Algebraic Concepts Characterized / architectonic articulations - the stanzaic structure of genuinely procedural forms / Projective Theory of Technology / Thinking as an Algebraic Mechanist

Neo-baroque Articulation of Columns: Modules, Solids, Units

We are familiar with the use of generative grammars, L-systems or other recursive procedural frameworks, similar to the subdivision generally applied here in the work of Michael Hansmeyer, mainly from the analysis of natural process and organic structures. What is extraordinary about these examples here is the fact that Hansmeyer does not seek to reference the same processes as analytical tools to investigate nature, but is instead directly interested in a genuinely synthetic treatment within the symbolic. As unswerving as he is provocative, his structures make reference to the foundational discourse of the architectural order of columns, in which systems of dealing with issues of articulation and junction have been negotiated from Antiquity through to the architecture of the early 20th century. And yet his approach is not intended to add criticism or to expand or modulate this discourse in any particular way – he does not seek a modified new order, but rather is interested in something like the orderability, the ability to arrange particular orders out of all potential ways of doing so. His columns thus bear witness to an architectonic dimension that could be understood as figurative, by means of which the entire field of the rhetoric, or of sophistic dialectics, now makes itself available for formal, architectural expression.

In his columns, Hansmeyer turns to a process referred to as subdivision, by means of which a surface represents the point of departure for iterative procedures of differentiation. This surface is not understood as a surface but as a threshold value, i.e. as an integral or mathematical primitive, and is utilized as a kind of pre-specific schema for infinitesimal differentiation and refinement over many generations. From an aesthetic point of view, any shape (not necessarily a basic geometrical form like the surface used here, for which reason I use the term shape) can be viewed, formalized and encoded as a mathematical primitive. Thereby they provide infinitesimal space to be analytically sounded out and stratified into figurations. In this stratification of shapes, derived elements are encoded and thereby encapsulated as operative ‘differential-elements’, extracted from the immanence of the outward shape. As a primitive, the outward shape turns into a kind of virtual matrix for the generation of many, differentially variable instances of such ‘differential-elements’. Using them it is now possible to symbolically ‘construe’. However, this construing is performed as a kind of symbolic proto-geometry, and needs to be rendered into analytical, geometrical systems for turning the symbolic construings into an actual, manifest construction. Such ‘frames for rendering’ can be articulated algebraically in infinitesimally many particular ways. In Hansmeyer’s work the constraints of his framework for rendering his construings into a coherent body are extracted from the Doric arrangement of columns.

As in the grammatical articulation of linguistic sounds, the algebraic articulation of ‘differential-elements’ is directly bound to a gradual spectrum of power by expression, which depends on the sophistication in articulating. If the digital procedures are utilized analytically and representatively for biomimetic replications, as is standard, this power in expression is limited by that which is to be representatively expressed – and itself, as a power of expression, remains unproblematic. However, if one is interested in applying this procedures synthetically within the symbolic (that is, within ordinality itself, or in its applied sense, within arrangeability itself), this power of expression or articulateness is then directly connected to the issue of a non-canonical specification of the shapes generated, or in short, the issue of a ‘specific formality’.


If one were to abstract the specifying features from a variety of representations, speaking of a ‘specific formality’ would be nothing less, in philosophical terms, than a categorical error. Forms indeed are forms, as well as types are types and kinds are kinds, because they are general (and not specific). Yet if one is interested in synthesis within the infinitesimal immanence of the symbolic, then the symbolic formality of the point of departure, that is, of the initial shape-as-primitive for the algebraic articulation, as described above, must be considered genuinely pre-specific. The development and establishment of that particular specificity, which will eventually be considered as characterizing the articulated structure once it is manifest and paid attention to, is itself an irreducible part of the generic, iterative process or architectonic articulation. Even if this manner of articulating shapes should only result in one manifest instance, this instance is nevertheless derived from many variably combined and construed instantiations which all differentiate the initial shape-as-primitive, and which all together embody a virtually evocated species that was bred throughout the generative computational process. I refer to them as virtual because, if the main objective is to articulate and not to exemplify, if the aim is to create a specific formal structure and not a general formal structure, then what serves as a frame of reference in the latter must be regarded a frame for rendering in the former. An articulation cannot be regarded as an exemplary instance of a class, family, breed, genus or species. Both, architectonic specificity-to-be-articulated as well as the articulation-to-be-specified must be conceived as coming about simultaneously.

