Category Archives: Mathematics

Architectural Geometry Informed by Static and Dynamic Loading Conditions. 01

Advanced computer aided design (CAD) techniques liberated architectural form, by enabling architectural designer to generate complex forms, such as freeform surfaces. In today’s common architectural practices, computational tools, associated with performance analysis and evaluation, are undertaken during a later stage of the design process, following the form generation. ParaMaterial aims to discuss how material can be integrated into a system in which architectural geometry, material and structural performance are interdependent to increase efficiency by identifying critical procedures towards manufacturing of complex forms in architecture. Mathematically driven surfaces are explored of which geometrical attributes can be altered parametrically. Because buildings are designed to withstand various complex loading conditions, simulations are undertaken for surfaces via Finite Element Method (FEM) and Computational Fluid Dynamics (CFD) analysis tools to investigate how material informs architectural geometry in respond to the static and dynamic loading conditions.

The use of mathematics in computational design process enables from simple to highly complex geometries with control parameters. Numerous different types of mathematically driven surfaces such as sphere, torus, cylinder, catalan, moebius strip, klein surface, catenoid helicoids, henneberg, elliptic paraboloid, enneper and many more can be structured in parametric systems by assigning their respective mathematical curve functions.


Material in Performance-Driven Architectural Geometry . 01

Variations for Panelization of the Architectural Geometry

To realize complex freeform surfaces, one has to segment the shape into simpler parts, so-called panels ( Schiftner, N. Baldassini, P. Bo, H. Pottmann, 2008). Various methods to subdivide the geometry have been investigated including quadrilateral, triangular, diamond shaped panels.

Architectural Geometry Generated Through Curve Functions

Klein Surface is investigated in terms of architectural geometry which represents a usable volume. It is a non-orientable closed surface which is homeomorphic to a connected sum of a number of copies of the real projective plane. 

//Parameters : range values (β, μ); re-built curve values

//Definitions of curve functions

Function X = (1 + cos(β /2)*sin(μ) – sin(β /2)*sin(2* μ))*cos(β)

Function Y = (1 +cos(β /2)*sin(μ) – sin(β /2)*sin(2* μ))*sin(β)

Function Z = sin(β /2)*sin(μ) + cos(β /2)*sin(2* μ)

Seminar@ Istanbul Technical University

 I was kindly invited to give a seminar as a part of the course called Contemporary Building Materials (MIM 328) at the Istanbul Technical University, Department of Architecture. ‘Material based Design Computing’  seminar is based on interpretation of my past work & current work by ParaMaterial in perspectives to the historical and contemporary framework of architectural design, as well as design computing.

Exhibiting @ Sigradi 2010 : Computing & Materialization. 03

Computational Surface Generation

Computational surface generation through the GH interface
f (x) = y * sin (z * x)
This research aims to investigate the generative process of components that belong to a double-curved computational surface, derived through the mathematical curve function f (x) = y* sin (z*x°) applied in two distinct directions. The surface parameters are able to vary and create time-based differentiated formal outputs in a digital parametric system. The parametric system is established through the set of rules and defined relations based on the McNeel Rhinocereos / Grasshopper platform. Geometry optimization becomes the integral part of the design process when the parametric model shifts into the architectural scale. Please visit: to watch the animation.