**Defining Emergence:**

**Complex behavior:** a system with multiple agents dynamically interacting in multiple ways, following local rule and oblivious to any higher level instructions.

[5]
**Emergent behavior:** a higher-level pattern arising out of parallel complex interactions between local agents.

[6]
**Adaptive emergent behavior:** the system would use local rules between interacting agents to create higher-level behavior well suited to its environment.

[7]
**Adaptive Emergent Architecture:** ‘Emergence’ is the scientific mode in which natural systems can be explored and explained in a contemporary context. It provides ‘models and processes for the creation of artificial systems that are designed to produce forms and complex behavior, and perhaps even real intelligence.’

[8]
**(Another Declaration of Our Contemporary Cultural Condition)**

“Perhaps, while we wish for more efficient, predictable systems where success is guaranteed, what we will get is an understanding of the benefit of a bit of chaos in the systems and the roll it plays in ‘progress.’”[9] (Steven Johnson, *Emergence*)

**Self-Organization: The Biological Model **

Experiments in self-forming processes emanate primarily from zoologist and mathematician D’arcy Thompson’s 1917 text

*On Growth and Form*[10] in which he speculated than the form of biological organisms is influenced by physical laws and mechanics as much as (or more than) than by Darwin’s “survival of the fittest” theory. Thompson identified similarity in the form of jellyfish and that of drops of liquid falling into a viscous fluid, for example, and in the form of the hollow bones of birds and engineering truss designs. (Around the same time as Thompson’s writings, early 20th century architect Antoni Gaudi was experimenting with catenary chain models to define the form of his Church of the Sagrada Familia.) Today, at least in part because of Thompson’s observations, biological organisms are understood as self-organizing systems: natural systems which organize material in space, over time, and under the load of gravity through the interactions of many simple components (such as sand grains, water molecules, and living cells) - a process known as

**morphogenesis**. It is the differences in the patterns of assembly of these simple components which result in differences in the form and performance of the organism (or

**system**)

[11]. (Siegfried Gaß, a student of Frei Otto, published an extensive analysis of typical forms resulting from self-forming processes in

*Form Force Mass 5: Experiments* published in Institut für Leichte Flächentragwerke (IL) 25. Of particular relevance to this paper is the section on structures in space and time - see

**structures in space and time**[12]). Extending this model to the design of buildings shifts the Modernist paradigm of form “rationalized for realization and superimposed functions” to a new paradigm where form is

*derived from *the capacities of materials and constructs.

[13]
To

*derive* form in this model, a process of

**differentiation** is necessary: the process of solving the (biological or architectural)

**system** for multiple variables, broadly defined by Achim Menges, architect and studio master for the Emergent Technologies program at the Architectural Association in London, as

**ecology**,

**topography**, and

**structure**.

[14]
**morphogenesis:**

the biological process that causes an organism to develop its shape.
(http://en.wikipedia.org/wiki/Morphogenesis)
**system:**

the part of the universe that is being studied, while the environment is the remainder of the universe that lies outside the boundaries of the system. Depending on the type of system, it may interact with the environment by exchanging mass, energy (including heat and work), linear momentum, angular momentum, electric charge, or other conserved properties. In some disciplines, such as Information theory, information may also be exchanged. The environment is ignored in analysis of the system, except in regards to these interactions.
(http://en.wikipedia.org/wiki/Environment_(systems))
**ecology:**

all the relationships between human groups and their physical and social environments.
**topology:**

the connections between all the material elements in an environment.
**structure:**

organizational capabilities above and beyond load bearing.
(Achim Menges. 2006. “Morphoecologies,” __AD__ 76, No2, p.73)
**structures in space and time **

(Siegfried Gaß)
Structures in space and time have typical phenotypes, depending on whether energy or matter is transported in space by the local motion of the particles.

Waves produce periodically recurring motions which transport energy as a function of time (left, from top, images 1 and 2).

If material particles are transported in space, motions of rotation are produced in an enclosed space which form vortexes.

Further characteristic families of forms are ring vortexes (left, image 3), Bénard cells (right, from top, image 4), helical vortexes (right, image 5), and vortex streets (right, image 6).

**Defining Differentiation:**

**Calculus: **the study of change, in the same way that geometry is the study of shape and algebra is the study of equations.

15
**Derivative:** in calculus ... the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point. For example, the derivative of the position (or distance) of a vehicle with respect to time is the instantaneous velocity (respectively, instantaneous speed) at which the vehicle is traveling. Conversely, the integral of the velocity over time is the vehicle's position.

16** **
**Differentiate:** to obtain the mathematical derivative of.

17
**Differentiate: **development from the one to the many, the simple to the complex, or the homogeneous to the heterogeneous.

18
**Differentiate:** modification of body parts for performance of particular functions (the sum of the processes whereby apparently indifferent or unspecialized cells, tissues, and structures attain their adult form and function).

