Fractal
W.I.T M. Arch
Ciro Podany
Space
Abstract
Gridded cities based on Euclidean geometry impose simplified order to mask the inherent complexity of nature, resulting in human’s disconnection from the excitement of variability and diversity. Human’s love of nature has been overruled by the desire to control nature. Technology and swift urban expansion, initiate the construction of artificial environments deprived of the spatial complexity necessary to inspire proper engagement of the human body and mind. With the emergence of fractal geometry- the first complete break from Euclidean geometry- spatial possibilities of architecture anticipate a deeper human connection to their constructed environments. Fractal patterns found in nature such as plants, clouds, crystals, and rivers, illustrate a taxonomy of conditions with the power to transform architectural space. In order to capture the beneficial qualities of fractal spaces, experiential perspectives and physical models can begin to legibly represent a recursive process. The application of fractal space can potentially challenge the simplified geometry of the city grid and harmoniously reinvigorate human’s interconnection to the complex geometry of nature.
Contents
Research
1-21
Question Research Essay Hypothesis
22-6
Visuals Frames Design Probe Site
9
Methodology
3
Conclusions
62-7
9
74-7
Appendix
Process Evaluation Research Discoveries Project Conclusions Intentions Design Speculation Future Criteria Objectives Timeline
Appendix Annotated Bibliography Endnotes Images
Fractals... “A family of shapes�, describe the geometric forms of clouds, mountains, coastlines, plants, seashells...and much longer list of naturally occurring forms, a continuous array of scale, growth, movement, density and space, fractals
Three-dimensional, or (Euclidean space) describes the boundary box around the shell, not the shell itself, rendering natures design without form. Formless objects in ambiguous cubes of space.
What happens when space no longer relies on the boundaries of simple planes... and has freedom to inhabit the breath of a fractal?
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Spatial Keywords Euclidean Geometry: Based on the axioms of Euclid the Greek Mathematician who is credited with the establishment of three-dimensional geometry and the box that withholds possibilities of fractal nature. Synonym: The GridÂŹ Mastery of Nature: The strong assumption that humankind can, and should, control natures systems, arguably leading to the destruction of both meaningful city form and human-nature interconnectivity. Synonym: Coherence: All spatial components or fragments, clearly manifest a framework of order and organization to ensure intrinsic relations with other organisms. Synonym: EnvironmentÂŹ Legibility: Way-finding of complex space and the ability to physically orient and navigate. Synonym: Perception Harmony: A combination of spatial conditions that compliment each other, resounding a seamless chord of invigoration. Synonym: Interdependency
Complexity Captures spatial detail to physically and psychologically excite the user, while invigorating components of exploration within the site environment.
Synonym: Diversity
Recursive An intuitive sequence of spatial growth, results networks of spaces generating other spaces creating a process loop and creating a self-conscience space enabling itself to reflect on itself . Synonym: Reproduction:
Self-Similar: Repetitive self-copying patterns that grow or diminish in scale (inhalation/exhalation) creating solid correlations between major structural elements and tiny details. Synonym: Breath Fractal Space: A composition of complex space, formed through the process of a recursive loop that self-similarly iterates different dimension of detail. The design language of, trees, clouds, crystals, and mountains is borrowed and translated to express coherence, legibility and harmony with all possible entities. Synonym: The Organism
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Research Paper Nature’s Mathematical Environment The application of mathematics to nature explains the humans desire to “Understand his environment, to give play to his artistic instincts, to engage in absorbing intellectual activity,”1 in addition to describes the relationship between humans and nature that reflect recent research of fractal geometry and biophilia. Mathematics, beginning “In pre-historic times, struggled to exist for thousands of years,”2 due to human fear of the natural world forces. It was the Classical Greek (500-336 BCE) atmosphere of reason that truly recognized the potential power of mathematics to conceptually grasp the workings of the universe. Greeks applied mathematics to nature in order to comprehend their environment “Because they enjoyed reasoning and because nature presented the most imposing challenge to their minds.”3 Abstract theories of nature allowed the Greeks to realize that all things encountered by the human senses worked by patterns discernible through numbers. Pythagoras and his followers asserted that, “nature is built in accordance with mathematical principles, and that number relationships reveal the order in nature.”4 The theories and axioms developed by the Greeks culminate humans curiosity, respect, and awe of nature and it is at this point of departure where most mathematics grows from. After the Greeks, the development of new societies in Europe transforms mathematics role to assist in the “mastery of nature.”5 The important English thinker Francis Bacon (15621662) makes this perspective clear by criticizing the Greeks, claiming, “Interrogation of nature 1 Kline, Morris., Mathematics for the Nonmathematician. (New York: Dover, 1985,1967.) Page 27. 2 Ibid. Page 14. 3 Ibid. Page 16. 4 Ibid. Page 189. 5 Ibid, Page 23.
