Mereology is a formal concept which enters architecture as an additional formal category. Form is a rather ambiguous concept in architecture. So in this essay, first an investigation is conducted by contrasting two closely related concepts: shape and content.
Hans Trusack criticises the problem of shape for its shallow formalism and historical-theoretical indifference as a defensive strategy that evades the disciplines and difficulties of past and future. The distinction between the terms “form” and “shape”, following Tursack’s argument, is a “matter of generative process”. Both terms point to the production of visual expression. Yet while shape refers to the appearance of an object, form reflects the logic of transformation and operation within historical and theoretical contexts such as political and religious ideology, economics and technological background. Tursack criticised the strategy of shape in architecture, stating its lack of reference, it being “plainly, and painfully, evident”, and incapable of moving forward. Whereas form is difficult, disciplinary and requires historical and theoretical study, and yet promises the future.
Form has the advantage of being able to deal with complex relations due to its deep and continuously evolving intervention with content. The term form derives from the Latin word forma, is understood as the combination of two Greek words: eidos, the conceptual form, and morphe, the physical form. The complexity of form can be attributed to these differentiated meanings, yet complexity is compatible with agencies and relations. This can emerge further by conducting a brief historical review.
Ancient Greek architecture pursues the ideality in mathematics and proportion. The efforts made by architects in designing the Parthenon provides evidence of this feature. These operations tried to approximate the physical shape of architecture to the “ideal” form. Form reflects the pursuit of ideality and perfection in this period.
For Gothic architecture, there were more concerns about structure, and matter was pushed to its maximum capability to build as tall as possible for religious appeal. Consequently, structures were designed to be rigid and lightweight, and solid walls were replaced by glass windows, while flying buttresses supported the main structure to grow even taller. Consequently, astonishing space and fascinating transparency emerged.
Modernism claims that “form follows function”, rejecting traditional architecture styles. The reality of matter and the logic of technology eschewed decorations, proportions, or any subjective distortion of matter. The emphasis on the term “function” illustrates an ideology of treating architecture as a machine. Each part is nothing more than a component that has a certain feature inside this machine, and redundant decorations and details are removed to deliver this idea clearly. Without distractions, space becomes evident.
In the shift to postmodernism, the uniformity and the lack of variety of modernist architectures were reacted against, and a great variety of approaches emerged to overcome the shortcomings of modernism. Parametricism, for instance, has been promoted by the thriving of digital technologies. Designers are capable of more complex formal production, and architectural elements have become variables that can be interdependently manipulated. In this formalism, rigidity, isolation, and separation are opposed, while softness, malleability, differentiation and continuity are praised.
From the examples above, form is the embodiment of the relations between architecture and its motive in specific historical scenarios, while for shape, only the results are accounted for – relations are ignored, and architecture is treated as isolated physical entities, incapable of producing new relations. Different methodologies of dealing with architectural form also imply the variation of ideology in compiling form with content.
Mereology – An Approach of Architectural Form
In recent philosophical texts, a third notion of form is brought forward. Contrary to a dialectic of form and content, here investigations deal with the resonance of parts: the description of objects by their ontological entanglement only. The writings of the philosopher Tristan Garcia are a strong example for such mereological considerations. In his treatise Form and Object: A Treatise on Things (2014), Garcia investigates the ontology of objects with two initial questions, “… what is everything compose of? … what do all things compose?” The first question interrogates the internal, the elementary component of everything. The second interrogates the external, the totality of everything. For Garcia, the form of a thing is “the absence of the thing, its opposite, its very condition,” form has two senses, the “beginning”, and the “end”, which never ends. Form begins when a thing ends, it begins with different forms; in the end, since it has “endless end”, form ultimately merges into one, which is “the world”. Garcia defines an object as “a thing limited by other things and conditioned by one or several things.” The form of an object depends on what comprehends or limits this object. Every object is “embedded in a membership relation with one or several things”, they can be divided by defining limits, which is also a thing distinguishing one thing from another. Garcia’s argument adapts the concept of mereology. Form has two extremes, one toward the fundamental element of matter, and the other toward the world, comprehending everything. All things can always be divided into an infinite number of parts, and they can always be parts of another thing. Identifying parts or wholes within a section we can operate on can establish a limit. The relevance between form and mereology opens a new opportunity to inspect architectural form from a different point of view.
