Fractals Geometry

Fractal is a term coin by their originator, Benoit Mandelbrot, in 1975. They are objects whose spatial form is not even nor fluid, hence irregular in nature, but in essence its irregularity as a non-smooth form repeats itself geometrically across in many different scales. The distinctive feature of fractals is that they are ‘self-similar’. It is the geometry of such object which is fractal, and any system which can be visualised or analysed geometrically, can be fractal if it inherit these characteristics.

Fractal geometry is a portrayal as geometry of nature. A geometry of nature evolved from the hypothesis which objects with geometry whose structure is irregular in terms of Euclidean geometry, but within this irregularity lays a pattern which is as order as those in simpler objects composed of straight lines.

A coastline or mountain ranges are examples of natural fractals. These fractals displays a sense of irregularity which characterises these objects, but is not entirely without order. Any part of a fratal, if enlarged or reduced in size, presents more or less the same appearance as a whole, the architectural forms or urbanistic models that are derived from them are characterised by a scalar ambiguity, by what we might call an “self similarity”[1].

Geometry is no longer composed and conceived by straight lines, the geometry of Euclid, but can now admit irregularity without abandoning continuity. Objects composed of a multitude of lines which are nowhere smooth may well manifest order in more accumulative terms than the sorts of simple objects which are dealt with in mathematics. This order can be discovered in fractals in terms of the following three principals.

Firstly, fractals are always self similar, at least in some general sense as this is the bases of its definition. On an enlarged or reduced scale, and within a given range you examine a fractal, it will always appear to be similar to the ‘whole’ with some degree of irregularity. The ‘whole’ will always be manifest in the ‘parts’. A great example is looking at a piece of rock that broke off from a mountain. You will see the mountains in the rock to some degree of familiarity[2].

Secondly, fractals can sometimes be portrayed in terms of a hierarchy of self similar components. Fractals are ordered hierarchically across many scales and the tree is a classic example. In fact, the tree is a literal interpretation of the term hierarchy and as such, it presents the most fundamental of fractals. Looking at the twigs on the branches of a tree and you can see the whole tree in these twigs, even thought at a much reduced scale. The organisation and spacing of cities as central places is such an order while the configuration of districts and neighbourhoods, and spatial distribution of roads and other communications are hierarchically structured[3].

The third principles relates to the irregularity of form. In terms of irregularity, we mean forms which are continuous but nowhere smooth, hence non-differentiable in terms of calculus. Take coastline as an example, “if you measure its length from a map, the map will have been constructed at a scale which omits lower level detail. If you actually measure the length by walking along the beach, you will face a problem of knowing what scale or yardstick to se and deciding whether to measure around every rock and pebble.” In the end, you will obtain a length that is relative and very dependent on the scale of measurement, and as the scales gets finer and finer down to the microscopic level, the length of the coastline will continue to grow. Therefore we are forced to conclude with the idea that the coastline’s length is ‘infinitely’ long or rather that its absolute length has no meaning and the length given is always relative to the scale of measurement[4].

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[1] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.59

[2] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.60

[3] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.60

[4] Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994) p.60

Parametric Modelling

“Transformation on Parametric Design Models” is an academic paper written by Carlos Barrios to present a research in progress in the development of parametric models for generation of complex shapes using parametric design, and introduces a methodology for exploration of possible designs generated from a single model. It also provides a case study on the designs of the Spanish architect Antonio Gaudi and examines the fundamental rules of form generation of the lateral nave columns of the Sagrada Familia temple in Barcelona. “A parameterization is presented as a fundamental tool for design exploration, which allows the reproduction of the original shapes designed by Gaudi, and the generation of a large set of new designs.”[1]

Barrios introduces the Parametric Modelling CAD systems as initially intended for the aeronautical industry, but later “making their way into the architectural domain since they provide a powerful framework for conception of design, allowing the description of multiple instances ad possible designs from a single modelling schema.”[2] Barrios further explains the breakthroughs posed by the introduction of parametric modelling system, as it overshadows the traditional CAD systems in terms of expanding the design process beyond current limitations, hence Experimental Modelling.

He states the principal advantage of a parametric model is the level of flexibility it provides to perform transformations that would result in different modifications and configurations from using one single geometrical components, creating an instance of a parametric model. It is important to note that parametric modelling creates an instance for each of the modified geometry component, thus allowing the designer to perform alterations and reconfigurations of the geometry without erasing and redrawing.

Barrios further explains the scope of design instances that a parametric model can generate, will be dependable on a balance between the parameters, the constraints of the geometry and the freedom or choices provided by the parameters. “The parameterization schema will determine which are the attribute subject to parametric transformations, in other words which components of the model will vary ad which components of the mode will be fixed.”[3] The range of the parameters can vary is determined by the step value of the parameters, along with its constraints.

Barrios moves onto discussing the parametric model for the columns of the Sagrada Familia. He observed Gaudi’s design rule are regulated by his architects’ intuition and interpretation of natural forms, as oppose to scientific knowledge of mathematics and geometry. Even though there are four initial shapes, the parametric model does not determine what kinds of shapes are valid. In terms of potential result and modification, the parameterized possibilities are infinite, as long as the shape is closed. He explains parametric combinations by stating “the number of design instances will not only depend on the number of shapes, but the possible combinations among them.”[4]

Barrios reinforces that the generation of a parametric model requires a level of design and planning deeper than just the presentation of an idea, but resulting a very powerful and versatile framework for design exploration. In the context of architecture, “most cases have used applications of parametric models in the context of design development, where most of the design decisions have been made”[5]. He then evaluates the purpose of parametric design and concluded that a well defined parametric model can serve as the means of creating designs in a particular purpose, while discarding other than do not fit the criteria.

