The
  Cornell
    Journal
      of
        Architecture
17
ARCH-Games, MATH-Games



Archie B. Mackenzie is associate professor of architecture in Cornell University Department of Architecture. He teaches, builds, and researches architectural games.
It has long been said that architecture is a game played with clear objectives, but no guiding set of rules. Mathematics, on the other hand, has forever been described by its believers as a form of knowledge best understood as a game with lots of rules but no clear objectives.[1]

The following piece has been developed as a language-game that plays upon three quotations. The first of these, which appears above, by Brett Steele in his preface to The New Mathematics of Architecture (nma), sets up a reciprocal relationship between MATH and ARCH (architecture) by suggesting that they share the common metaphor of games: MATH, a game of many rules, as it awaits problems to solve; or, in the case of ARCH, forms to produce. The book contains an exceptional range of mathematically generated architectural forms and a very intriguing array of MATH-to-ARCH applications. How apt, however, is the aphorism?

In order to examine it, two boxes have been designated to hold the constituents of MATH on one side and of ARCH on the other. It is easy enough to insert a short list of concepts, representing Steele’s “rules” into the MATH box. These clearly belong to MATH and are on loan—as MATH is to any donor of “objectives”—to ARCH. Almost immediately, however, there is a sense of imbalance between the two sides. How does architecture, stripped of its multitude of historical failed attempts to claim architectural Truth, provide “objectives” that are comparable with the immutability of mathematical language?

In the second quotation representing the MATH side, David Hilbert forcefully adds to the unequal equation.

We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.[2]

This does not invalidate the game metaphor for MATH; it simply validates the integrity of MATH-languages that bear Steele’s rules.

The third quotation is from Ludwig Wittgenstein, whose later works refuted “essence” in language, preferring to move away from what he calls, and I paraphrase, the frictionless ice of the logical, to the rough ground of common language that he states clearly does not refer to the natural world. He does embrace the notion that fungible languages may occur in the works of humankind, not simply in its verbal transactions. He considers in the following “the Ideal”:

We misunderstand the role of the ideal in our language. That is to say: we too should call it a game, only we are dazzled by the ideal and therefore fail to see the actual use of the word ‘game’ clearly.[3]

The purpose of the essay is to redress the perceived imbalance between the MATH-side and the ARCH-side of the equation by demonstrating that game-building/game-play enables the dialogue between the two sides.


Move 1.0 EMPTYING OUT the ARCH and MATH BOXES

Move 1.1 NATURE as MATH’S Ultimate AUTHORITY

As Nature is the ultimate authority for MATH, sanctioning the premise that it is “a form of knowledge … with lots of [immutable] rules” (from Steele quote, above), then ARCH, as the other side of the equation should be possessed of immutable “objectives.” The opposite seems true, that Architecture has been progressively emptied out and is now either void or awaiting the new contents. What role does MATH now have in determining, warranting, and delimiting these objectives?

Move 1.2 ARCH and MATH Boxes

To look more closely at Steele’s conjoined aphorisms, two receptacles are provided—an ARCH Box and a MATH Box—into which items from the lists may be placed and dumped out, like dice in backgammon cups. There are two lists, one of mathematical concepts (from nma) and one a short-list of “rejected polemics.” The arrow to the right in the “MATH Box” loads up the container with a sampling of mathematical procedures, or “rules,” from nma. The arrow to the left shows a sampling of the many polemics from the last century that was rejected by theorists. These acts are mimicked here—by removing the same from the ARCH Box—as an expression of the imbalance between the unchanging rules of MATH and the conditional “objectives” of ARCH.