should be pursued not to delight scholars, but to relieve suffering, to better the mode of life, and to increase happiness.”6 Rene Descartes (1596-1650), the French philosopher, adds that “Knowing the force and action of all bodies that environ us, we can thus render ourselves the masters and possessors of nature.”7 Differing completely from the Greeks in motive, these ideas still recognized the natural environment as the powerful life force that must be understood to better life and sought the application of trades and arts to do so. It has taken humankind until recently to look at its relation with nature scientifically and critically. The Biophilia Hypothesis (1993) embodies the intersection between the Greeks love of nature and subsequent civilizations love of human life with the aim to “probe what an affinity to life can mean for all of us.” 8
Fractal Space
Mandelbrot establishes fractal geometry and publishes Fractal Geometry of Nature (1982) to initiate discussion around emerging definitions and concepts. The content of Morris Kline’s Mathematic for the Nonmathematician (1985) provides a brief but necessary historical backdrop to Mandelbrot’s innovation, and crystallizes the connection between man, nature and mathematics. Rhonda Roland Shearer, in her article “Chaos Theory and Fractal Geometry: Their Potential Impact on the Future of Art” (1992) investigates the role of human perception to distinguish the conceivable benefits of fractal space. Holmes Rolston III breaks the concept of sexual reproduction down into pieces that can instill the continuous diversity of an artificial human 6 Ibid. Page 280. 7 Ibid. Page 280. 8 Kellert, Stephen R., and Edward O. Wilson. The Biophilia hypothesis. Washington, D.C.: Island Press, 1993. Page 17.
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environment section titled “Biophilia, Selfish Genes, Shared Values,” (1993) and depict plausible reasoning for humans innate comprehension of natural space. Michael Batty and Paul Longley’s Fractal Cities (1994) addresses modern urban design criticism by proposing fractal space as a meaningful closeness of parts that are perceptually pleasing and architecturally legible. Ron Eglash’s book, African Fractals: Modern Computing and Indigenous design (1999) analyzes vast cultural applications of fractal geometry drawing intriguing results from keywords that capture the true to life behavior of fractals and the complex spaces they can generate. Stephen Kellert’s, Building for Life: Designing and Understanding the Human-Nature Connection (2010) revisits spatial design theories, and attempts to further develop and define these components into contemporary design methodologies. A chapter titled “super organisms” within Tim Flannery’s book Here on Earth, (2010) compiles research on the incredible effectiveness of super organisms through their interconnected existence. Claim: Fractal space embodies elements of life with enough conviction to anticipate the psychological and physical progression of human consciousness.
Complexity: The birth of fractal theory The discovery of fractal geometry by the mathematician Benoit Mandelbrot describes the systematic design of both humans and nature; therefore describing the mathematical components of human’s love of life. Abstract qualities found in fractals are best explained through natural elements. This is mainly because once a fractal seed shape is planted ,it functions and grows as a living entity. Mandelbrot uses this truth to reinforce the fractals ability to describe what Euclidean geometry cannot, including, clouds, mountains, coastlines, plants,
and snowflakes. Mandelbrot tests fractals ability to define natural elements by computer generating fictional landscapes 3
from the number sets, (fig. 1). A Peano curve, (fig.2) also exemplifies certain geometric conditions of nature, such as “rivers, watersheds, botanical trees, and human vascular systems,”9 The Koch curve, (fig. 3) which begins as a triangle connected to a line must be thought of as a living organism
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according as it iterates itself continuously, “It would not be possible to do away with it without destroying it altogether, for it would rise again and again from the depths of its triangles, as life does in the universe.”10 Recursion:
Ron Eglash’s discovery of African villages designed with fractal methods
builds upon Mandelbrot’s theories. Eglash studies the psychological, social, and physical interconnection these African’s have with their natural environment. Two examples of fractal shapes established by Mandelbrot contain the overarching
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themes of: scaling, and self-similarity employed by Africans in their use of fractals. Similar to the Peano curve, it has been found that the paths through certain African settlements are “like the bronchial passages that oxygenate the round alveoli of the lungs, the routes that nourish circular settlements often take branching form.”11 It is not only the African settlements 9 Mandelbrot, Benoit B. The Fractal Geometry of Nature. San Francisco: W.H. Freeman, 1982. 10 Ibid. Page 43. 11 Eglash, Ron. African fractals: modern computing and indigenous design. New Brunswick, N.J.: Rutgers University Press, 1999. Page 34.