Barrios finishes off with speculate how many possible designs a parametric model can generate, since all of the designs are obtained through altering and changing the initial shapes of the parametric model. “Design space (DS) of the parametric model equals to the sum of number of parameterized entities (PE) , and the step value of the parameters (SV), multiplied by the number and types of constraints (CN) which determines the degrees of freedom. “[6] He concluded that the number of designs generated by the parametric model is directly proportional to the number of parametric entities. “If we were to consider that design representations, such as plans, elevations, and 3D models, are geometrical models in an explicit representation, we must conclude that they are subject to parameterization. This proves that there is a great potential for applications in architectural design that has yet to be explored.”[7]

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[1] Carlos Barrios, Transformations on Parametric Design Models: A Case Study on the Sagrada Familia Columns (2006) Department of Architecture, Massachusetts Institute of Technology, USA p.1

[2] Carlos Barrios, Transformations on Parametric Design Models p.1

[3] Carlos Barrios, Transformations on Parametric Design Models p.2

[4] Carlos Barrios, Transformations on Parametric Design Models p.7

[5] Carlos Barrios, Transformations on Parametric Design Models p.7

[6] Carlos Barrios, Transformations on Parametric Design Models p.8

[7] Carlos Barrios, Transformations on Parametric Design Models p.8

10 sources

2 Major Sources

• Carlos Barrios, Transformations on Parametric Design Models: A Case Study on the Sagrada Familia Columns (2006) Department of Architecture, Massachusetts Institute of Technology, USA
• Michael Batty, Paul Longley, Fractal Cities: A Geometry of Form and Function (London, Academis Press Limited, 1994)

8 Sources

• Carl Bovill, Fractal Geometry in Architecture and Design (Birkhäuser Boston, 1995)
• Shaoming Lu, Hidden orders in Chinese gardens: irregular fractal structuve and its generative riles (Environment and Planning B: Planning and Design 2010, volume 36, pages 1076-1094)
• Stephen Demko, Laurie Hodges Bruce Naylor, Construction of Fractal Objects with Iterated Function Systems (New York, USA, Proceedings of the 12th Annua conference on Computer graphics and interactive techniques, 1985)
• Jonathan Chapuis, Evelyne Lutton, ArtiE-Fract: Interactive Evolution of Fractals (International Journal on Artificial Intelligence Tools, 2006)
• Mai Abdelsalam, Digitizing Architecture: Formalization and Content [4th International Conference Proceedings of the Arab Society for Computer Aided Architectural Design (Manama, 2009) p. 297 - 304
• Tristan Al-Haddad, Parametric Modulations in Masonry, (CAADRIA 2008) p.221 - 228
• Fabrizio Apollonio; Marco Gaiani; Cristina Corsi, A Semantic and Parametric Method for 3D Moels used in 3D Cognitive-Information System (Zurich 2010) p.863 - 872
• Ayman Almusharaf; Elnimeiri Mahjoub, A Performance-Based Design Approach for Early Tall Building Form Development (Morocco, 2010) p.39 - 50

Fractal - Lexicon

Hierarchical Structure

Evolving

Self Similar

Family Tree

Relations

Ratio

Design Inspiration

Federation Square

Lab Architecture Studio directed by Donald Bate, Peter Davidson and Bates Smart

Melbourne

Red Location Museum

South African architectural partnership Noero Wolff Architects

New Brighton Township of Port Elizabeth, South Africa

Seoul National University Museum

Seoul, South Korea

3 technical aspects of complex geometry

Rhizome

divided into two main principles:

• Principles of connection and heterogeneity, by which point of the rhizome can and must be connect with any other point.
• Principles of Multiplicity, which the rhizome can break all contacts wit the central unit and treat multiples as a substantive, ie: multiplicity

Genetic Algorithm

Several equilibria can coexist and different problems can be resolved simultaneously. Break down project can create multiple starting point that in turn generate new series of orders.

Fractural

A more adaptive system of exchange and integration of information that can be used to model dynamic scenarios of development, such as dynamic scenarios of development, such as generative process or the definition of programs, formalised as analogical models of natural phenomena.

Pierluigi Nicolin, Lotus International quarterly Architectural Review (Via Santa Marta 19/a 20123 Milano) edition 127, Diagrams

3 Design Aspects

Gothic Architecture

Gothic Architecture is a style flourished between the Romanesque Architecture and the Renaissance Architecture period, very successful within the Medieval Period. This type of architecture is mainly used churches and castles in that period. It places a very heavy emphasis on height, to create a majestic feeling to all worshippers, demonstrating might of God or its related religion. One of the main stylistic feature that is shown predominating on most Gothic Architecture is the use of arches, you can see the evident showing of arches on every aspect of the architecture.

Modernist Architecture

Modernism is a period formatted during the early 1900s. "This notion of 'modern' architecture was in turn roots in developments of the late eighteenth century, in particular the emphasis on the idea of progress."

Blob Architecture

Blob Architecture was first to sense and to investigate the creative possibilities of solid modelling software, characterised by its strong emphasis given to complex forms.

Sources

William Curtis, Modern Architecture Since 1900, (London: Phaidon Press, 3^rd edition, 1996)

Pierluigi Nicolin, Lotus International quarterly Architectural Review (Via Santa Marta 19/a 20123 Milano)