are computed. Each work attributes symbolically an un-reasoned, (entropically ichnographic) elementariness to an architectonic form of thinking (in Hansmeyer’s work: platonic solids, columns, grotto), upon which it operates and from which it engenders a symbolical solid: the fully rational (operation of subdivision) and practical (aesthetic articulation) manifestation of beauty as a promise.

Platonic Solids (2009) attends to the most primitive forms, the platonic solids, and repeatedly employs one single operation – the division of a form’s faces into smaller faces – until a symbolic form is articulated in »agreeable« manner.

Subdivided Columns (2010) focuses on the Doric Column and subdivides it iteratively eight times according to its own order, thereby producing 5,8 million faces. One such pre-specific typicality of this order is elected here and plotted into a cardboard model (1 mm sheets).


The Digital Grotesque (2013, [together with Benjamin Dillenburger]) is a walkable room produced by 3-D printing. A simple input form is recursively refined and enriched, culminating in a geometric mesh of 260 million individually specified facets. Every detail of the architecture in this project is generated through customized algorithms, without any manual intervention. And yet it is not only that during the computation every point (spatial or temporal) is latently connected to all the other points; rather, each one actively reflects them all at once at every instant, and this in an individual manner that in each case responds to and communicates with all the others – the actuality of this happening, this is what the architects settle.



[1] „The Foundations of Digital Architecture: Peter Eisenman,“ in conversation with Gregg Lynn on the opening day of the Archaeology of the Digital exhibition at the Canadian Center for Architecture (CCA). Available online:

[2] as the project’s presentation at the Canadian Center for Architecture (CCA) puts it:

[3] Michael Hansmeyer:

[4] This means creating a conceptual hybrid between architecture theory (Vitruv’s triplet of addressing ‚disposition’: ichnography, scenography, orthography) and and drama theory (Aristotle, De Poetica (335 BCE)).

[5] Mathematics literally means all that pertains to mathema, Greek for ‘learning’.

[6] The French Academy of Science, for instance, has declared right after the French Revolution that work on these problems by mathematicians will no longer be considered as legitimate contributions to the corpus of academic knowledge (by argument of being a waste of the nation’s intellectual resources). “The Academy took this year the decision to never again consider a solution for the problems of doubling the cube, trisection of an angle, squaring the circle and of a machine of claimed perpetual movement. Such sort of researches has the downfall that it is costly, it ruins more than one family, and often, specialists in mechanics, who could have rendered great services, and used for this purpose their fortune, their time and their genius.“ Cited in: American Mathematical Society, The Millennium Prize Problems, published with the Clay Mathematics Institute, Cambridge Massachusetts, 2006. ( Indeed, solutions to these purely abstract problems have often been brought forward in suggested “demonstrations” (in the axiomatic sense of formal proofs) of the existence of an infinite magnitude (like “God”), cf. Hudry, Françoise (ed.), Le Livre des XXIV Philosophes (Latin text and French translation), Millon, Grenoble, 1989.

[7] Methodically speaking, the point of mistrust towards manners of treating these problems is that suggesting resolutions to these purely mathematical problems is not rooted in “elementary” mathematics (the Euclidean elements), but involve what is somewhat helplessly called “superior” mathematics—which means that any resolution to these problems manifests in constructions (axiomatic demonstrations of theorems) that cannot be achieved and reproduced with compass and ruler alone. This „superiority“ is an ambiguous term, as in the Greek (Platonic) origin, the term was used in reversed manner and attributed to geometry for precisely the reason that it did not involve numbers, and hence proposed a way to bypass the problems of incommensurability that arise for example from demonstrations dealing with the irrational diagonal of a square. Eg. cf. John Tabak: Geometry: the language of space and form, Facts on File Math Library, New York 2004.