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that take on fractal concepts and forms, Eglash points to “traditional hairstyling, textiles, (fig. 4), and sculptures, in painting carvings, metalwork, in religion, games, (fig. 5), and practical craft, in quantitative techniques, and symbolic systems, Africans have used the patterns and abstract concepts of Fractal geometry. “12 A distinction noted by Eglash between African Fractals and The Fractal Geometry of Nature is the concept of “recursion”13 which is not recognized
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once in Mandelbrot’s text. This is the understanding that fractals are created “by a circular process, a loop in which the output at one stage becomes the input for the next,”14 just as biological and geological change occurs. This concept also captures how a physical object can be fractal only a limited range of scales. The recursive properties of a particular village allowed for women to express “social control” by implementing the next iteration, of space by building an addition to a current living space. Social power can potential empower women through difficulties like divorce or a changing family structure.15 Dr. David Hughes, an architect and professor, notes the African settlements to be environmentally harmonious as well as the product of intentional design and semiotics, in contrast to a vernacular approach
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“unconsciously”16 adapting to its environment. Another example of fractal scaling occurs along a royal passage where as the spaces scale smaller you go up in social ranking and must act more 12 13 14 15 16
Ibid. Page 7. Ibid. Page 109. Ibid. Page 17. Ibid. Page 195. Ibid. Page 218.
formal, until you reach the room of leader and must take your shoes off and speak with complete formalities, (fig 6.) Eglash attributes the recursive, selfsimilar, scaling properties of indigenous African design to an open perception of sexual reproduction, a close relationship to agricultural cycles, and the belief in sand divination techniques, maintaining theory that recursion holds “some kind of universal ethical or social value.”17 Similar patterns to fractals can be found in the “grand masters paintings and beaux arts architecture,” because
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of the implementation of many scales of length and self-similarity, (fig 7). Algorithmic games, traditions, and artwork leads to the conception of spaces using similar recursive methods, interconnecting notions of culture with architectural space.
Reproduction 7 Research shows evidence that humans are stimulated by nature because of an innate biological dependence on it. The artist Rhonda Roland Shearer explains that the, “Visual experience of plants exists somewhere between direct perception (subjective) and memory (universal).”18 Experiencing plants is distinct from unnatural pattern perception because it connects humans to the larger context of life systems on Earth. The understanding of collective origin and time “universal” is juxtaposed with singular emotions, and reactions to the “universal”. Shearer references experts like neurologist Antonio Damasio who supports “that the brain perceives human-made objects and natural objects using distinctly different processing.”19 Perceptual distinction touches on the trans-formative spatial implications plant geometry, or 17 Ibid. Page 192. 18 Rhonda Roland, Shearer,. “Chaos Theory and Fractal Geometry: Their Potential Impact on the Future of Art.” Leonardo 25 (1992): Page 148. 19 Ibid., 148.
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fractals, hold in the world of art and architecture. Conditions of visual interest occur at every scale due to growth. Human perception of this particular aspect provides assimilation of different stages of life within a larger cycle demonstrating the self-similar “relatedness”20 of the parts to the whole, (fig. 8).
Human’s awareness of sexual reproduction, portrays 8
the vital effects of plant perception.
Holmes Rolston III’s
article Biophilia, Selfish Genes, Shared Values describes the arrival of an organism “In the world as a beneficiary of past variation, and it inhabits a natural system in which it can cope only if it can make variant copies of itself.”21 Humans recognize the existence of fractals in the patterns of variant recursive copies in plants and in this subtle exchange a deeper understanding of place is possible. Potentially, the forms of plants stimulate in order to facilitate the interaction of organisms. The human Importance of interconnectedness is exhibited well by superorganisms Tim Flannery’s Here on Earth identifies how super organisms are “Emblematic of the coevolutionary capacity to create an entity that is more competent and productive than the sum of its parts.”22 The coordination of systematic groups of people, exhibit some of humankinds greatest achievements. Construction of a building requires an extensive coordination and human power possible only through the strength of systematized groups. Human’s love and desire connection to systematized groups of people because chances of survival and evolution increase drastically. This behavioral pattern draws directly from the theory that “every creature, 20 Ibid., 151. 21 Kellert, Stephen R., and Edward O. Wilson. The Biophilia hypothesis. Washington, D.C.: Island Press, 1993. Page 391. 22 Tim Flannery,. Here on earth: a natural history of the planet. (New York: Atlantic Monthly Press :, 2010.)