On a concrete, technical level this happens when the variable instantiations are provided with traits that steer the weighting in the iterative selection of instances over generations. Here a trait is not even a full sign, but merely the index for a virtually-specific feature, which, however, is genuinely artificial and cannot be derived from any essential nature, or any other structure of non-negotiable necessity. Using symbolic frameworks such as this, the characteristics of structures can be generically specified without them being ‘naturally present’ or ‘familiar’ in advance as a characteristic. In this sense we are talking about ‘virtual’ characteristics that can be generated directly from the symbolic. Thus, these specifying features also require to be abstracted by comparison of instances, yet this comparison is operative, not representational. This essentially means that the schematic forms applied while comparing need to be articulated, competitively, and cannot merely be administrated. It is also from this, that the aesthetic quality emerges. A virtually specifying feature can only stabilize into a tangible trait of the instance(s) finally rendered into actual architectural constructions, if it is successfully selected over many generations. Through condensating and mutually referencing and modulating these virtually specifying features, an entire species needs to be evocated as a virtual species. Such virtual species provide a kind of simulacra fauna within the symbolic, and can thus serve as an abundant reservoir of pre-specific differentiation. They can be bred to populate the symbolically stratified space of immanence within analytical geometrical axiomatics, as a kind of socially domesticate-able, intersubjective fantasm-creatures, to inspire architectonic articulations. The greater the ability of sophisticated articulation of what has been evocated, the more captivating – perhaps the more ornamental, in a neo-baroque-ish sense – is the architectural product. The ornamented columns by Hansmeyer are an impressive demonstration of such sophisticated ‘craft’.

Neo-Baroque ornamentation as architectonic articulations

Upon first encountering Michael Hansmeyer’s computational architecture, one may indeed be reminded of Baroque rationalism, and regard them as instances of neo-baroque or digital baroque ornamentation. In the following I would like to present some more theoretical arguments regarding the pros and cons for conceptually moving digitally produced architectural closer to Baroque rationalism. The Baroque period was undoubtedly a striking example of a time of constructive-speculative curiosity in dealing with the vast space of real numbers, which had been symbolized and made accessible to formal thought by Newton and Leibniz. Through the capturing of the infinitesimal in procedures of integration and differentiation, people started to encounter a veritable infinity piercing between each natural whole. Especially perhaps in baroque ornamentation, rational reason at that time is celebrated. This also implies that rational reason is elevated beyond itself, and considered in relation to something “greater.” In any case, the tangible embellishing arrangement of the Baroque’s celebration of this abstractness is something that is lived in a comprehensive sense, and that at the time followed a cultural mood that perhaps today is somewhat rashly perceived to be comparable to ours. Baroque rationalism undoubtedly involved more than the pragmatic exploration of the power of applying calculus to problem solving, which today is performed by computers and has become so commonplace, as digitally encoded and algebraically mechanized analytics.

