The
  Cornell
    Journal
      of
        Architecture
9
Challenge the Order



Maria Hurtado de Mendoza is a Spanish architect and founding partner of the Madrid-based practice estudio.entresitio. She has taught architectural design at the Architecture School of Madrid (ETSAM) since 1998, and was a visiting critic at Cornell University Department of Architecture in the Fall of 2011. Challenge the Order is the title of Hurtado de Mendoza’s visual representation seminar at Cornell, based on the ongoing research of her Ph.D. thesis, “Systems of Order.”
The order of a system is equal to the amount of information necessary for the description of this system.

J. Monod, Le Hazard et la nécessité (Chance and Necessity)

In contemporary practice, architects have tended to reproduce and code certain states of order/disorder that occur in nature or arise from creative intuition, describing them as complex because they are beyond immediate comprehension. The seductive graphical representations of this complexity, of what is apparently random, inevitably intrigue and distract one’s attention. This obsession with randomness has been described as “a paradigmatic anti-scientific attitude”[1]—a deserved comment indeed, if randomness occurs without meaning or under-standing. With the intention of going beyond the superficial, “Challenge the Order” investigates the underlying systems in search of a methodology to reproduce a certain configuration, more generally applied to architecture as a discipline, and in particular, to the construction of architectural space. In this sense, the objective is not only to think about geometry as the beginning of form, but to think about geometry as the beginning of space.

Scientist Christopher Langton believes that between order and randomness there is an intermediate space called complexity. As a result of the interactions between elements, new properties emerge that cannot be explained by the properties of isolated elements. A complicated system, in contrast, is also composed of several parts, but the links between them do not add additional information. We need to know each of them to understand the system. In a complex system, however, there are hidden variables that prevent us from analyzing the system accurately. Thus, a complex system has more information than what each part is giving independently, and to describe it requires the understanding not only of the performance of the parts but of their relationship to each other.

The Spanish psychiatrist Nicolás Caparrós refers to this intermediate space of complexity as “the fertile place that resists the limits of a definition, diffuse in nature, where the proportion of order to disorder reaches a limit that is called the border of chaos, the score or hinge moment or even phase transition. It is the place of the new, the unusual, creation and emergence.”[2] His approach is related to the creative clutter notion of Ilya Prigogine, who proposes “non-equilibrium” as a source of the new.[3] For Caparrós, complexity does not mean the decline of classical science, but a natural extension of it; some old problems might be relegated or declared unscientific, others, previously nonexistent or trivial, may, with the new point of view, become significant archetypes.

Feng Lin’s (M. Arch. ’13) interpretation focuses on the transition from the square to a circle in a linear system. Using 3d software allowed her to arrive at the hybrid form. By placing the same number of points in both shapes, an algorithm calculated their intermediate shapes. The midway curve between a square and a circle is called “squircle” (to name is to know).
Feng Lin’s (M. Arch. ’13) interpretation focuses on the transition from the square to a circle in a linear system. Using 3d software allowed her to arrive at the hybrid form. By placing the same number of points in both shapes, an algorithm calculated their intermediate shapes. The midway curve between a square and a circle is called “squircle” (to name is to know).


Hua Ye (M. Arch. ’12) used both figures as a way of making “randomness.” The diagram above shows a mathematical way to analyze the ratio of a hyperbolic pentacle, which was a sacred geometry symbol in ancient times. Using the diagram for the Children’s Center for Psychiatric Rehabilitation, Hokkaido, Japan by Sou Fujimoto, the basic geometric object was duplicated, rotated, and overlapped to create a system in which randomness was found by making the parts visible in a selective way.
Hua Ye (M. Arch. ’12) used both figures as a way of making “randomness.” The diagram above shows a mathematical way to analyze the ratio of a hyperbolic pentacle, which was a sacred geometry symbol in ancient times. Using the diagram for the Children’s Center for Psychiatric Rehabilitation, Hokkaido, Japan by Sou Fujimoto, the basic geometric object was duplicated, rotated, and overlapped to create a system in which randomness was found by making the parts visible in a selective way.


