January 31, 2015

## ‘FracShell’: A Grid Shell Design based on Fractal Geometry

‘FracShell’ is an output of a computational design workshop held in Politecnico di Torino, Turin in Italy in 2014. This workshop was based on a theme which was to find a novel form for designing a shell structure using unique geometric shape, and ‘fractal geometry’ was the best choice we had for representing radically a new type of form system and exhibiting its potency to produce an innovative structural system.

Archimedes’ famous ‘midpoint displacement method’ was used in our project to automatically design the basic form of the shell structure. Usually, Archimedes’ method is one of the classic mathematical tools to produce a parabola in the two dimensional space and a paraboloid in the three dimensional space. In Archimedes’s method, the value of height factor ‘w’ (midpoint displacement value) is 1/4 which is the key responsible for producing a parabola and a paraboloid from a straight line and a flat polygon respectively.

Later, in 1910, Teiji Takagi modified Archimedes’ method and replaced the ‘w’ value by 1/2, and found a remarkable change. The smooth curve of the parabola or the smooth surface of the paraboloid turns into unsmooth and rough with statistical self-similarity, which was later defined by Mandelbrot (1983) as a ‘fractal’. This new fractal curve is known as ‘Takagi curve’ and its three dimensional counterpart is known as ‘Takagi mountain’. (Mandelbrot 1987) Much later, George Landsberg further modified the Archimedes’ and Takagi’s method, and generalized the ‘w’ value to a changeable floating number that ranges from 0.25 to 1.0. Parametrically, the changing of ‘w’ value changes the texture of the surface of a paraboloid. In grasshopper (parametric) software embedded in Rhinoceros3D (CAD software), it can be shown animated way that ‘w’ value exploits the dimension of the paraboloid’s smooth surface, thus changes the fractal dimension ( which is fractional and non-integer) of the surface from 2 to 3, especially when ‘w’ ranges from 0.5 to 1.0.

Based on this geometric venture of exploiting a regular geometric shape into a fractal shape, we intended to apply this shape morphology for structural form design as a gridshell structure, and see how it mechanically reacts and changes structural behavior with the changing of ‘w’ value.

However, after finite element analyses process, as a structural feasibility check, we had decided to make its real scale physical prototype by taking ‘w’ value as 0.5. ‘FracShell’ was a design team which devoted their times to fabricate the structure by facing some critical challenges. However, in the end, we were able to complete the job, and see the physical realization of a mathematical entity as an innovative structural form in architecture.

Ref:

1. Mandelbrot, Benoit B. The fractal geometry of nature. Macmillan, 1983.
2. Mandelbrot, Benoit B. “Fractal landscapes without creases and with rivers.”The science of fractal images. Springer-Verlag New York, Inc., 1988.

Note:

Special thanks to , Bruno Iorio, Elisa Pitassi, Gabriele Bonnet, Gabriele Fusaro and Samuele Marino

August 16, 2014

## FracTruss: An Application of Fractal Geometry in Truss Design

The lattice configurations of conventional trusses follow the Euclidean geometric system. However, in the nature and in mathematics there exists another new and an interesting geometric system, known as ‘fractal geometry’. Here, the fractal geometric system has been applied to design a structural truss, named as ‘FracTruss’.
The geometric model of the ‘FracTruss’ is a transformation of a simple mathematical function that is based on the notion of fractal geometry lied in the Hausdorff metric space. Iterated Function System (IFS) has been used as a device for this transformation.

The model has been made parametric such a way that (a) the overall geometry can be deformed by the changing of its base angles, (b) the height can be adjusted and (c) the lattice can be made denser or lighter using the parameter of iteration. With the changing of the above parameters, the Hausdorff Fractal Dimension is also changed. Play the following video to see how the parameters changes the model and its fractal dimension.

The geometric model has been transformed into a Finite Element Model for the structural analysis. The Karamba has been used for the finite element analysis. Different parameters result different stiffness of the truss. Galapagos, a computational search algorithm component, has been used to get the best configuration in terms of high stiffness of the structure.