One of the first discussions about parts and wholes in modern philosophy was posed by Edmund Husserl, in Logical Investigation (1st ed. 1900-1901, 2nd ed, 1913), but the term “mereology” has not been put forward until Stanisław Leśniewski used it in 1927 from the Greek work méros (parts). Mereology is considered as an alternative to set theory. A crucial distinction lies between mereology and set theory in that set theory concerns the relations between a class and its elements, while mereology describes the relations between entities. The mathematical axioms of mereology will be used as the fundamental theory of developing the method of analysing architectural form.
Following Roberto Casati and Achim Varzi, the four fundamental mathematical formularisations of mereology are: “Relations are reflexive, antisymmetric and transitive. (…) First, everything is part of itself. Second, two different objects cannot be part of each other. Third, each part of a part of a whole is also part of that whole. Fourth, an object can be a part of another object, if both exist.”
Mereology can be a promising approach also for the reading of architectural form, as it emphasises relationships without reducing buildings to their appearance or function. However, such philosophical descriptions consider wholes and parts as mostly abstract figures. Therefore, a supplement could be developed to properly categorise the mereological relations in the field of architecture. Having the relations between form and mereology addressed, methodologies can be developed to access the analysis of architectural form. Mereology as a specific methodology for architecture is quite new. One of the first introductions can be found in Daniel Koehler’s book The Mereological City: A Reading of the Works of Ludwig Hilberseimer (2016). Here, Koehler departs from the modern city, exemplified through the work of Ludwig Hilberseimer to illustrate mereological relations in the modernist city. From the room to the house to the city to the region, Hilberseimer canonically drew the city as a hierarchical, nested stack of cellular spaces. However, through the close reading of its mereological relations it becomes clear that political, economic or social conditions are entangled in a circular composition between the parts of the city. Recalling Garcia’s discourse, and resonating with Leon Battista Alberti’s thesis, Koehler shows that the cells in Hilberseimer’s modernist city are interlocked. A house becomes the whole for rooms; a city becomes the whole for houses. By considering the city and its individual buildings equally, “the whole is a part for the part as a whole.”
Architectural Relations Between Parts and Wholes
Parts are not only grouped, packed and nested through different scales, but also in different relations. Specific relationships have been developed in different architectural epochs and styles. Mathematically, four general classes of relations can be drawn: whole-to-whole, part-to-part, whole-to-parts and parts-to-whole, while more specific subclasses can be discovered from each.
According to the mathematical definition, between wholes there exist complex relations, the whole could exist on any mereological level, and the complexity of relations between multiple levels are also accounted for. Whole-to-whole relations can become complex when considering multi-layer interaction, and more relations can be identified: juxtapose, overlap, contain, undercrossing, transitivity, partition, trans-boundary, intact juxtapose, compromised juxtapose.
A first glance of New York, gives the impression that it is quite heterogeneous, but underneath there is a city grid underlying the heterogeneity, and while the relations displayed in the grid are rather simple, all wholes juxtapose with one another. In comparison, in Siena, an Italian city, the urban space is quite complex, where boundaries of all wholes negotiate with others, the gaps in between are carefully treated, the nesting relations are extremely rich, and multiple relations from the diagram above can be found.
The whole-to-parts relation studies what the whole does to its part, namely in terms of top-down rules. The mathematical definition does not involve specific situations that a whole-part condition holds. Distinctions within individual contexts make a significant difference in clarifying an explicit relation. The situations for the whole can generally be classified into following types: fuse, fit and combine.
One of Zaha Hadid’s projects, Heydar Aliyev Centre, indicates the fusing relation. Architecture is represented as a smooth, fluid volume. The distinction between elements disappears, and this dominating power even extends to the external landscape. In order to maintain a continuous whole, parts are fabricated into a particular shape, having their unique unchangeable locations. The continuous whole excessively overwhelms the parts, yet not all parts are reshaped to fuse into the whole, and because the parts are small enough in relationship to the whole, the control from the whole is weakened, and parts are fit into the whole.
The third type is combining. An example for this relation is Palladio’s project Villa Rotonda. In this case, parts are obvious. The whole is a composition of the parts’ identities. However, the whole also holds a strong framework, in a rigorous geometric rule that decides positions and characters of parts. The arrangement of parts is the embodiment of this framework.
The parts-to-whole relation studies what the parts do to the whole, or the power of bottom-up relationships. The different situations of parts are also key parameters in validating a given relation. The classification of situations for parts are as follows: frame, intrinsic frame, extrinsic frame, bounded alliance, unbounded alliance.