Simin Wang (M. Arch. ’13) also described these objects showing how a square and a circle look contradictory in 2D, yet can produce a harmonious condition in a 3d object, since all different views of these peculiar objects are either a square or a circle (top, front, and left).
Simin Wang (M. Arch. ’13) also described these objects showing how a square and a circle look contradictory in 2D, yet can produce a harmonious condition in a 3d object, since all different views of these peculiar objects are either a square or a circle (top, front, and left).


Alison Nash (M. Arch. ’13) studied quasicrystals and aperiodicity, and how, related to a network of primitive figures, there can be another less evident construction underlying it, with greater potential for a controlled randomness.
Alison Nash (M. Arch. ’13) studied quasicrystals and aperiodicity, and how, related to a network of primitive figures, there can be another less evident construction underlying it, with greater potential for a controlled randomness.



The ways in which the irregular and fragmented is controlled, and turbulence or self-similarity are described are open questions for architecture and its particular way of thinking. While it is certainly risky to establish a classification of different systems of order that interest architecture, this classifier’s desire arises as an attempt to understand, or using Koolhaas’s words, to “undo the ignorance” about what is interwoven in itself. The terms of mathematics are pervasive in the contemporary verbiage of architects: words such as parametric, topological, dynamic, emergent, fuzzy, and algorithmic have become relatively commonplace, however, their consideration should not imply the denial of some other words, such as the ideal.

The seminar was set up as a series of explorations investigating the space between order and disorder, beginning with a puzzle known for its impossible resolution“squaring the circle.” To square a circle has been considered, since ancient times, the question of how to draw a square with the same area of a given circle using only basic drawing tools: the impossibility of its resolution is due to the number π (pi) being involved in the circle’s dimension. The challenge introduced the world of geometric paradox and the possibility of working away from certainty, and opened up the possibility of redefining problems based on contemporary knowledge and tools.

Students were asked to find their own place of encounter between ideas of order and architecture, to reach an original and personal set of ideas, and to consider how their discoveries could be shared as a source of knowledge for the collective. Each chose a single system of order and after coming to an in-depth understanding of that system, proposed a possible translation into architecture.

Architects today are familiar with pure ordering devices such as the grid, or the radial. As architecture embraces more complex systems, new knowledge is required to interact with these (often concealed) systems and to be able to incorporate exceptions, deviations, or transformations. By understanding the ordering systems, the behavior, and the tools implied, architects are freer and solutions are more open and coherent. Geometry is not meant to be the solution, but the frame in which the architectural solution evolves—and to know the rules gives us the required margin to incorporate contingency in a rigorous way. While some may question such a desire for rigor, this work pushes toward a new understanding of rigor in which the absolute might be confronted with the inexact.


Order, whether it is evident or not, is a key to architecture, and contemporary thinking gives us the freedom to consider both extremes; there is no necessity anymore to choose between one thing and the other: no right or wrong. In fact, these extremes, the orthodox and the innovative ways of understanding order might not be so far apart. Yet still, our ideas of beauty are related to nature, where proportion plays a major role: the golden ratio defines the shape of the human body and face, the Fibonacci series defines the shapes of shells, flowers, fruits, and so on, but we also define as “natural” the indescribable beauty of a harmonious yet chaotic and seemingly random arrangement that arises from rules or patterns embedded within it.

Architecture understood as a way of establishing relationships and organizations represents the holistic meaning of the “whole” as something other than the sum of the parts, and order can be considered what holds things together. To consider how new ideas of order imported from the sciences of complexity can be translated and edited into architecture is something that architects must take time to investigate. This can only begin with questions, and albeit preliminary, this selection of graphic inquiries that seek to understand the mathematical system underlying the fixed image already begins to provoke possible spatial and formal consequences for architecture.


Endnotes

1. R. Thom, Halte au hasard, silence au bruit, Le Débat no. 3 éditions (Paris: Gallimard, 1980).

2. Nicolás Caparrós, Introduction to the encyclopedia Viaje a la complejidad (unpublished)

3. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (New York: Wiley, 1977).




Go back to 9: Mathematics