August 20, 2013

## Design by Coding: Parametric Wave Pavilion using Python Script

‘Water Wave Pavilion’ was an entry for an international design competition ‘City by Dreams’ hosted by New York City Councils. The concept was to create a pavilion literally under the water by using garden hose pipes. The intention was to make a waveform, and to give it a shape a flexible and transparent water pipe was coiled by wrapping up three main structural wooden curves.

To do it, there was a need of parametric approach so that by changing the parameters of structural curve geometry and their thicknesses, and by altering the number, radius and curved depth (due the weight of water inside) of hose pipe, I could get a desirable design output based on the needs and suitability.

It was also the occasion for me for attempting to apply scripting knowledge first time to create some algorithmic architectural design.  After some efforts, I was able to write a simple scripting code, i.e., an algorithm in Python language which could be visualized in Rhinoceros. Python component ‘GhPython’ in Grasshopper was an advantage for me which helped me to offer different design outputs by changing the parameters very easily.

scripts_scho

May 25, 2013

## Fractal Tower: A Mixed-Use ‘Bridge-Tower’ on Hooghly River (an Utopia?)

This design proposal was my entry for the ‘CTBUH International Tall Building Design Competition‘ in 2012. It was not selected as a winning entry 😦 .But it was a very good experience for me for daring the first time to externalize an Utopian design idea for the urban regeneration, considering a case of Kolkata, India. I know, there are so many things to be criticized; but as I have said, it is after all an ‘UTOPIAN’ IDEA 😉

The aim of designing the ‘Bridge-Tower’ is to REGENERATE the city Kolkata (in India) by improving the riverfront of holy Ganges, to bring back the glory of the ‘CITY OF JOY’, and to connect the people from city to river in contemporary way but by keeping its tradition and culture alive. Kolkata, once known by ‘Calcutta’, is one of the largest metropolitan cities in India, HAS BEEN SUFFERING from a number of urban related serious issues since mid-twentieth century. Once, it was one of the major commercial and intellectual (art and literature) hubs in 19th and early 20th centuries.  Hooghly River (a branch of Ganges river) crossing the city centrally was the main artery of Kolkata in every sense. During high time, the energy of Hooghly RIVER WAS PULSATING THE URBAN life of the city; it was the major mode of commercial transportation and shipping; jute mills along the river was the busy commercial place; TRADITIONAL GHATS (open platform with series of steps leading down to holy river) with Hindu temples along the river were the main community places (religious and cultural). There was an innate RELATIONSHIP BETWEEN URBAN PEOPLE of Kolkata and HOOGHLY RIVER. But, the city started decaying from the second half of the 20th century because of several unfortunate reasons (geographical, political, economical, etc.). Jute mills, which were one of the main sources to fuel the city economically, were shut down. Gradually, urban people had started isolating from the river and concentrated towards urban centres away from river, and the ghats and river banks had become dirty places. River water has become severely polluted by city’s sewage waters, garbage and drains. It was the river which pulsated the city, and it is the only RIVER WHICH CAN BRING BACK THE BEAUTY AND GLORY OF THE CITY again. I have selected this location of ‘Sobhabazar-Banda’ Ghat area across the river, because this LOCATION has the POTENTIALITY TO ATTRACT URBAN PEOPLE, connect again with the river and revive the city once more.

Only landscape improvement of the riverfront is not sufficient to truly connect the people with the river, because people use to come and relax in the parks and at open spaces only at free times and in weekends. But if we put some DAILY AND REGULAR URBAN ACTIVITIES on the river then we CAN PHYSICALLY CONNECT the people with the river. That is the reason a MIXED-USE HABITABLE BUILDING TOWER has been proposed for the various immediately needed programs like residence, office, shopping, hotel, restaurant, cinema, cultural events, health, etc. In addition, VERTICAL COURTYARDS FOR COMMUNITY ACTIVITIES has also been proposed. At present, these immediate needs of infrastructures cannot be accommodated inside the city because of overcrowd and lack of available urban spaces inside the city. For building a river bridge only for SINGLE USE – TRANSPORTATION, It costs huge amount of money as well as structural materials and labors. So, why don’t we think about utilizing this massive structure for MIXED-USE purpose too?