This more is also addressed today when, taking into account computer-supported algorithms and the forms generated thereby, the talk is once again of ornaments and not merely of patterns. Strictly speaking, ornamentation always refers beyond the formal systematization by which it was configured and in whose formality it had found its tangible expression. Baroque is, though perhaps pointedly phrased, that which canonizes the possible path integrals for determining variable values as specific values abstractly, in their modalizing and operational – i.e. geared towards effect – quality. The Baroque period is the period before the philosophies of spirit and nature of the Enlightenment and the Romantic period. Only afterwards did the focus increasingly concentrate on binding the baroque-ish celebration of the progressions of numbers, ebullient in their behavior as they were at first, into actual pathways of determination, sequences, chains of continued fractions, and series of the infinitesimally approximate-able processuality of these progressions. Today we remember of these early adventures mainly the analytically traceable and analogously reconstructable continuities, explicated into symbolically encapsulated relationships (functions) that need to be optimized and integrated pragmatically, realistically. Yet the Baroque period itself was at least as interested in the multiplication, or even the proliferation, of integrals. It was in precisely this manner that the Baroque was able to relate the ordinal cosmological and geometrical perceptions set in motion in an infinitesimal way to a virtual orthogonality, and thus to start thinking not only movement, but also the conservation of potential for movement (later to be symbolized via thermodynamics as ‘energy’). The Monads of Leibniz, what do they incorporate but the idea that there is an inexhaustible, infinitesimal potential for movement and change in every single point, instance, or moment – a potential which can unfold in infinitesimally variable manners? The predominant theme for Baroque mentality, which also expresses itself in Baroque ornamentation, seems to be the inconceivable yet suddenly ‘countable’ infinite, in its expressions as exuberance. In a societal and political sense, this exuberance developed as décadence – in an irritatingly confident manner, as far as Enlightenment thinking is concerned. Speaking today of neo-baroque ornamentation inevitably evokes any of these aspects.

Nevertheless, or perhaps also precisely because of this, speaking of neo-baroque ornamentation lends itself today for characterizing computational architecture. Also today, we are challenged in sorting out, as we knew it, the produces we can shape when dealing with digitalized numbers and scripts. Many of the architectural shapes that are computed using digital procedures, such as the columns designed by Michael Hansmeyer, get their captivating aesthetic quality from explicating unusual figurative combinations, extracted and encapsulated by iterative differentiation, from the actualization space of the integral from which it was generated. The exciting thing about architectural dialogue surrounding ornamentation, as opposed to a dialogue regarding biomimetic or ecomimetic morphology, is the fact that here, the structures generated are fascinating in the discontinuities they exhibit (there is hardly any repetition, if you look closely) which are, first of all, of interest because of their aesthetic qualities. From the initial integral these computations depart from, distinct segments of many possible derivative paths are encapsulated symbolically, and used as distinct elements for construction that can be combined with enormous freedom to manifest, in the end, a singular, figurative Gestalt. When the interest in calculus were not geared towards the aesthetic quality that can thus be achieved, but rather towards identifying biological or material regularities, then it is perfectly valid if one were to pursue only those derivatives that could be considered as smoothly integrable into the formula for one continuous segment – as only they can depict “natural” progressions. Regarding the continuity of natural progression, Leibniz once stated: Natura non saltum facit (“Nature makes no leaps”). Subsequent to the insights gained by thermodynamics and the population sciences, and supported by the rise of information technology, complexity theory, the theories of dissipative structures and self-organization, this idea has of course been challenged for quite a while now. Nevertheless, it remains strangely incomprehensible why for computer aided architectural design, the exploration of the potential of such striking manifestations of intellect, as computers undoubtedly are, remains tied so strongly to an interest in ‘building naturally’.

The shapes of Michael Hansmeyer present themselves, as ornamented columns, very self-confidently as the produces of artificiality – even though there is a strong touch of an alien organicity proper to them. Only in this way – in that they do not merely mimetically depict some hidden ‘nature’ but are genuinely computed – can they, in a ‘neo-baroqueish’ manner, be regarded as ornamental. They articulate the differential relation between structure and form in a figurative manner. Comprehended like this, as genuinely procedural shapes that articulate a certain figurality of the form-structure relation, they indeed share some key features of Baroque rationality – namely the radically abstract interest in aesthetics by calculation. Architectural articulations, so the argument thereby, in their ornamental baroque, ebullient sophistication in expression, embody the pre-specific more of the virtually actualizable path integral from which their computations start out. The idea, thus, is to regard them as articulations of this initial integral. It is important to stress out the distinction between figurative and formal language, because the tricky thing will be to bypass the trap of binding the articulation at stake here, which is an operationally symbolic articulation, back into a representational framework. If one were indeed to tap into this trap, we would find ourselves in all the  muddy grounds which Nietzsche has so importantly cleared in his critique on idolatry. Binding genuinely procedural shapes back into a representational framework of any sort would mean nothing less than worshiping them as an incarnation of an analytically idealized (nature/material/spiritual) divinity, so to speak, as the form of a false idol. In architectural theory, the introduction of the category of tectonics, as a result of Boetticher and Semper, has been able to relieve architectural design at least to some degree of such cultic appreciation. It too has related construction/form and language anew. Yet speaking of architectonic articulations in regard to computational design, instead of characterizing them as tectonic forms with varying styles, points to a further modification of our understanding of formality within aesthetic practice.