Architectural Polemics/Math Procedures. Image by Heriberto Rodríguez Valenzuela
Architectural Polemics/Math Procedures. Image by Heriberto Rodríguez Valenzuela

Move 1.2.1 REMOVING POLEMICS FROM ARCHITECTURE

Nothing occupies the ARCH Box. It has been emptied out. Not only is it without a guiding set of rules, but also there are no “Objectives” cards in it. This makes sense because at the level of generality attributed to MATH—comprising its immutable languages of symbols and rules—there are no equivalent archetypal objectives left in ARCH. Its baseline has come into question. It has no “form follows function,” no firmness, commodity, and delight, no first, second, and third typologies, no program as authority, no true modernity, no allegory, no acceptable phenomenology, no perpetual social project, no such thing as archetypal taste. And it is vulnerable to consumption. Moreover, even the validity of “plan” has been removed; likewise “architectural space.” As traditional constituents of ARCH leak away, it begins to appear that ARCH has no immutable principles on a par with the rules of MATH.


Move 2.0 THE GAME FOR FORMULATING A DEFINITION OF “GAMES”

Move 2.1 DEFINING “GAME”

Wittgenstein’s notion of the “ideal” derives from the philosophical project to use language to define the essence of things—and ultimately the essence of being. In Philosophical Investigations, he denies the efficacy of this endeavor in favor of an alternate activity, which he calls “language-games.” To define the ideal, to calculate the essential, is not to discover immutable Truth but is itself a language-game woven into the culture and methodology of philosophy. The following considers how the author’s of Rules of Play researched and crafted a definition of “game,” a process that is both paradoxical and pragmatic: paradoxical, because profound definitions must maintain engagement with their exclusions; pragmatic because academic fields of inquiry must have boundaries, even if these are incessantly gamed by competing ideas.

As game-builders themselves, the authors of Rules of Play were motivated “by the feeling that despite the breathtaking pace of recent technical and commercial advancement, games have remained creatively stunted,”[4] to build a critical discourse for game design. Not to be confused with Game Theory, the mathematical method often employed in economic theory, their work intends to construct a conceptual framework for serious research into the nature of game-building/game-play. They felt the necessity to not only bring conceptual order to the burgeoning not-quite-yet-field of game design but to ask what is holding it back from what it could become.

The formulation that follows is the product of their efforts:

A Game Is a System in Which Players Engage in on Artificial Conflict, Defined by Rules That Results in a Quantifiable Outcome.[5]

A full discussion of their terminology is beyond the scope of this piece; instead, the meaning of the terms is exemplified by what is usually called “circular” but more recently would be regarded as “recursive” thinking—that is, to accept the formulation of a definition for “game” as a language game in its own right. Or, more directly: to consider the work of the authors of Rules (Salen and Zimmerman) as an academic game to define a critical terminology needed to enable the field itself to coalesce.

Move 2.2 APPLYING THE TERMS OF THE AUTHOR’S DEFINITION OF “GAME” TO DESCRIBE THE GAME THEY PLAYED TO FORMULATE IT

One might employ their terms to describe their game-to-define-games. Firstly, the Rules’ game system could reasonably be identified as the book itself, the covers of which act as a symbolic boundary to the game-space. (To construct this space, of course, the authors had to explore an even larger one; still, the book has digested that search into a focused presentation.) Secondly, the players are the authors, specific supporters and potential antagonists, the members of the field in general, commercial game-building professionals, and a broader public to which the field may appeal. Thirdly, the desired outcomes of the definition-game are echoed in the evocative fragments: breathtaking pace, creatively stunted, build a critical discourse, and focusing the burgeoning field. Fourthly, the artificial conflict is variously: the competing ideas of the players, the anarchy of the field, which hitherto had resisted conceptualization, and the difficulty of finding a potent yet durable terminology. Fifthly, the resulting definition must both engender debate yet persist against its inevitable challenges. And, finally, the outcome is not purely quantifiable but rather must be judged by the quality of its survival or demise.

Perhaps the most defining term here is game system, because in many ways Rules embodies a field and is achieved by a design process. In this language-game to define “game,” the words game and design are nearly synonymous. Similarly, game-building is a design process; and designing is, in many but not all ways, a game. The authors of Rules built a (re) search space to create—to design—the core “rule” of the field: the definition of “game.”