‘NAMASTE’, the unique hand gesture of GREETING, is the inspiration of the building’s overall shape which represents the Indian long-history of tradition, and welcoming of visitors as well as the holy Ganges. The FRACTAL-LIKE BRANCHING PATTERN at the lower part of the bridge-tower is symbolizing the river branches of Bengal’s delta, and inspired by the the tree’s root-branches which supports the weight of the whole tree. The branches of the DIVERSITIES of different cultures, languages, foods and lifestyles of India MERGE INTO AN UNITY and make an unique harmony. This design is the MANIFESTATION OF THIS UNITY IN DIVERSITIES of India.

India’s traditional wrestling ‘KUSTI’ is the STRUCTURAL CONCEPT of the main form of this structure. Besides, inspired by the structural phenomena of root-branches of a tree, INVERTED BRANCHING STRUCTURE at the lower level of the building has been designed which will AVOID THE STRESS CONCENTRATION of the main tower load on the arch-road and DISTRIBUTE THE FORCE FLOWS UNIFORMLY. Because of the HIGH HUMIDITY of the local climate in Kolkata the tower has been designed with PERFORATIONS at lower part where main public activities are allotted. These perforations are like VERTICAL COURTYARDS which will allow to pass RIVER BREEZES. These vertical courtyards are the places for traditional (folk arts, baul, etc.), cultural (drama, dance and music) as well as contemporary COMMUNITY ACTIVITIES. The lost treasure of the traditions of art and literature of Bengal and Kolkata can be revived through these HANGING COMMUNITY-COURTYARDS above the holy Ganges ..  !!

March 17, 2013

## Fractal Forest (‘Monalisa’) Pavilion: MadeExpo2012, Milan

Last year in 2012, it was an amazing experience on working in a workshop conducted by ‘Wood Lab’ of ‘Politecnico di Torino’.  A poplar plywood company financed WoodLab for designing a pavilion to exhibit and promote their architectural, sculptural and furniture products. It was a heavy creative exercise to come up with a unique but sensible design idea. Finally, an interesting design concept came to mind and took shape as ‘why don’t an architectural piece can be a manifestation of a story of poplar tree itself’? Poplar grows from its seed, and then gradually it becomes young plant and finally turns into a perennial woody tree.  Poplar trees altogether live in a family making a forest, and finally they are used for making plywood needed for building construction, furniture, and so on. This whole story had to be turned into a shape, a design, a pavilion.

The challenge of composing this story was done by using two different mathematical design vocabularies –  ‘Algorithms’ and ‘Fractals’. Algorithms helped us to represent the ‘growth’, whereas  fractals helped us to represent the ‘nature’, the forest by tree branches.

The representation of seed was a small ply panel. Then the panel started growing by increasing its size, then it was budding with two new small branches, it kept on growing and then started taking shape of a tree with the increasing of more branches. Functionally, the seeds were designed for sitting benches, whereas the networks of branches provide the feeling of shelter under the shadow in a poplar forest. Poplar fruited us with furniture and it was represented by benches.  Curvilinear shapes on plan represented the sense of nature.

To realize the concept into design form, we used computational and parametric techniques in computer by using ‘rhinoceros’ with the help of ‘grasshopper’ and ‘python’. In ‘python’ we scripted the shape of trees with growing branches. Then ‘grasshopper’ was used for making algorithmic benches and arrangement of trees.

However, after the exhausting creative and designing phases, the next main challenge was how to construct the pavilion by showing the versatility of poplar ply. For this, a structural analysis program ANSYS was used for assessing its stress and bending strength behaviors.

We started making small scale model to show the freeform character of poplar ply. We made the model in FabLab by using CNC cutter. After that, we made a real scale prototype of one tree module to test its bending strength including other structural potency.

In October 2012, finally the pavilion was installed in prestigious international ‘MadeExpo’ Exhibition in Milan, Italy. An awesome outcome was ready, and our hardworking were ended by attracting and fascinating a large crowd of visitors !!