is in some sense, is connected to and dependent upon the rest. “23 Humans make up a network of genetic copies and compose a groups inherently strengthened by their interconnection. Each reproduction of a fractal pattern can establish individual spatial relationships while simultaneously strengthening the whole system, (fig. 9)
The Grid
Research at the scale of city, has reinforced the value
of fractals inherent form that describes the geometric essence of life and nature. Cities reflect the human desire for a survival 9
through interconnection, and require opportunities for diverse form in order to maintain longevity. The two main approaches to city design extend tensions between nature’s organic growth and modern cities artificial harshness. Michael Batty and Paul Longley’s Fractal Cities considers the mastery of nature as a point of departure, joining an ongoing critique of artificial geometry’s role in urban design theory. Ideas of the journalist, activist, and architectural critic Jane Jacobs, borrowed by Batty and Longley, illustrate the “destruction of visual order and harmony,”24 inflicted by gridded cities .25 One possible attribute to this argument is the generic aspect of scale immediate established by a cities size.
Coherence, Legibility, and Self-Similarity 23 Ibid., 55. 24 Batty, Michael, and Paul Longley. Fractal cities: a geometry of form and function. London: Academic Press, 1994. Page 7. 25 Ibid., 7.
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Older cities, like London, and Paris, (fig. 10) according to Batty and Longley encapsulate a greater level of spatial awareness . Without the goal of controlling the entirety of a cities visual form, each portion of the city can complete another portions imbalance or create contrast in order to instill coherence. Looking at An issue underlying the origins of human’s inclination to master nature is the inability to properly comprehend the complexity of a natural environment at the scale of a planned city condition. The feasibility to construct larger cities spurred humans to introduce simpler geometries than urban models like citadels, fortresses, and castles had previously called for. At a larger scale, an immediate way to understand built form can be efficiently executed by forcing geometric order upon an environment. Fractal’s self-similar scaling can possibly resolve the scalar issues of the grid largeness due to diversity of scales accommodation of growth and decay. Generating spatial richness of scale, connecting to the surrounding environment and its conditions, suggests an architectural methodology beginning to, “identify, understand, and treat cities as problems of organized complexity,”26 and legibility.
Closeness of Parts and Adaptation
When looking at more complexly composed cities, the opportunity to understand a higher dimension of the living components through “spatial hierarchies and successive scaling, where elements of the urban structure are repeated in diverse 10
ways.”27 When the emphasis of space forming challenges the notion of simplification 26 27
Ibid., 7. Ibid., 47.
and focuses on adaptation, site is realized as a geometric model from which fractal growth of space appropriates the built form with true meaning. Historic cities and towns depended on geometries growing from the site for survival and longevity, (fig 11). In the classic style of fortification, repetitive angular displacements of the main outer wall “increase wall space available for their defense and as such represent a kind of space filling phenomenon reminiscent of the regular fractal called the Koch curve.”28 Cities built with this attentiveness to existing natural geometries successfully transfer local interests into clarity of human’s
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dimensional relationships to their landscape. The interconnected development of urban spaces supports self-affinity through closeness of parts and their ability to adapt.
The over-simplification of unnatural geometry represents an overarching trend of disengagement with the natural environment as a whole. Stephen Kellert’s Building for Life considers the psychological, and physical consequences disconnection from nature has on humans. Many of the theories are based on research done by experts like Nobel Prize Biologist Rene Dubos who affirms, “The functioning and maturing of the human body, mind, and spirit depend on the quality of people’s ongoing experience of nature.”29 Among many of the inherent benefits discussed, problem solving, intellectual performance, social ties, and relationships, highlight the difference between fractal space and typical green architecture, that fails to recognizes geometry as a key tool of integration. Green architecture instead focuses 28 Ibid., 22. 29 Dubos, quoted in, Kellert, Building for life designing and understanding the humannature connection. (Washington, DC: Island Press, 2005.) Page 11.