Move 3.0 LEAKY CONCEPTS, ASSOCIATIVE PATHWAYS

Move 3.1 Still Rules CONSTRUCTING a WEB of INTRA- and EXTRA-MURAL OBJECTS

In Rules, the authors researched the work of seven prominent writers on the nature of games; selecting critical features from the definition search gave for “games.” They then plotted game features* to authors** in a table, in order to compare the seven definitions to each other and to their own formulation. Finally, they extracted a definition they believed was sufficiently broad and concise to both stand up to academic scrutiny and yet to include a significant range of exemplary games. Their final definition is paired below with a version cobbling together fragments from five numbered features—from the original 15 that they determined relevant to their conception of “game.” The second version reflects upon the parsimony of the first, both suggesting the author’s struggle to posit a core terminology for a new field and yet hinting at exclusions: for example, “quantifiable outcomes” as opposed to “goal-oriented” play. The diagram following immediately restores the 10 rejected features, not in an attempt to undermine the Rules definition but rather to suggest a rhizome-like relationship between the core and the creative, if marginal, potential of the satellite concepts engendered by the language-game to define “games.”

A Game Is a System in Which Players Engage in on Artificial Conflict, Defined by Rules That Results in a Quantifiable Outcome. (See Move 2.1)

A game is (14) a system in which players (2) engage in conflict or contest that is (8) artificial (and) outside normal life, that (1) proceeds according to rules that limit players and that is directed towards (3) a goal-oriented/outcome oriented result. (See note *.)



Family resemblances. The diagram shows a network of the game theorists (red nodes) and possible game features (blue nodes). The purple node represents Salen and Zimmerman’s de nition; the  ve solid lines emanating from a single node indicate those features that are consistent with their de nition. The red nodes show dashed black lines connecting to many but not all of the same features. This cluster bears substantial agreement between authors but with differences. The lighter lines of varying colors designate connections of game de nitions and features substantially at odds with the main group—the former, immediate family of similarities; the latter, distant relatives. Image by Heriberto Rodríguez Valenzuela
Family resemblances. The diagram shows a network of the game theorists (red nodes) and possible game features (blue nodes). The purple node represents Salen and Zimmerman’s de nition; the ve solid lines emanating from a single node indicate those features that are consistent with their de nition. The red nodes show dashed black lines connecting to many but not all of the same features. This cluster bears substantial agreement between authors but with differences. The lighter lines of varying colors designate connections of game de nitions and features substantially at odds with the main group—the former, immediate family of similarities; the latter, distant relatives. Image by Heriberto Rodríguez Valenzuela

Move 4.0 A WEB OF RELATIONSHIPS

The network (above) is biased toward its dominant object, which is the final definition of “game.” In other words, the network is rooted in its most successful element. Like a banyan tree, however, the diagram of relationships may be thought of as growing outward from its trunk—while all of the tendrils of previous researchers’ definitions are cross-linked one to another by features they share. This makes a network of relationships that grows the way banyan trees grow, whose branches project aerial roots that drop to the ground and later become accessory trunks. The central trunk is equivalent to a key concept, or dominant methodology; sustaining the trunk even while it fails is not calamitous to the tree, if accessory elements can take over growth. Such a shift in conceptual loading to an accessory truck system is similar to the gestalt structural heuristic called re-centering. Linear, cumulative design procedures are committed to successive moves, previous ones becoming the foundation for subsequent ones. Unlike re-centering (above), such processes are inherently brittle, as they are not rooted in fields of competing ideas in the way games are.

Move 4.1 RECENTERING upon EXCLUDED FEATURES

The following statements exemplify what the definition of “game” from Rules of Play would exclude from game-building if it were employed as a way of designing. Each statement has two parts: excluded activities (EXCL), i.e., design modes not considered game features, and included ones (INCL), or how those same modes might usefully serve game-building and game-play.

EXCL. Purely cooperative games: activities without conflict are not games. INCL. Cooperation may occur between certain players to gain advantage over others; when the game pits the “designer” against the “problem,” team members may sometimes compete to have an idea accepted or cooperate to develop various aspects of a scheme.

EXCL. Qualitative games: a true game must have a quantifiable outcome. INCL. ARCH-games “always” have conditional features that require qualitative judgment.

EXCL. Games in search of rules: a game must be a game throughout play.
INCL. Searching for rules is a form of structural play.