March 15, 2013

## Portable Chamber: Womb to Habitable Container

Portable womb to parambulator

The significance of portability relies on the sustainability, customary and the identity of the thing which is transported – be the medium is place or time or both together. For this the chamber which contains the portable, living or non-living objects, has the major key role to sustain the objects. The chamber may be a process or system of the porting agent. But during transporting, while on the one hand, the identity of the stuff should be unchanged, on the other hand it should be acclimatized with the new place or new time.

Lukin Fisheries Pty Ltd in Australia, a renowned tuna fishing company, developed a very unique method of transporting tuna fishes from Australia to Japan and to other nations. Dinko Lukin was the first who came up with the idea to transport this fast swimming ocean fish not by conventional method of catching the fishes from ocean and putting them in containers, but by farming them in the ocean where he can be able to transport the fishes flown fresh, straight from the sea to the market.  The company has four giant tuna vessels in their Australian fleet. First, they fish for tuna in the Great Australian Bight. Then capture the fish with large nets. Finally by using “tow cages” slowly bring them to Port Lincoln. Before, tunas were ported not fresh but dead and frozen and it took several days to months for shipping to different countries. But after Dinko’s new idea, this time tunas are transported alive and fresh. When fishers first catch the tunas, the tunas are small in size in group. After arresting them in large tuna vessels, they are transported underwater. While transporting, after a long days tunas become grown enough and fisher earned double profit as the size and weight of tunas become almost double. The key of this huge success is laid behind the unique transporting system where the environment for living and growing of tunas were not disturbed rather than just redirect this fast ocean fishes to the destination. Here the portability coherently consists the shifting of place and time, and sustains the function of tunas’ accustomed living environment and identity.

Left – Aerial view of tuna vessels of Lukin Fisheries Pty Ltd, Australia. Right – Inner view of the tuna vessel

Likewise, human beings are in the ocean of mother’s womb before our eyes open to the earth. As a single embryo we start to develop and grow inside the dark womb. For the embryo the womb itself is his first world where it gets everything whatever it needs and grows. But when the pregnant woman moves from one place to another the embryo is also transported with her body but the process/function of its growing continues without any disturbance. Furthermore, as modern technology has dramatically progressed, human wombs will never be the same. What about artificial womb that act to mimic the functioning of the womb? As a matter of fact, scientists have experimented to create artificial wombs with animals where human embryos implanted in an artificial womb to yield a pregnancy. For this, the artificial womb had been treated with the provision of chemical and organic fluids in a chamber that needs for the fertilization of fetus as in the actual womb.  However, as far as moving is concerned, may it be possible to bring the artificial womb anywhere as required in a house where along with the sustaining of fetus inside the other aspects of portability for the porting is considerable.

The notion is also applied to a shorter timeframe chamber. In long-distance train some cabins are designed for the families. One cabin is like a small house unit having all the basic necessary facilities like toilet, kitchen, sleeping and so on so that a single family, while traveling, can easily spent their time for one or two days as they spend in their own house. In this case the family is the object which is transporting to the other place by train. But if we keenly observe the fact then we can find the main purpose of cabin in which it is designed as the prototype of time that they spend in house rather than the house as an object or chamber of family. While in cabin, if the time and life style for one or two days are not affected and changed then the transportation of the family could be seemed success, but if fails for another family, then for that family the transportation of the family is not success, for which the family paid before traveling as train ticket. But if there is such a flexibility of the settings (furniture, beds, etc.) then any family according to their lifestyle they can arrange the cabin according to their own lifestyle for one day, no matter how short the day is, rather than compromising the money of ticketing and forcefully spending the times while transporting. Accordingly, the cabin should be made such a way that it would act as the womb of the whole train where the family value and their identity of lifestyles are the embryos of the cabin.