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on applicatory methods of green walls, solar panels, and gray water systems. These elements lower the energy consumption of a building, the form is largely incongruent with fostering a cultural “recognition of symmetry and harmony.”30 Distillation of natures conditions into language applicable to design bridges the gap between both the natural and built environment . According to Rachel and Stephen Kaplan , a more meaningful geometry embodies, coherence, mystery, complexity, and legibility. Coherence presences the human mind. Understanding time and place, coherence “discerns order and organization in nature. “Mystery suspends normal interest. It motivates care, development, growth, and knowledge. Mystery enables “investigation and examination of the complexities in nature.” Complexity describes not only nature’s mathematical form, but where it is oriented in time and place because of the laws of growth and interdependence. For humans it is the “ability to identify and respond to diversity and variability in the natural world.” Legibility is visual clarity and “reflects the human capacity to orient and navigate natural settings.”31 The applicability to design through a language of geometries from the psychological properties of nature can be appropriate because it seeks not to make an exact copy of nature but to extract conditions that convey the essence of nature. In the language of architecture, form and space epitomize the main tools that must relate to the “archetypal image of the natural world.”32 Frank Lloyd Wright began to architecturally employ spatial theories intuitively from an understanding of natural conditions. He designed “visual connections between interior rooms, with outside views and few closed interior spaces as a way to “destroy the box,” of artificial space. Due to the recent understanding of fractal geometry, the design of architectural space has the potential to revolutionize humans understanding of the built environment on the same level that nature can scientifically be understood. 30 Ibid., 15. 31
32
Ibid., 15.
Ibid., 127.
Harmony
Seeking to control nature by imposing simple geometry misunderstands science and a falsely assumes the assurance of progression. The negative impact of built environment, suggests that “if we do not strive to love our planet as much as we love ourselves, then no further human progress is possible here on Earth.”33 The state of the urban gridded environment, doesn’t successfully extend beyond the limits of it sets upon itself. The capability to transcend conceptual and physical gridded boundaries intersects “the heart of mathematical, computational, limits of our subjective experience of consciousness.”34
With a desire to
recursively love and grow instead of control and dictate there are opportunities to strengthen the interconnected environment of humans. Discovery of fractal geometries represents deeper self-reflection of humans and the “revelation that beyond the three-dimensional space of Euclid, new spaces are possible.”35 Through the implementation of complex geometric spaces, playfulness, discovery, and appreciation brings humans closer to harmony within vastness of life connecting all environments and living creatures, (fig 12). Beginning as a simple shape, a fractal iterates itself, growing out “into the community it inhabits, until the self has come to focus on not-self, but on other selves.”36 Fractals form and structure natural elements existing in a multiplicity of complex environments, channeling power through architectural space and 33 Griffith, as quoted in, Flannery, Tim F.. Here on earth: a natural history of the planet. New York: Atlantic Monthly Press :, 2010. Page 47. 34 Eglash, Ron. African fractals: modern computing and indigenous design. New Brunswick, N.J.: Rutgers University Press, 1999. Page 219. 35 Shearer, Rhonda Roland . “Chaos Theory and Fractal Geometry: Their Potential Impact on the Future of Art .” Leonardo 25 (1992): Page 151. 36 Rolston III , as quoted in, Kellert, Stephen R., and Edward O. Wilson. The Biophilia hypothesis. Washington, D.C.: Island Press, 1993. Page 17.
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envisioning human life in a truly congruent harmonious interplay with its world.
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Hypothesis
If a fractal space is planted in the interstitial space of oversimplified Euclidean geometry, than it will catalyze a harmonious coherent and complex architectural organism, continually strengthened by its interconnected relation to its host.
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Overlay Fractal Geometry geometry infinite irregularity small and large details the same endless non-integer dimension describes form of nature algorithm equation measurment organization computable tool of understanding method human invention applicatory
evolution reproduction naturally occurring affinity super-organism genetic biology fear/pleasure naturalistic vernacular adaptation life/death survival origin instinct
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Methodology
Chaos and the Cave: Abstractions of fractal space.
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Visuals
Iterations: Methodology, transformation, described through the lens of nature and the computer. Where do they meet?
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Visuals
Mind Map 1: A profound interest in nature lead me to the field of Biophilia. It is a rich area of research stretching back to human origins and combining findings across all scientific disciplines. Psychological implications of Biophilia examine natural forms and environments and their potentially beneficial properties.
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Visuals
Mind Map 2: Intersection of mathematical constructions of nature with human’s psychological love for nature. Portrays that fractal forms can activate biophilia. The cerebellum of the brain deals with fear, pleasure, and coordination, and features elements of fractal geometry.
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Visuals
Mind Map 3: Begins to introduce the idea of art as a bridge between the world of nature and humans. Fractal art, and cubism, contrast with abstract art because it is much more difficult to relate to.
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Visuals
Recursion: Every form is derived from the central pentagon by a rule of sub-divisions, demonstrating that complexity can be found through repetition and
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Visuals
Fractal Space The outlines along the boundary are the forms of leaves. As the path extends, the space iterates increasing degrees of folding.