19** **
**Differential Equation: **a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics and other disciplines.

20
**The Mathematical Model...**

D’Arcy Thompson first applied mathematics to biological form to quantify his theory of morphogenesis. This conceptual leap paralleled mathematician and philosopher Alfred North Whitehead’s theory which argues that “organisms are bundles of relationships that maintain themselves by adjusting their own behavior in anticipation of changes to the patterns of activity all around them.”

21 From these two theories (collectively arguing that form and behavior emerge from process)

22 has emerged a rich discourse in mathematical and computational models for complex systems including

**cybernetics** (Norman Weiner), geometrical phyllotaxis and the a

**ttraction-diffusion **model (Alan Turing),

**systems theory**,

**complexity theory**,

**genetic algorithms** (John H. Holland), and most recently, mathematical simulations of genes acting in

**Boolean networks** (Stuart Kauffman - see

**boolean networks**23).

24

**cybernetics:**

organises the mathematics of responsive behavior into a general theory of how machines , organisms and phenomena maintain themselves over time. It uses digital and numerical processes in which pieces of information interact and the transmission of information is optimized. Feedback is understood as a kind of ‘steering’ device that regulates behavior, using information from the environment to measure the actual performance against a desired or optimal performance.
**attraction-diffusion:**

a hypothesis of the generation of a pattern from a smooth sheet of cells during development in the formation of buds, skin markings and limbs. Chemicals accumulate until sufficient density is reached, then act as morphogens to generate organs.
**systems theory:**

the concepts and principles of organization in natural systems are independent of the domain of any one particular system.
**complexity theory:**

focuses on effects produced by the collective behavior of many simple units that interact with each other, such as atoms, molecules and cells.
**genetic algorithms:**

initiate and maintain a population of computational individuals, each of which has a genotype and a phenotype. Sexual reproduction is simulated by random selection of two individuals to provide ‘parents’ from which ‘offspring’ are produced. By using...random allocation of genes from the parents’ genotype and mutation, varied offspring are generated until they fill the population.... The process is iterated for as many generations as are required to produce a population that has among it a range of suitable individuals to satisfy the fitness criteria.
(Achim Menges. 2006. “Morphogenesis and the Mathematics of Emergence” AD 76, No2, p.11-15)
**boolean network:**

a set of boolean variables (a primitive data type having one of two values: true or false) whose state is determined by other variables in the network (through comparison operators: >, ≠, AND, &, *, OR, |, +, EQV, =, ==, XOR, NEQV, ^, NOT, ~, !). elementary cellular automata are particular cases of boolean networks, where the state of a variable is determined by its spatial neighbors.
(http://en.wikipedia.org/wiki/Boolean_network)
**boolean networks **

(Stuart Kauffman)

boolean networks with low connectivity or with certain biases in boolean switching rules exhibit unexpected, spontaneous collective order.

sites within a network can affect one another's behavior: nearby sites communicate frequently via many small avalanches of damage; distant sites communicate less often through rare large avalanches.

networks on the boundary between order and chaos may have the flexibility to adapt rapidly and successfully through the accumulation of useful variations.

**...as applied to Architecture**

These mathematical and computational models (all based on

**calculus,** the mathematical study of change) have so far found their way into built architecture through

**topological **surfaces functions in CAD software (which are so prolific they appear as the

**sandbox tools** in the free and popular Google SketchUp software), time-and-force modeling attributes in animation software and

**parameter-based modeling**.

25 Architectural research and experimental design is attempting to instrumentalize the most recent mathematical advances primarily through investigations into phylogenesis, morphogenetic design techniques, and material emergence/performance.

**Phylogenesis** is the development of an evolutionary history of architecture within a practice,

**morphogenetic ****design **is the integration of ecological, topological and structural performances using generative design strategies and

**material emergence** is the exploration of forms with inherent potential for self-organizationduring fabrication.

Architects working within these discourses are beginning to tackle how architecture can best absorb a shift in thinking actualized by our contemporary cultural condition; the shift from

*top-down *to

*bottom-up*.

These three approaches differ principally in their

*application* to architecture: morpho-ecologies are formal, structural and programmatic solutions to architectural problems, while phylogenesis is used to generate

*diagrams* for architectural design, and material emergence develops the architect’s

*intuition *for

**animate forms **(Greg Lynn - see

**animate form**26). In

*Animate Form*, architectural theorist Greg Lynn argues that “even in the most scientific applications of computer simulations ... first an intuition must be developed in order to recognize the non-linear behavior of computer simulations.”