The fittingness of chamber with the changing behavior of porting agent (as time goes) reflect the ability of the chamber for fostering both the physical entity of the agent and the time associated with it. When we consider the portability with the portable thing and the time together, then the relativity must come automatically. For example, sailors move from one region to another where they spend several months to years on a ship. The ship as a chamber of the sailors associated with other objects shifts them not only by means of place but also, throughout the journey, it carries them through the different time seasons – summer to winter to summer. If the ship continues to move within a same climatic zone for years, then time shifting is more significant for making the chamber time-responsive to customize the sailors and objects according to the time season. On the other hand, if the ship moves from one climatic zone to different climatic zone but within a short period, then the place shifting becomes more important for making the chamber place responsive. Thus, the porting chamber is time and place responsive as the time and place are closely related to each other.

Accordingly, the action of flexibility, expandability and customary of the chambers for porting agent to allow continuing the agent’s usual or spontaneous process can be easily traced. The customary of tuna vessels, the fostering of mother’s womb, and the flexibility of family cabin in train are the physical key factors for the meaningful portability.

A number of strong creative efforts of implementing the meaning of portability by considering both the place and time together including aptness, compliance and identity of the porting people have been applied to the architectural project. Among a range of attempts, portable house1 is particularly noteworthy in which a group of families can compose a portable community as new nomadic.

By using the flexible and expandable architectural elements, in terms of tectonic and spatial, the houses can be made smart by means of place and time responsive which customize the life of new nomadic people with the change of place and time as well. An another appealing effort is “Aeromads”2, in which the inflatable, malleable and movable house is designed that is as much moldable and flexible for customizing the dwellers need as a large airy blanket which is used by any means for comforting or fitting the human body.

Accordingly, when the chamber contains some portable objects then its responsibility is not finished just after keeping the objects properly, but its awareness to take care the objects becomes also the principal conscientiousness.

Mother’s womb not only contains the baby inside, but also it becomes fully responsible for all the aspects of growing, protecting and fostering the baby. When a baby is born, s/he is started to get provision of the environment as in the womb in terms of caring, protecting, and so on. As the baby grows up, the place womb is taken by baby carrier and then perambulator as pseudo womb for moving the baby from one place to another.

Accountability of the chamber is not only inside the parameter of containing and shifting the porting agent, but also in the territory of the mood and reverse response of the porting agent. Sometimes may the chamber provide all the requirements to the porting agent by considering the place, time and its identity, and to provide these all to the agent the chamber does transform itself at the moment of agent’s needs. But this transformation of the chamber should be such that it will not affect the agent’s convenience rather than fit with the agent’s comfortableness. While on one side, during transporting the responsiveness of the chamber to the porting agent is very much significant, on the other side, the process of taking the portable agent from one place and then placing to the new place and again repacking for the new destination, plus the caring and sustaining of the agent throughout the whole process is also very much imperative.

Note

(1) A good example of the portable house has been well manifested in Jennifer Siegal’s project. The project has central kitchen and bathroom core, sleeping area in one side and on the other side an expandable and contractible living space. According to climatic condition the house can be reoriented to take advantage of sun and natural ventilation. See Jennifer Siegal, Mobile: The Art of Portable Architecture, Princeton Architectural Press; 1 edition (June 1, 2002)

(2) The “Aeromads” was designed by architect Alexis Rochas and his students at the SCI-ARC in Los Angeles. It is an idea of a home that is a malleable and movable environment that can be deflated and fit into a suitcase, then travel to a new location with its owner. It uses air pressure as a building material to exist as its own and independent structure.

March 15, 2013

## Fractal Architecture and Nature’s Geometry

Fractal geometry, a branch of mathematics developed in 1970s [Mandelbrot 1975, 1984, Edgar 1993] studies abstract configurations characterised by self-similarity patterns and recursive growth [Mandelbrot 1984]. Fractal objects show the properties of being exactly or nearly the same at every progressive scale. From the mathematical point of view, fractal objects are sets that have fractional dimension, so that they are intermediate objects between one and two dimensional shapes (as lines and surfaces) or two and three dimensional forms (as surfaces and solids) [Batty 1985, Falconer 2003]. Recently, thanks to the development of advanced computers, the domain of fractal geometry applications has covered a wide set of scientific discipline, ranging from mathematics [Berkowitz 1998], natural sciences [Vicsek et.al. 1994, Sornette 2004], pure and applied sciences [Peitgen 2004], biology and medicine [Losa & Nonnenmacher 2005], to engineering [Dekking, et. al. 1999, Leung 2004, 2011] and architecture[Bovill 1996, Ostwald 2001]. Fractal geometry is specifically used as theoretical as well as technical tools for the analysis, interpretation and description of complex, natural and human phenomena, where continuous or Euclidean geometry are failed to describe.