[8] While in Antiquity, mathematicians and mechanical artists didn’t have the symbolic notations we have today, the Hippocrates’ and Archimedes’ of former times did have technical devices (like spirals, feeding from what we today call “angular momentum”) that operate on the same physical principles we have learnt to index by symbolical notation only much later. This is for example the strong emphasis underlying Michel Serres’ suggestion to relate Lucretius’ clinamen to contemporary physics: Archimedes’ mathematics, which underlies and gives coherence to Lucretius poem (according to Serres) translated into hydraulic mechanics; it is the blindness of modern science to conceive of mechanics as static, and fluid mechanics as a derivative thereof. “We mix experiment with equations. And we accompany the protocol, step by step, with formalism and with metrics: Without this continual proximity, no experimentation, no law. The Greeks would, I believe, have been strongly repulsed by this mixture. They did not have, as we do, a unitary mathematical physics. Theirs was double. They produced rigorous formal systems and dissertations upon nature like two separate linguistic families, like two disjunct wholes. And, since they are often signed with completely different proper names., no one dares to think that they are structurally isomorphic.” Michel Serres, The Birth of Physics, Transl. by Jack Hawkes, Clinamen Press, London 2000, p. 13. That nature and mathematics are indeed to be regarded as disjunct, and that they ought to be reasoned as isomorphic, is Serres key point also in The Natural Contract (1990): the real and the rational are to be considered equipollent (equality in force, power or validity); one cannot be subjected to the other, this in-subordinatory relation is the Natural Contract is supposed to formulate.

[9] cf. Alfred North Whitehead, Treatise on Universal Algebra with applications, Cambridge University Press, Cambridge 1910. In relation to this cf. also Michel Serres: „At the beginning of the seventeenth century, when what we came to call the applied sciences first made their appearance, a theory spreads that one can find in several authors, although none of them is its sole source, which seeks to account for a harmony that is not self-evident. This discourse may be found in the work of Leibniz, Descartes, Pascal, Fontenelle and so on, but even before them in Galileo, and perhaps in a number of alchemists. What spreads is the idea that nature is written in mathematical language. Language here is too strong or too weak a word. ln fact mathematics is not a language: rather, nature is coded. The inventions of the time do not boast of having wrested nature’s linguistic secret from it, but of having found the key to the cypher. Nature is hidden behind a cypher. Mathematics is a code, and since it is nor arbitrary, it is rather a cypher, Now, since this idea in fact constitutes the invention or the discovery, nature is hidden twice. First under the cypher. Then under a dexterity, a modesty, a subtlery, which prevents our reading the cypher even from an open book. Nature hides under a cypher. Experimentation, invention, consist in making it appear.” Serres, Birth of Physics, ibid., p. 140.

[10] It is interesting to remember that for Vitruv, the subject matter of architecture included (1) buildings, (2) machinery, and (3) clocks (gnomons). Vitruvius: De architectura libri decem, book one.

[11] Cf. especially René Girard, I See Satan Fall Like Lightning. Maryknoll: Orbis Books, 2001; Georges Bataille,Theory of Religion, MIT Press, Cambridge MA 1992, also Jean-Luc Nancy, “Myth Interrupted” in The Inoperative Community, University of Minnesota Press, Minneapolis 1991, p. 43-70; for a more generally historical documentation cf. Robert Hamerton-Kelly (Ed.), Violent Origins: Walter Burkert, Rene Girard, and Jonathan Z. Smith on Ritual Killing and Cultural Formation, Palo Alto, California: Stanford University Press 1988.

[12] ‘Action’ derives from the Latin translation of the Greek term energeia (and not dynamis) for a reason: energeia meant activity in infinitive mode, and it was reserved by Aristotle for the prime mover because it denoted what exceeds rational grasping via the sequencing of this infinitive activity into finite elements of processes cf. Ludger Jansen, Tun und Können. Ein systematischer Kommentar zu Aristoteles’ Theorie der Vermögen im neunten Buch der Metaphysik, Springer, Wiesbaden 2016 [2002].

[13] Georges Bataille, The Accursed Share (vol. 1 & 2), MIT Press, Cambridge MA 1993; Maurice Blanchot, The Infinite Conversation, University of Minnesota Press, Minneapolis, 1969, and The Unavowable Community, Station Hill Press, Barrytown, 2006 [1983], Maurice Blanchot, “Literature and the Right to Death” in The Work of Fire, transl. by C. Mandel, Standford: Stanford University Press 1995; Jean-Luc Nancy, The Inoperative Community, University of Minnesota Press, Minneapolis 1991.

[14] Vitruv, ibid., book one.

[15] Cf. the captions to the images in this text.



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