Mathematics : (as a)Mediating Link in Architecture
Assoc. Prof. Dr. Arzu Gönenç Sorguç
METU Department of Architecture
arzug@metu.edu.trhttp://archweb.metu.edu.tr!2

Why computational modelling?
▪ Need to solve a problem, curiosity, inspiration, to learn, tocompare, to re-experience.
▪ Computational models should not be perceived by theirparameters and what they represent alone, but also with theprocess of modelling consisting of several mappings, that is,algorithms constructing the end model.
▪ They are the mediating links from nature to architecture or▪ De-coding and Re-coding in architecture▪ AN ARCHITECTURE SOMETIMES WITHOUT AN
ARCHITECT!

A groundbreaking paradigmatic change inarchitecture: computational design▪ Particularly after the Information Revolution, architecture has
been going through a shift from an interest in form
production towards form generation and form finding.
▪ The basic accelerator of such a progress is the intention ofderiving an explicit structure of generative design (system),
where the architectural product is regarded as the associative
outcome of an integrated process that intakes knowledge/
information from various disciplines▪ FROM FORM CENTERED TO PROCESS CENTERED

Architecture has always been in communication with nature throughout the humanhistory. Some architectural products interacted with nature environmentally, and someinternalized nature in the form and generation of design. Common to both, theinformation existing in natural phenomena has been modelled in a particularity andstructure in order to be transferred to architecture. Depending on the complexity andcapability of the model, the knowledge extracted and implanted had various inferences
for architecture in addition to a variety of scale, proximity, technicality, complexity, andscope.

!

THE PROCESS-BASED INTELLIGENCE OF NATURAL PHENOMENA IS THE
PRIMARY SOURCE OF LEARNING ABOUT SYSTEMS, WHICH CAN INSTRUCT/CONDUCT ARCHITECTURAL INQUIRY AND THE FORM FINDING
EXPERIMENTATIONS/EXPLORATIONS WITHIN THE GENERATIVE PROCESS ITEMBRACES.

!

Re-visiting nature with computation

CASE 1

From Sea Shell to Man-made Shell
Arslan Semra, Ph.D Thesis, Supervisor: A.Gönenç Sorguç,What is shell? Why shell?
▪ the word shell is commonly used to describe external, usually
hard, protective, or enclosing case or covering in nature.
▪ Similarly, in man-made products the word shell is used todefine a rigid covering that envelops an object or a frameworkor exterior, as of a building.
▪ Shell structures are greatly superior to conventional columnbeam structures when seeking to cover large spans withminimum material. spans, carrying capacity compromising live
and dead loads is very efficient in shells.
▪ Although shells have the most complex structural behaviors
they are still the most effective form.

Decoding and Re-coding Seashell▪ A shell is a three dimensional curved structure which resists
load through its inherent curvature.
▪ There are numerous shell structures in nature; eggs, skulls,nuts, turtles and seashells, are notable examples▪ A shell’s structural behaviour is derived directly from its form,▪ This not only dictates the æsthetics, but the overall efficiency
and behaviour under load of the structural system.

Firstparameters…anda new parametricmodel
S: Section; Shape of the aperture or
shape of the shell’s tube cross section
H: Horizontal Displacement; Departurefrom the coiling axis of the section in thehorizontal direction
V: Vertical Displacement; Translation
along the vertical direction of the coilingaxis
G: Growth: Aperture expansion or the rateof increase of section sizeAbstracted simple seashell models
generated by the initial model ….

We should
understand more!!!
Seashell growth…
Seashells are formed in nature by growthat the shell’s free leading edge.
Their increase in overall size is achieved
purely from successive addition ofmaterial to one end only [From inspection of actual seashell crosssections, the older previously formed
parts of the shell remain, on the whole,unaffected and geometrically unchanged
once produced

Cut cross-sections of a selection ofseashells found in natüre (METU)

Let us focus
more!
Simple models are not good enoughto understand structure-form
relationBe more precise!
B-Splines to model
seashells…
A parametric model of the shell was set
up using the cross section of a single
whorl as the input. In a similar approachto Fowler et al [the shape of a single
typical whorl was defined as a B-spline
The shell surface geometry is definedusing cylindrical polar coordinates
X=rcosΘ	y=rsinΘ	

Ζ=z

B-spline and its control polygon(right) single whorl cross-section
Geometry of thelogarithmic spiral.

Towards a
realistic model…
From observations made from real
seashells the growth rings, whichcorrespond to the whorl cross-section, are
not radial and do not even lie in one
plane.
They are curves in three dimensions. Thismeans that the shell’s rate of spiral, λ are
φbegin on the current leading edge crosssection must lie coincident with φend on thepreceding section after slightly less than
one revolution about the major z-axis, i.e.
Δθ = 2π - θjoin,

Geometric relations of the shell surface

New model…
Thus λ is controlled by therelationship:

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The growth constant is applied tothe whorl by transforming its
coordinates using the formulae
below, based on the cylindrical polar
coordinate transformation

Scan of the seashell found in nature (left) computer
generated shell model (right) by the algorithm
developed for this research.

The model is generative enough…

Demonstration of the flexibility of theprogram to experience different coiled
seashell forms.

Stress-Form relations
Compression tests in lab

FEM results

Some Non-dimensional Parameters toRepresent a Shell

What we learnt?

CASE 2Cactus to highrise….
M.Sc Thesis by Elif Erdoğan Supervisor Arzu Gönenç SorguçWhat we will learn from cactus?
▪ “Thus, the architects wishing touse this new tool[geneticalgorithms] must not only become
hackers(so that they can create
the code needed to bringextensive and intensive aspectstogether) but also be able “tohack”biology, thermodynamics,
mathematics, and other areas ofscience to tapinto necessaryresources.”!

Manuel DeLanda. “Delueze and the Use of theGenetic Algorithm in Architecture.” in Phylogenesis:
foa’s ark. Edited by Foreign Office Architects.
Barcelona: Actar, 2004. p.529.

Self-shadowing, self-ventilation

The ‘shared body plan’ of the cactus family. A schematic representationof parameters, operations, relations and the generative set-up at a highlevel of abstraction

Testing the Correspondence of the Script OutcomesWith The Cacti Species.

Populations of the Script With Different Initial
Polygonal Geometries, and Different DivergenceAngles

Populations of the Script With Different AreoleHill Depths

Three different architectural projections of the Cactus
Script: (a) High-rise buildings, (b)built terrains, (c) double façades.

A Set of Possible Architectural Inferences of theScript for High-rise Buildings.

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Case 3: Human Crowd
Simulation
Muge Krusa M.Sc Supervisor A.Gonenc Sorguc#32

Case 4: Human Crowd
Simulation

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Basic Assumptions

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