Architecture is closely associated with geometry, and that is the reason this new concept of fractal geometry can be used for the advancement of architectural and urban designs. In a very wide range of phenomena, the geometry of nature displays fractal-like properties [Mandelbrot 1975, 1984]. Any form, shape and pattern of a natural object are its phenomenological outcomes [Bertol 2011] and therefore, it is believed that there is a strong correlation between biological forms and mechanical phenomena [Thompson 1917, Turing 1954, Durgun 2007]. Accordingly, fractal geometry of nature possibly has a connection with nature’s structural and mechanical behavior. But, there is a recent debate about the fractal geometry and its definition to explain the form and pattern of nature. Adrian Bejan critically argues in his much acclaimed ‘constructal law’ that it is the ‘laws of thermodynamics’ which decides the geometry and form of the natural objects [Bejan 1994], and there is no logical connection between nature’s forms and fractal geometry [Bejan 2000].

For many centuries, a variety of nature’s forms, which in many cases present fractal geometry in their structural appearance, such as trees, cells, crystals etc., have been creatively used by architects and engineers in projects like shells, light-weight structures, arcs, tents and bridges (e.g. Stuttgart Airport, Stuttgart; Galleria & Heritage Square, Toronto; Heart Tent, Riyadh) [Blanco 2001, Otto 1995, Portoghesi 2000]. In the past, several technical ways were exercised to connect fractal concepts with architecture by the method based on physical modelling process. But, nowadays, a procedural generative approach based on a composition of mathematical functions can be practiced by using the advantages of contemporary computer technology for connecting the fractal concept with architecture (e.g., Federation Square, Storey Hall in Melbourne; etc.) [Huylebrouck & Hammer 2006].

The main intention of my research is to increase the knowledge and understanding of nature’s fractal phenomena and forms, and try to apply the results for a better comprehension of human and social behaviour and to the architectural design. Biomimetics is the study of the structure and function of biological systems as models for the design and engineering of materials and machines. Therefore, the area of my research is oriented towards ‘Biomimetic Architecture’ but by means of computational and algorithmic techniques, used as advanced tools for the study, analysis and forms generation.

References:

ENGINEERING:

BARNSLEY M. F. (1988), “Fractals Everywhere”, Academic Press, Inc.

BATTY M. (1985). “Fractals – Geometry between Dimensions”. New Scientist (Holborn Publishing Group) 105 (1450): 31.

BEJAN, A. (1997) “Advanced Engineering Thermodynamics”. (2nd ed.). New York: Wiley.

BEJAN, A. (2000), “Shape and Structure from Engineering to Nature”, Cambridge University Press.

BERKOWITZ J. (1998), “Fractal Cosmos: The Art of Mathematical Design”, Amber Lotus.

DEKKING M., et. al. eds. (1999), “Fractals: Theory and Applications in Engineering”, Springer.

DURGUN M.E. (2007), “Geometric Generalization of the Structure of Nature: A theory of everything and a mathematical formulation of a philosophy”,http://www.unitytheory.info/ggsn.pdf.

EDGAR G. A., ed. (1993), “Classics on Fractals”, Addison-Wesley.

FALCONER K. (2003), “Fractal Geometry, Mathematical Foundations and Applications”, 2nd ed. Wiley, London.

LEUNG A Y T, , WUB G R, ZHONG W F. (2004), “Exterior Problems of Acoustics by Fractal Finite Element Mesh”, Journal of Sound and Vibration, Volume 272, p 125–135.