27
**topology:**

a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects (deformations that involve stretching, but no tearing or gluing).
(http://en.wikipedia.org/wiki/Topology)
**triangulated irregular network (TIN):**

a vector based representation of a surface, made up of irregularly distributed nodes and lines with three dimensional coordinates (x,y, and z) that are arranged in a network of non-overlapping triangles.
(http://en.wikipedia.org/wiki/Triangulated_irregular_network)
**sandbox tool (google sketchup):**

use the Sandbox From Contours tool to create a TIN from contour lines.
(http://download.sketchup.com/OnlineDoc/gsu6_mac/Content/P-Terrain_Modeling/TerrainTool-Intro.htm)
**parametric modeling:**

using the computer to design objects by modeling their components with real-world behaviors and attributes. A parametric modeler is aware of the characteristics of components and the interactions between them. It maintains consistent relationships between elements as the model is manipulated.
(http://www.pcmag.com/encyclopedia_term/0,2542,t=parametric+modeling&i=48839,00.asp)
**animate form:**

animation is a term that differs from, but is often confused with, motion. While motion implies movement and action, animation implies the evolution of a form and its shaping forces; it suggests animalism, animism, growth, actuation, vitality and virtuality.
(Greg Lynn. 1999. Animate form. Princeton Architectural Press. New York.)
**animate form **

(Greg Lynn)

“If there is a single concept that must be engaged due to the proliferation of topological shapes and computer-aided tools, it is that in their structure as abstract machines, these technologies are animate.”

a keyboard is an actual machine, it is technological therefore it is a concrete assemblage.

the distribution of letters on keys in space is an abstract machine, it is a virtual diagram designed to limit the speed of typing; no particular test word or sentence exists, and it applies to an indefinite series of existing and future words.

**On Walking**

Francesco Careri, Italian architect and member of the Stalker urban art workshop, writes at length about the history of walking as a form of architecture and as an aesthetic practice in Walkscapes, defining the path as the first architecture, a creation of nomadic world rather than the sedentary, settled world. “The act of crossing space stems from the natural necessity to move to find the food and information required for survival. But once these basic needs have been satisfied, walking takes on a symbolic form that has enabled man to dwell in the world. By modifying the sense of the space crossed, walking becomes man’s first aesthetic act, penetrating the territories of chaos, constructing an order on which to develop the architecture of situated objects. ... This simple action has given rise to the most important relationships man has established with the land, the territory.”

28
flocking:

the behavior exhibited when a group of birds, called a flock, are foraging or in flight. There are parallels with the shoaling behavior of fish, or the swarming behavior of insects.

From the perceptive of the mathematical modeler, "flocking" is the collective motion of a large number of self-propelled entities and is a collective animal behavior exhibited by many living beings such as birds, fish, bacteria, and insects. It is considered an emergent behavior arising from simple rules that are followed by individuals and does not involve any central coordination.

(http://en.wikipedia.org/wiki/Flocking_(behavior))

Muybridge

“The term ‘path’ simultaneously indicates the act of crossing (the path as the action of crossing), the line that crosses the space (the path as architectural object) and the table of the space crossed (the path as narrative structure). ... In this century the rediscovery of the path happened first inliterature (Tristan Tzara, Andre Bréton, and Guy Debord are writers), then in sculpture (Carl Andre, Richard Long, and Robert Smithson are sculptors), while in the field of architecture the path has led to the pursuit of the historical foundations of radical anti-architecture in nomadism, and has not yet led to a positive development.”

(Process Philosophy)

“If you abolish my consciousness … matter resolves itself into numberless vibrations, all linked together in uninterrupted continuity, all bound up with each other, and traveling in every direction like shivers. In short, try first to connect together the discontinuous objects of daily experience; then, resolve the motionless continuity of these qualities into vibrations, which are moving in place; finally, attach yourself to these movements, by freeing yourself from the divisible space that underlies them in order to consider only their mobility – this undivided act that your consciousness grasps in the movement that you yourself execute. You will obtain a vision of matter that is perhaps fatiguing for your imagination, but pure and stripped of what the requirements of life make you add to it in external perception. Reestablish now my consciousness, and with it, the requirements of life: farther and farther, and by crossing over each time enormous periods of the internal history of things, quasi-instantaneous views are going to be taken, views this time pictorial, of which the most vivid colors condense an infinity of repetitions and elementary changes. In just the same way the thousands of successive positions of a runner are contracted into one sole symbolic attitude, which our eye perceives, which art reproduces, and which becomes for everyone the image of a man who runs.“

(Henri Berson, Matter and Memory)

To permit a material experiment which incorporates occupation as a generating force in the form (and gradient of performances) of an architectural surface, I propose broadening this discourse to include two more primary references: Greg Lynn’s Animate Form and Leon van Schaik’s Spatial Intelligence.

Weinstock cites Antoni Gaudi (tangentially) and Frei Otto (substantially) as the progenitors of this new area of material experimentation and credits “new mathematical models”

as providing the experimental framework for next-generation investigation into self-organizing systems.