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Frame 1: Program
World of action: 3d
World of complexity: Fractal
P r o g r a m is growth, discovery, and intersection.
Point
Space
Line
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Frame 1: Program
Positive and negative extractions of a leaf’s fractal geometry.
The forms are pulled apart and joined at structural overlaps abstracting the spatial qualities within the leafs geome-
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Frame 2: Tectonics
Building in an iterative process, the largest triangle’s dimension was continually divided to intuitively grow structure,
The resulting form brings forth an idea of section multiplicity that could portray a alteration and iteration through time, as well as self-similarity and recursion.
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Frame 2: Tectonics
The Koch Snowflake Curve was used to generate the exterior and interior tectonic study. The exterior of the module is on iteration 1 while the inside is on iteration 3, illustrating contrast of complexity.
Solid folding planes embody density at the top and bottom of the models interior. Members of structure give the feeling of openness and perforation.
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Design Probe: Space
Recursions of triangles grows from the ground plane and uproots the simplicity of a three dimensional, Euclidean
Though the iterations of triangles are smaller in size and built with lighter structural pieces, the interconnection of them transforms a simple space and potential establishes a setting of spatial harmony.
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Design Probe: Space
The following set of charcoal drawings explore deconstruction, bifurcation, and transformation of a three dimensional solid into an abstraction of fractal space.
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Design Probe: Space
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Design Probe: Space
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Design Probe: Space
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Site: Discoveries
Deceased space examines abandoned sites of grand interior spaces with no longer any sense of life or activity. This site analysis is tangential to the idea of interstitial space but still provides clues to the development of fractal space.
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Grain Terminal
Grain elevators structurally interconnected could create a framework for developing spaces.
Structure decaying into water.
Inside storage pipe for grains. The veins of the building.
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Power Plant
Columns, windows, balconies, and trussing are joined to divide and network ideas of space.
Exterior demonstrating nature’s power over artificial structure. Close relation to plants could generate specific fractal patterns
Interior structure is exposed by the decomposition of the walls, ceiling, and floor.
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Power Station
Horizontal and vertical members create new connections and bring geometric interest to an overly regulated interior.
Smoke stacks illustrate opportunity to use vertical growth to liberate disconnected interior spaces.
Module of space in the lower levels of the plant.
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Sugar Refinery
Trusses, columns, and beams extend down a vast horizontal space that can be divided to add complexity and multiplied to bring human scales and inhabitable space of self-similarity.
Industrial functions, arrange parts of building to efficiently work as a network of production.
Large columns create modules to potentially iterate.
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Asylum
This room was constructed with bars over the window to keep the psychologically ill patients from getting out. Fractal geometry could grow from large rectangular openings to join the exterior conditions of natural life to the interior of artificial restraint.
Regular windows and structure make for a possible framework of regular geometry to be transformed.
Almost completely covered with nature, fractal patterns crop the view of the asylum.
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Intersection sequence
different scales
Fract
algorithm affinity
lia hi
op Bi
human proportion
Fractal
self-awareness deeper human connection with nature
cycle repeating
tal
pa
tte
rn
method
Space Sp a
ce
2d patterns
nic
flow
a g r o
ing
3d groundplane 67
Timeline Find architectural precedents relevant to site selection and spatial conditions Research programmatic implications Develop algorithm methodology Learn grasshopper research limitations/constraints
Dec.
Draft schema to mitigate constraints and maintain human value
Site Discovery
F
Re-vise
Contact Prof. Gospidinov expert in fractal geometry
Direct focus
Select fractal condition from (recursion, self-similarity, and scaling) to narrow scope of research
question, ab
Read Godel Escher Bach: An Eternal Golden Braid
Begin mode
Search for site in Manhattan (interstitial spaces)
Experiment
Architectural sources to direct fractal’s spatial input (Wright,Corb, Mies, Loos.)
Analyze spe
Submit abstract for WIT symposium
Conditions i
Employ algo
Define fractal seed shape and module Plan final presentation experience Consider possibility to create fractal space 1:1 scale installation or piece of intervention? Finalize the algorithm methodology
Feb.
Visit site for final documentation
Writing
s of claim,
Apr.
Spatial Impact Mar.
Write narrative of final presentation
bstract, research
Re-visit claim and hypothesis
el building
Capture idea of space in all drawings
ts digital/physical
ecific spatial
Emphasize fractal element in design Thesis connects to continual discourse on architectural space.
in architecture and nature
orithm to direct cohesive simple results of findings
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Conclusions Process Evaluation
Program
0Research began with an interest in biophilia, humans natural love of nature, but transformed into an exploration of fractal geometry and space. This transformation occurred when it was realized that fractal geometry constructs spaces naturally interesting to humans...