LEUNG A Y T. (2011), “Fractal Finite Element Method for Thermal Stress Intensity Factor Calculation”, ICF11 Proceeding.

LOSA, G. A.; NONNENMACHER, T. F., eds. (2005). “Fractals in Biology and Medicine”. Springer.

MANDELBROT B. (1982), “The Fractal Geometry of Nature”, San Francisco: W.H., Freeman.

PEITGEN H, JÜRGENS H, SAUPE D. (2004), “Chaos and Fractals: New Frontiers of Science “ 2nd ed. Springer.

SORNETTE (2004). “Critical Phenomena in Natural Sciences: Chaos, Fractals, Self-Organization, and Disorder: Concepts and Tools”. Springer. pp. 128–140.

THOMPSON, D W. (1992), “On Growth and Form”, Dover reprint of 1942 2nd ed. (1st ed., 1917)

TURING, A. M. (1954). “The Chemical Basis of Morphogenesis”. Philosophical Transactions of the Royal Society of London 237 (641): 37–72

VICSEK T, SHLESINGER M, MATSUSHITA M. (1994), “Fractals in Natural Sciences”, World Scientific Publishing Co Pte Ltd

VON B. P. (2009), “A Geometric Comparison of Branching Structures in Tension and Compression versus Minimal Paths”, University of Michigan, Ann Arbor, MI, USA.

VYZANTIADOUA M.A., AVDELASA A.V., ZAFIROPOULOSB S. (2007), “The Application of Fractal Geometry to the Design of Grid or Reticulated Shell Structures’, Computer-Aided Design, Volume 39, p 51–59.

ARCHITECTURE:

BEN-HAMOUCHE M. (2011), “Fractal Geometry in Muslim Cities, Succession Law Shaped Morphology”, Nexus Network Journal, Volume 13, No 1.

BERTOL D. (2011), “Form, Geometry, Structure – From Nature to Design”, Bentley Institute Press.

BOVILL C. (2000), “Fractal Geometry as Design Aid”, Journal for Geometry and Graphics, Volume 4 No. 1, p 71-78.

BOVILL C. (1996), “Fractal Geometry in Architecture & Design”, Birkhäuser, Boston.

HUYLEBROUCK D, HAMMER J. (2006), “From Fractal Geometry to Fractured Architecture: The Federation Square of Melbourne”, The Mathematical Intelligencer – Volume 28, No. 4, p 44-48.

JOYE Y. (2011), “A Review of the Presence and Use of Fractal Geometry in Architectural Design”, Planning and Design, Volume 38, No. 5, 2011, p 814-828.

JI Z., ZIYU L., XIAOZHOU L. (2011), “Remote-Sensing Expert Classification of Land Use/Land Cover Types Using Fractal Dimensions Over A Subtropical Hilly Region in China”, Fractals, Volume 19, No. 4 p 407–421.

LORDICK D. (2009), “Architectural Fractals”, Rutgers University, New Jersey.

LORENZ W E. (2004), “Fractal Geometry As An Approach To Quality In Architecture”, 1st International Conference on Fractal Founder 21st Century Architecture and Environmental Designations f.

OSTWALD M. J. (2001), “Fractal Architecture”: Late Twentieth Century Connections Between Architecture and Fractal Geometry”, Nexus Network Journal – Volume 3, No. 1. p 73.

PADRÓN V., SALINGAROS N. A. (2000), “Ecology and the Fractal Mind in the New Architecture: a Conversation”, RUDI — Resource for Urban Design Information.

PORTOGHESI P. (2000), “Nature and Architecture”. London: Thames & Hudson.

SALINGAROS N. A. (1999), “Architecture, Pattern and Mathematics”, Nexus Network Journal, Vol. 1, p 75-85.

SALINGAROS N. A. (1998), “A Scientific Basis for Creating Architectural Forms”, Journal of Architectural and Planning Research, Volume15, p 283-293.

VAUGHAN J., OSTWALD M. J. (2012), “Using Fractal Analysis to Compare the Characteristic Complexity of Nature and Architecture: Re-Examining the Evidence”, Architectural Science Review, Volume 53, No 3.