Emil Kaufmann thoroughly investigated the innovative works by Claude Nicholas Ledoux in Three Revolutionary Architects: Boullee, Ledoux and Lequeu (1952). According to Kaufmann’s study, Ledoux’s works developed new compositional relations of elements from the Baroque. The characteristics of parts in Baroque architecture are rich, but tend to regulate the identities of all the elementary parts and fuse them together to serve the harmony of the whole, presenting the intrinsic framing. Ledoux’s work is an extrinsic framing, where the parts are relatively independent, with each element maintaining its own properties, and while consisting of the whole, they can be replaced with other identical components.
One of my projects in discrete aggregation of elements presents an unbounded alliance relation. The aggregation as a whole shows a form that is discretised (Figure 12), and does not pass any top-down instructions to its parts.
Part-to-Part Without Whole – The Ultimate Parts
For part-to-part relations, local interactions are emphasised, and interactions occur at multiple levels of compositions, where the part-to-part relations in some cases are similar to that between wholes. It has following classifications: juxtapose, interrelate, contain, partition, overlap, trans-juxtapose, over-juxtapose, over-partition, over-overlap.
Architects have been working on the possibility of removing the whole by studying the part-to-part relations. Several approaches have been developed, mainly through computation. Neil Leach considers the city as a “swarm intelligence”, bringing forward the potential of developing urban form with computational method. Leach encourages swarm intelligence for the interactions between agents (parts), which “offers behavioral translations of topology and geometry”, while fractals, L-systems or cellular automata are all constrained by some limitation. However, although swarm intelligence is based on the interaction of individual agents, it is always treated as a whole; all cells of CA are fixed in the background grid, which is also a whole. For fractals and L-systems, they can be subdivided into infinite parts, a transcendent whole where all parts grown from still exist. In the mereological sense, none of these cases can escape the shadow of the whole – strictly speaking, they are part-to-whole relations. To discuss the part-to-part relation in more depth, more investigation is needed to clarify the concept of part.
In The Democracy of Objects (2011), Levi Bryant claims that objects constitute a larger object by establishing relations with others, but this doesn’t alter the existence of objects, as he says, “all objects equally exist, but not all objects exist equally.” In Bryant’s discourse, this independence suggests the dissolution of the whole. Bryant proposes a concept of “regimes of attraction”, that includes the “endo-relation” and the “exo-relation”. The endo-relation indicates that the proper being of an object consists of its powers or what an object can do”, not the “qualities” emerging within an exo-relation. An object possesses “volcanic powers”, the stabilisation of the regime of attraction actualises it into a specific state. The concept of the whole reduces objects to this state, which displays only a section of their proper beings. The concept of regimes of attraction is against this reduction.
The regime of attraction can be linked to the notion of “assemblage” from Manuel DeLanda, however, there is a distinction between the two. Assemblage holds only the relation of exteriority, whereas regime of attraction maintains both relations of interiority and exteriority. In Assemblage Theory (2016), DeLanda reassembled the concept “assemblage”, which was originated from the French agencement. Created by Gilles Deleuze and Félix Guattari, this original term refers to the following meanings: the “action of matching or fitting together a set of components” – the process, and the “result of such an action” – the product.
DeLanda emphasised two aspects, heterogeneity and relations. As he indicated, the “contrast between filiations and alliances” can be described in other words as intrinsic and extrinsic relations.
The nature of these relations has different influences on the components. The intrinsic relation tends to define the identities of all the parts and fix them into exact location, while the extrinsic relation connects the parts in exteriority – without interfering with their identities. DeLanda summarised four characteristics of assemblage: 1) individuality, an assemblage is an individual entity, despite different scale or different number of components; 2) heterogeneity, components of an assemblage are always heterogeneous; 3) composable, assemblages can be composed into another assemblage; 4) bilateral-interactivity, an assemblage emerges from parts interactions, it also passes influences on parts.
DeLanda then moved on to the two parameters of assemblage. The first parameter is directed toward the whole, the “degree of territorialisation and deterritorialisation”, meaning how much the whole “homogenises” its component parts. The second parameter is directed toward the parts, the “degree of coding and decoding”, meaning how much the identities of parts are fixed by the rules of the whole. The concept of assemblage provides us a new lens of investigating these mereological relations. With this model, the heterogeneities and particularity of parts are fully respected. The wholes become immanent, individual entities, existing “alongside the parts in the same ontological plane”, while parts in a whole are included in the whole but not belonging to it, and according to Bryant’s discourse, the absence of belonging dispelled the existence of the whole.