What is important for arguing in this direction is pointing out that with computers, there has been established a ‘short circuit’ between the hitherto separated and mutually complementary dealings with language, and our dealings with numbers. To put it very briefly, and bluntly in its consequences for our context of neo-baroque ornamentation: The formal dimension of language, that of logical assertions, can be treated arithmetically, like numbers. The figurative dimension of language cannot. So far so good, it feels safe to account for this as a truism today. The interesting situation arises from the following. By its algebraic encoding, the ‘basic’ body of rules for that which is ‘computable’ is arranged in systematizations that allow to bypass the separation of formal from figurative language. Yet this does not mean that the figurative dimension now too becomes ‘calculable’ as an assertion, in the sense of ‘determinable in its causality and effects’. Rather, it means that construction and figurative expression are no longer separate realms. Figuration needs no longer to be added-on, or attached, to structure. We can calculate not only form by engaging with functions, but also figurality. We can articulate structure in various manners, by constructing within the symbolic. For computer-aided design this altered formality means, for the time being, nothing more but also nothing less, than that the manipulation of symbols, in the universal format of digital code, is prior to designing with geometrical forms, and can provide the constraints for such design within a much wider scope of feasibility.

When constructing within the symbolic, the focus is not primarily on what the symbols stand for, and mean respective to what we are used to see represented in them. Prior to any such consideration, the focus is on what can be done with them by releasing an immanent dynamics that is proper to symbolic systems. An example to illustrate this difficult thought which might sound somewhat esoteric at first (speaking of a ‘dynamics’ that is immanent to the abstract formality of symbols) might be gained by thinking about the ‘body’ of a real number opposed to that of a natural number. The number 2 was simply the undifferentiated magnitude 2, until people realized that depending on whether they use 2 in the form of e.g. 8/4, 46/23, or any other of the infinite possibilities, they can indeed do different things with it. They realized that depending on the particular form, two-ness can be integrated into different analytical procedures. This has accommodated the analytical-functional order, which, speaking in terms of technology, allows for transformation, and not merely for transportation as our formerly geometrical-formal thinking did. Within the real number space, the ‘thing’ we deal with when we say ‘2’ is not the same anymore as before, because now it suddenly makes sense to choose, from infinitely many ways of putting ‘2’ formally – out of all the two-nesses conceivable – a particular one and see what can be realized with it. Today, so the hypothesis, with the algebraic approach to symbols, we have begun to accommodate a similar shift in what dealing with numbers means to us, a shift which, so the suggestion here, might be referred to as an articulatory orderability.[1]

The tradition of tectonics had also discussed the constructive elements under the primacy of their connectivity, and in this sense as ‘articulations’ (from the Latin articulatio meaning “separation into joints”). The assumption of a language of forms (Formensprache), and its ‘grammatical’ order of articles, or junctors (Glieder) was constitutive both for Boetticher’s dialectical interplay between core form (Kernform) and art form (Kunstform) as well as for Semper’s technology-oriented modularization of this dialectic by relating art form (Kunstform) to work form (Werkform). Yet both subordinate tectonic articulation, in exclusion of the dimension of a rhetorical/figurative discourse, under a deliberately mystically inspired idea of construction, as exemplified by Antonio Gaudi, Frei Otto, and, perhaps, in its digitalized form today, by Cécile Balmond.[2]