EXCL. Games that produce surpluses without use: non-engageable elements do not make elegant games.
INCL. Auxiliary elements, like this excluded feature, may be used later in the game, just as they may in design.



Move 4.2 WITTGENSTEIN’S MULTIPLICITY

The critical ingredient in Wittgenstein’s aphorism on “multiplicity” is the emphasis on the activities into which language is woven. He illustrates common language-games as everyday activities in which the task and the context of language usage allow words to “mean” specifically something. He imagines such common practices as games, because we make “moves” in situations and are limited and directed by the grammars of the activity. In this regard, ARCH-design may be seen as innumerable language-games played within the evolving milieu of the project. Below, Wittgenstein lists common tasks-as-games to stress the ubiquity of game-like procedures in everyday life.

Review the multiplicity of language-game in the following examples, and in others: Giving orders, and obeying them—Describing the appearance of an object, or giving its measurements—Constructing an object from a description (a drawing)—Reporting an event—Speculating about an event—Forming and testing a hypothesis—Presenting the results of an experiment in tables and diagrams—Making up a story; and reading it—Play-acting—Singing catches—Guessing riddles—Making a joke; telling it—Solving a problem in practical arithmetic—Translating from one language into another—Asking, thanking, cursing, greeting, praying … It is interesting to compare the multiplicity of the tools in language and of the ways they are used, the multiplicity of kinds of word and sentence, with what logicians have said about the structure of language. (Inducing the author of the Tractatus Logico-Philosophicus.)[L.W.’s parentheses]

While Wittgenstein’s examples are not specifically about ARCH design, they suggest the myriad activities that comprise architectural projects—like the “rejected” heuristics above that nonetheless play roles in design games.

Move 4.3. THE WEB AS A LATTICE OF COMBINATIONS

Wittgenstein understood the body of all games as “a complicated network of similarities overlapping and criss-crossing….” Instead of a common thread that all others obey, they are more like “family resemblances:” what we might refer to today as the consequent similarities and differences resulting from genetic combinations. Siblings, for example, will have some unique mix of the pool of eye and hair color, facial features, build, or left or right-handedness.

This, of course, is very much like the network described above; moreover, it identifies a set of morphological language-games that account for various historical architectures.


Move 5.0 GAMES that BRIDGE ARCH and MATH

The history of architecture is replete with games, whether Renaissance gardens or postmodern “French Hotels,” that play with narrative or precedent. While scores of examples have been produced in the many years that this thinking has been the subject of studio investigations at Cornell, a single example will suffice to demonstrate the potential trajectory into an ARCH-game future.

Move 5.1 THE DENSITY GAME

The Density Game (below) is both system of production and rhetorical language that focuses an ARCH problem and offers a rule-based system for describing a MATH task should it be needed. Rather than beginning a priori with the assumption of MATH modeling, it nonetheless requires the formulation of a game appropriate to the emerging problem. This stresses the importance of the uniqueness of the problem rather than one defined by an exploration of a MATH procedure. If efficiency and optimization are key issues, then these might be built into the game—but without preempting the search for an appropriate language.

The image—showing blocks stacked in a pattern that barely remains stable—
is not even, not yet, perhaps not ever, a building. It is, however, a kind of tipping point in a design project that poses interesting questions about the MATH-side and the ARCH-side of the discussion so far. Has it reached the state where a designer might ask, “Can this game be modeled and so manipulated by an algorithm?” In other words, has it reached that point where it is so well articulated as an OBJECTIVE that the mathematician can calculate it?