From the study of regime of attraction and assemblage, this essay proposes a new concept – “the ultimate parts” – in which a proper “part-to-part without whole” is embedded. A part (P) horizontally interacts with its neighbouring parts (Pn), with parts of neighbouring parts (Pnp), as well as interacting downwardly with parts that compose it (Pp) and upwardly with wholes it is constituting which are also parts (Pw). This concept significantly increases the initiatives of parts and decreases the limitations and reductions of them. It doesn’t deny the utilities of the whole, but considers the whole as another independent entity, another part. It’s neither top-down, nor bottom-up, but projects all relations from a hierarchical structure to a comprehensive flattened structure. The ultimate parts concept provides a new perspective for observing relations between objects from a higher dimension.
One application of this concept is TARSS (Tensegrity Adaptive Robotic Structure System), my research project in MArch Architectural Design in B-Pro at The Bartlett School of Architecture in 2017–2018. This project utilises the features of tensegrity structures of rigidity, flexibility and lightweight. The difference is that rather than fixing parts into a static posture and eliminating their movements, the project contrarily tries to increase the freedom of parts as much as possible. The tensile elements have the ability to adjust their lengths collaboratively to change the general shape of the aggregation. Reinforcement learning is employed to empower the parts with objective awareness. The training sessions were set up toward multiple objectives that are related to architectural concerns, including pathfinding, transformation, balance-keeping, self-assembling and structural load distributing. This approach brings obvious benefits, as architecture design in this sense is not only about an eventual result, but about the dynamic process of constantly responding to the environmental, spatial or functional requirements. The premise is to treat parts as ultimate parts whilst retaining their objectivity and being able to actively interact at all mereological levels without limitations.
The concept of ultimate parts brings forward a new relation of “part-to-part without whole”. This new relation belongs to a higher dimension. The details and essence of objects are simultaneously displayed, without being obscured by the compositional structure. Analogised with spatial dimensions, a 3-dimensional cube simultaneously shows all its faces and interior in 4-dimensional space. The significance is that it opens vast new perspectives and operational methodologies in the architectural design realm. Especially with the advancement in robotics and artificial intelligence, this type of new relationship enables greater opportunities by regarding machines as characters with immense potential to work with us, instead of for us. The role of designers would be very much like “breeders of virtual forms”, who do not rule the form, but guide it towards the demands. This moves away from anthropocentric design by overcoming part-to-whole with part-to-part.
 H. Tursack, "The Problem With Shape", Log 41 (New York: Anyone Corporation, 2017), 53.
 Ibid, 50.
 L. Sullivan, "The Tall Office Building Artistically Considered", Lippincott's Magazine (1896), 403–409.
 T. Garcia, M. A. Ohm and J. Cogburn, Form And Object (Edinburgh: Edinburgh University Press, 2014), 19.
 Ibid, 48.
 Ibid, 77-78.
 Ibid, 145.
 E. Husserl, Logical Investigation (London: Routledge & K. Paul, 1970).
 Stanisław Leśniewski, O podstawach matematyki [trans. On the Foundations of Mathematics], I-V, 1927-1930, Przegląd Filozoficzny, 30 (1927), 164–206; 31 (1928), 261–291; 32 (1929), 60–101; 33 (1930), 77–105; 34 (1931), 142–170.
 R. Casati and A. C. Varzi, Parts and Places: The Structures of Spatial Representation (Cambridge, Massachusetts: MIT Press, 1999).
 L. Hilberseimer, The New City: Principles of Planning (P. Theobald, 1944), 74-75.
 D. Koehler, The Mereological City: A Reading of the Works of Ludwig Hilberseimer (Transcript, Verlag, 2016), 182.
 E. Kaufmann, Three Revolutionary Architects, Boullée, Ledoux, And Lequeu (Philadelphia: The American Philosophical Society, 1968).
 N. Leach, "Swarm Urbanism", Architectural Design, 79, 4 (2009), 56-63.
 L. Bryant, The Democracy Of Objects (Open Humanities Press, 2011), 290.
 M. DeLanda, Assemblage Theory (Edinburgh: Edinburgh University Press, 2016), 2.
 Ibid, 19-21.
 Ibid, 12.
 L. Bryant, The Democracy Of Objects (Open Humanities Press, 2011), 273.
 M. DeLanda, "Deleuze And The Use Of The Genetic Algorithm In Architecture" (2001), 3.