The interest in the theoretical perspective of architectonic articulations aims at opening up the somewhat Delphic articulations-with-calculated-forms of the early masters – and this is in no way intended as a pejorative characterization – to a discursive criticality regarding the (aesthetic) articulability of computed solids, shapes, or any other rendering. Only such a criticality can provide space for cultivatable sophistication. Today, the various CAD and scripting environments provide a generally available playground for simple and unsophisticated application by everyone, and this is fantastic. Yet the dissemination of judgments of taste – Kant had defined the power of judgment as an individual ability and thereby explicitly as taste – without the possibility for cultivation through reflection and discursivity must inevitably commit itself either to a habit of mediocrity or of tyranny. In order to regain such a reflective distance today, it seems to be crucial to broaden the notion of articulation in a way that is suited for the ‘algebraic-symbolic formality’ digital construction is concerned with.[3] Thus, somewhat surprisingly perhaps, it is through the tradition of ornamentation, in its roots in Baroque rationality and its celebration of exploring various ratios (lat. for reasoning, procedure) with various capacities, experimentally, that we might find a way to virtualize the tectonic dialectic in such a way as to make the ability in dealing with forms/mechanics/technology, i.e. the training of an individual capability regarding astuteness and slyness, accessible from a reflective, and thus cultivatable, distance.


© Michael Hansmeyer 2011,

[1] A brief digression at this point. The New Materialism movement postulated first and foremost by Manuel deLanda fails to recognize, in all awareness of the significance of the shifts discussed here, that the relationship between information and material/structural performance can only become accessible to us as a symbolic relationship (i.e. as measurement). Negating this symbolism as a symbolism, as consistently as he does, comes close to invoking a deity of some apparatus-like materiality and certainly misuses, in architecture for example, sophisticated products of human intellect as false idols for the assertion of an (effectively political) efficiency paradigm that is said to follow a dynamic declared to be the intrinsic logic of the material.

[2] I am especially thinking of his beautiful book: Element, Prestel 2008.

[3] The Hjelmslevian and Deleuzian/Guattarian noetic scheme of a ‘double articulation’ provides such an ‘operative-elementary’ way of thinking. It suggests to conceive what they call the form of content and the form of expression as a mutually co-constitutive differential. Cf. Louis Hjelmslev, Prolegomena to a Theory of Language. Trans. Francis J. Whitfield. University of Wisconsin Press, Madison 1961. Cf. the adaptation of this noetic figure by Deleuze and Guattari: “The first articulation concerns content, the second expression. The distinction between the two articulations is not between forms and substances but between content and expression, expression having just as much substance as content and content just as much form as expression. […] There is never correspondence or conformity between content and expression, only isomorphism with reciprocal pre-supposition. The distinction between content and expression is always real, in various ways, but it cannot be said that the terms pre-exist their double articulation. It is the double articulation that distributes them according to the line it draws in each stratum; it is what constitutes their real distinction.” Gilles Deleuze, Félix Guattari, “The Geology of Morals”, in: A Thousand Plateaus. Capitalism and Schizophrenia II. Trans. Brian Massumi. Minnesota University Press, Minneapolis 1988, p. 39–74, here p. 44. Cf. for a more detailed discussion of my argument about architectonic articulations, cf. Vera Bühlmann, “Primary Abundance, Urban Philosophy. Information and the Form of Actuality”, in: Vera Bühlmann, Ludger Hovestadt, Metahilithikum I Printed Physics. Applied Virtuality Series, Springer Vienna 2012 (forthcoming in December).

The main line of argument in this text has been published as: “Genuinely procedural shapes. Neo-baroque ornamentation, architectonic articulations”. In: Neil Leach, Philip Yuan (Eds.), Catalog to the DIGITAL FUTURE EXHIBITION which took place in Shanghai (August 13th – September 15th 2011), 2012.

As well as at the International Deleuze Conference in Kaifeng 2012.

This post is the first in the category Architectonic Articulations – The Stanzaic Structure of Genuinely Procedural Shapes 

One thought on “Neo-baroque Articulation of Columns: Modules, Solids, Units

  1. Pingback: architectonic articulation – the stanzaic structure of genuinely procedural shapes | monas oikos nomos

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