Program was abstractly based on Bernard Tschumi’s La Villette principle where he overlays (and equates) ideas of activity, movement, and appropriation with the architectural elements of points, lines, and spaces. The axon in frame 1 translated this approach by using the architectural line as a fractal path to collide with cubes symbolizing space that create a tension and release of activity and movement.
Critiquing the role of geometric space in architecture is no small or simple task. That fact considered, this research topic has sustained the great challenge it poses because of its exciting spatial implications. Due to the vastness and complexity of fractal geometry, process has focused on gaining an understanding of what a fractal is , how it relates to humans, and its influence on the space occupied by humans. The inherent difficulty to encapsulate this complex relationship calls for mitigation by extracting certain characteristics of fractal geometry that can inform an architectural space. For the purpose of direction and =clarity, conditions of recursion, selfsimilarity, and scaling have been selected for further focused investigation. With the knowledge of fractal geometries basic components of behavior, research has begun to evolve towards connecting its properties to architectural spatial theories.
Site The question of site requires the critic of pre-existing spatial elements both of interior and exterior nature. For an alternate theory of space to be tested, it must be in the context it seeks to transform and enhance. These parameters resulted in the investigation of abandoned sites of dead space in New York. Each site desperately needs revitalization and exists as a shell of emptiness and nothingness. The dead architectural space within each of these sites could be enlivened and transformed by introducing a fractal form that grows from the interior to the exterior. Re-integration to the surrounding context would be possible as well with the intervention of fractal spaces that stem from the original growth within the dead space.
Tectonics Models have contributed valuable information concerning translatable tectonic design elements. Deconstructing the fractal patterns of a leaf, lead to investigation of structural implications and focused the project on an urban context epitomized by its euclidean form. The most important discovery thus far has been the iteration and sequence of growth in the design probe model. The method of bifurcating a triangle transforms the three-dimensional euclidean framework into a composition of fractal spaces. The result uproots the original structure and envisions a diverse interconnected network of spaces blurring interior and exterior boundaries. Architectural tectonics holds the consequential elements and characteristics of a fractal space because it must structurally conceptualize space and convey ideas of construction and assembly. A fractal space is defined by growth and evolution, architecturally providing the understanding of space constructing itself and other spaces through the use of a spatial tectonic. A method of generating and designing architecture of fractal space requires a set of rules to accommodate the needs of site program and user. An algorithm must be developed to establish clarity of process, cohesion of concept, and comprehension of tectonic fractal space.
Goals 1. Find site that sets up a tension between traditional space and architectural fractal space. 2. Clarify the extent of fractal characteristics creating space. 3. Connect projects main claim and hypothesis to existing spatial discourse. 4. Create method of physically realizing fractal space. 5. Design spatial theory supplementing advancement and transformation of a typical context of simple geometric space.
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Future Development Design Speculation Going forth, I am going to use the method of physical modeling to materialize and design fractal spaces. Since the subject matter I am exploring is abstract, ambiguous, and complex, a working process of physical constructions can yield tangible results that document a clear evolution of the idea itself. As the models develop towards a physical construction of space, experiential perspective drawings will be utilized to capture the essence of spatial transformation portraying, engagement, delight, and stimulation. The development of an algorithm as a generative tool is a critical aspect of methodology that brings meaning and execution the concept of a fractal. Studying a possible alternate method of constructing space requires the re-appropriation of an algorithm to best suit the development of this thesis project. Questions about complexity will continuously come up with the design and development of the spatial models and drawings. In its use and definition complexity captures positive qualities simplified spaces lack but the word complex, by its own nature signifies problematic design situations that cannot be understood or read. The development of an
Future Criteria The work of this thesis should be judged by its, relevance, clarity, and originality. Do the findings matter at all in the context of architectural spatial discourse? Are the theories of architectural fractal space legible and understandable in relation to human scale? Most significantly, has the research advanced the pre-existing notions of architectural space?
Inspiration
Nature is the oldest architect in the world. The beautiful patterns designed by nature have a life and a death. Humans most elemental perception of self in space begins with the understanding of the life cycle and the sense of time it conveys. Fractal geometry only occurs through growth and decay, embodying time in its manifestation. Fractal architectural space would allow the user to experience frames of time inbetween the past (initial seed shape) future (spaces not yet conceived). Encountering an architectural vein of space and time induces a heightening awareness of the present moment.