The sides of the basic diagram, MATH and ARCH, might be renamed temporarily as the activities LANGUAGE CALCULATIONS and LANGUAGE GAMES to reflect Wittgenstein’s move from his seminal study of the logic of language in Tractatus[6] to language games in Philosophical Investigations.[7] From the perspective of the former, one’s task would be to create a generative grammar of forms that could account for the pile of blocks and its textual relationships. From the latter, we would consider what sort of form game(s) might do the same. Clearly, the latter strategy would be easier and more flexible, but would presumably lack the capacity to automatically generate the block pile. Similarly, George Stiny has long sought to subsume architectural production through logical operations of his “shape grammars,” his version of the MATH-side of this discussion. However, he has recently published a competing argument in a book called Shape.[8] Citing Wittgenstein in references to both Tractatus and Philosophical Investigations, he appears to be retracing the philosopher’s own repudiation of “logical philosophy” in favor of systems of meaning embedded in the activities of usage. At the same time he searches for “ways of counting” and “calculating” with shapes.

With Stiny’s shape grammars in mind, that is, a formal, mathematically driven object language, we might take the pile as an OBJECT to be modeled as a grammar. Unfortunately, without further development, this would not only be an impossible task—we would know neither the meaning of all placements nor be able to review multiple variations that would reveal the pile’s conditional logic—but would miss the process, the game, that created it. The discussion that follows, on the other hand, speculates on such further development of the ARCH-pile’s language to bring it to a point at which MATH modeling would be feasible.


Move 5.2 A GAME SIMULATION

While one cannot do justice to this design process in the space left in the essay, a great deal comes into focus even when the game is briefly described. The imperative that preceded the studio exercise, for example, was to discover scales and patterns in a small town that would suggest any addition to the town fabric. In this case, the designer initially chose the party wall texture and the massing of its buildings along a main commercial street. The buildings forming that pattern, while dominant, were undistinguished and when appropriated for her design, merely reproduced the pattern. She invented her game, not because she understood the logic of the site but because she did not. What she did have, on the other hand, was the conviction that some grain, some mixture of scales, was needed to set her building apart even while embedding it in the urban fabric. To her “density” meant all these things.

To shorten the story: she created an inventory of many small blocks and, placing these in a Plexiglas container, experimented with producing a “natural” pattern. The strategy, in retrospect, was to search for a mix of scales relevant to the artifacts of the town—its porches, rooms, windows, furniture, even paintings, rugs, and the like—rather than its urban fabric. For the tourist town, the pile-of-blocks would become cabinets of wonder, inventorying town fragments and breaking through the party wall obstacle. Openings in the pile would give access to a new lateral, day/night, summer/winter, and tourist/citizen street cutting through perpendicular to the party wall rhythm.

Gradually, using gravity and the dynamics of collision and settling as a “natural” way to order the pieces, she devised means for influencing the process. By mixing blocks of different sizes and proportions and by coaxing “architectural order” with formworks of rods, wedges, channels, planes, and so on, she imparted indirect control to the “natural phenomenon.” Interestingly, this indirect manipulation of behavior is similar to cellular automata: the blocks are simple passive “cells” governed by gravity and the shaking of the box. The rules were also passive in the sense that they were strictly physical conditions. The designer intervened by determining starting patterns, by progressively introducing formworks, and by analyzing multiple outcomes to determine behaviors. Like cellular automata, the design was evolutionary, in which good starting patterns, formwork tactics, shaking schemes, and updated ratios and densities of aggregate blocks were monitored. As the process evolved, additional conditions and artifacts were identified and included: block size-to-usage; groupings of blocks to ARCH function; space-to-mass for occupancy; movement, light, and air; density/scarcity; and structural latency in block arrangement versus separate systems.

The accompanying image captures the density-game as it approaches a plausible balance between natural and architectural orders. The process of defining an emergent architectural language has just begun. It is a moment of confidence and a branching point for the way forward. Because game-building develops rule systems for ARCH languages in whatever representational mode, it provides a way to manage methodological decisions dependent on time, resources, and knowledge. The Density Game described above may by-pass sophisticated methods for small projects or low budgets. It can be ramped-up to create a MATH-driven simulation, or proceed as a to-and-fro process of alternating algorithm-driven and manual operations.

(L) An Architectural Pile. This is the last of three-steps, in which the pile-of-blocks becomes a plausible basis for a study for an urban Architectural Cabinet of Wonders.<br />
<br />
(R) A Gravily Game. This the fist of three-steps, in which a gravity game—comprising random blocks shaken in a game-box—challenges the player to achieve architectural order within a “natural” procedure.
(L) An Architectural Pile. This is the last of three-steps, in which the pile-of-blocks becomes a plausible basis for a study for an urban Architectural Cabinet of Wonders.

(R) A Gravily Game. This the fist of three-steps, in which a gravity game—comprising random blocks shaken in a game-box—challenges the player to achieve architectural order within a “natural” procedure.

Gravity Game Box. Cut-away of a plexi Gravity Game Box, into which rods, wedges, channels, etc. are inserted in order to herd “natural” behavior into architectural order. All models by Camille Vélez (B. Arch. ’14).
Gravity Game Box. Cut-away of a plexi Gravity Game Box, into which rods, wedges, channels, etc. are inserted in order to herd “natural” behavior into architectural order. All models by Camille Vélez (B. Arch. ’14).


Concluding remarks

Although Steele’s paired aphorisms have ARCH presenting OBJECTIVES to MATH, the last century saw most polemics stripped away from architectural discourse. MATH, in its guises of morphogenics, biomimetics, cellular automata, and so on, has become a surrogate for the authority of Nature. As such, mathematical procedure sets ideological OBJECTIVES, not architecture. Hence the dis-balance in Steele’s pairing.

The thrust of this essay has been to introduce and demonstrate the validity of a way of working called the Architectural Game and to redress the imbalance perceived in Steele’s aphorism about the relationship between architecture and mathematics. The essay asserts that game-building/game-play is a natural mediator between the language games of architectural design and the “conceptual system possessing internal necessity”[9] that is mathematics. ARCH-games—because of their inherent rule-building capacity, their propensity to become language games, and their characteristics as systems that are bounded, interactive, structured, and flexible—are suited for gathering, mixing, and resolving the exigencies of ARCH and MATH.

Closing note

While the central topic, Mathematics: From the Ideal to the Uncertain, has been subordinated in the essay to the ARCH/MATH dialog, the spirit of the IDEAL and of the UNCERTAIN is captured in the image of the Density Game, in which a randomness is herded, by successive moves, into a relative order. The player must abide by the rules of the game, even as its ingredients are nudged toward a balance between a NATURAL and an ARTIFICIAL order. Simulated Nature provides the host phenomenon; artifice intervenes. By cosmological analogy, the IDEAL is a consequent, not a predetermined form; it is a consequence of a universe structured in ways that depending on celestial activity perfect limits are approached. The argument here is that the ideal is MADE, or more accurately, achieved as a limit; that it may also be felt, because the sensorial subject is also MADE—possibly to be harmonic.



Endnotes

1. Brett Steele, “War of the Gods,” in Jane Burry and Mark Burry, The New Mathematics of Architecture (New York: Thames & Hudson, 2010), 6–7.

2. David Hilbert; Paul Bernays, Grundlagen der Mathematik (Foundations of Mathematics) (Berlin; Heidelberg; New York: Springer, 1968).

3. Lois Sawver, Commentary on Wittgenstein’s Philosophical Investigations (Internet: Rhizomeway.com, November 2011), Aphorism 100.

4. Frank Lantz, from the Foreword to Rules of Play, Katie Salen and Eric Zimmerman (Cambridge, MA;, London, UK: The MIT Press, 2004).

5. Lois Sawver, Commentary on Wittgenstein’s Philosophical Investigations (Internet: Rhizomeway.com, November 2011), Aphorism 23.

6. Ludwig Wittgenstein, Tractatus Logico Philosophicus (Germany: Annalen der Naturphilosophie, 1922). English translation published by Kegan Paul, 1922.

7. Ludwig Wittgenstein; G.E.M. Anscombe; Elizabeth Anscombe, Philosophical Investigations (New York: Wiley-Blackwell, 1991). The German Text and a Revised English (US) Translation.

8. George Stiny, Shape (The MIT Press: Cambridge, MA, 2008).

9. David Hilbert; Paul Bernays, Grundlagen der Mathematik (Foundations of Mathematics) (Berlin; Heidelberg; New York: Springer, 1968).




Go back to 9: Mathematics