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Appendix 1. Benoit B. Mandelbrot, The Fractal Geometry of Nature (San Francisco: W.H. Freeman, 1982). Benoit B Mandelbrot’s book The Fractal Geometry of Nature (1982) explains and demonstrates a plethora of fractal number sets that describe the form and function of elements found in nature. Mandelbrot applies fractals to snowflake’s, rivers, clouds, plants, and mountains. He also uses human’s lungs and vascular system as fractal examples. His main claim is that fractals can define a more complex dimension of form that includes characteristics of growth, decay, and reproduction. Mandelbrot’s The Fractal Geometry of Nature is important for this thesis because it provides concrete evidence, formulas, and data, of the inherent connection nature’s form has with fractals. 2. Morris Kline, Mathematics for the Nonmathematician, (New York: Dover, 1985). Morris Kline’s book Mathematics for the Nonmathematician (1985) supports the claim that mathematics is essential for understanding the natural systems of the world and human’s relation to these systems. The book reveals a dichotomy between mathematics applied for the understanding of nature and mathematics applied for the mastery of nature. By revealing this tension, the book affirms inherent connections between man, nature, and mathematics. Kline’s book imparts upon this thesis precedents of mathematics use for humans to better assimilate their environment.
3. Rhonda Roland Shearer, “Chaos Theory and Fractal Geometry: Their Potential Impact on the Future of Art ,” Leonardo 25 (1992): 143-152. The article, “Chaos Theory The article, “Chaos Theory and Fractal Geometry: Their Potential Impact on the Future of Art” (1992) by Rhonda Roland Shearer makes the claim that the next major innovation in art (comparable to the Renaissance, and Modern Art) can happen through the use of fractal geometry. Shearer borrows examples from architecture and urbanism to critique simplified geometries and uses examples from nature to reinforce the value of complex fractal geometries. Shearer references the discoveries of new geometries in history as the key to human comprehension of space. The article looks at the psychological properties of fractal geometry and the opportunity it poses to see the world from a different perspective. Shearer’s article holds the spirit and direction of this thesis with its goal of capturing the exciting complexity fractal space has for humans. As an artist who investigates the potential of fractal geometry, Shearer provides creative methods for further investigating questions of fractal space. 4. Stephen R. Kellert and Edward O. Wilson, The Biophilia Hypothesis. (Washington, D.C. Island Press, 1993). Stephen R. Kellert and Edward O. Wilson’s edited book The Biophilia Hypothesis (1993) compiles the voices of expert scientists from different fields to examine
what a human affinity to nature means for the world. The book claims that humans are connected to nature on a deep, “aesthetic, intellectual, and spiritual” level that extends from humans physical dependence on their natural environment. The research complied in The Biophilia Hypothesis informs this thesis of the human love of nature and how natures form and growth can act as a model for the design and implementation of fractal space. 5. Michael Batty, and Paul Longley. Fractal cities: a geometry of form and function, (London: Academic Press, 1994). The book Fractal Cities: A Geometry of form and function (1994) by Michael Batty and Paul Longley, researches the connection between cities complex form and fractal geometry. It theorizes that cities can be better organized, understood, and designed through the usage of fractal geometry. Batty and Longley claim their results to overlap with urban design theory and use the critical voice of Jane Jacobs to guide their research. Most importantly, this source offers a scope and method of applying fractal geometry to urban design. The methodology of this thesis can be approached with rules and constraints appropriated from Batty and Longley’s research. 6. Ron Eglash, African Fractals: Modern Computing and Indigenous Design (New Brunswick, N.J. : Rutgers University Press, 1999).
Ron Eglash’s book African Frac-
tals: Modern Computing and Indigenous Design (1999) makes the claim that fractal patterns are very similar to African designs and African knowledge systems. Eglash unpacks these similarities through his research while living in African fractal villages and speaking with the people who build using fractal patterns. An overarching aspect of importance is the usage of algorithms in culture, as a way to organize the space of people physically, psychologically, socially and spiritually. This source provides architectural fractal examples and real world design techniques that are of great use for the research for this thesis. 7. Stephen R. Kellert, Building for life designing and understanding the human-nature connection. (Washington, DC: Island Press, 2005). Stephen R. Kellert’s book Building for Life: Designing and Understanding the Human-Nature Connection, studies architecture’s ability to “restore the basis for a more compatible and even harmonious relationship with nature.”(3) The discipline of building and design should surpass typical constraints according to the theories of Kellert’s book, and must capture conditions found in a natural environment. Kellert translates characteristics of nature that benefit humans into a language that can bridge the gap between design and biophilia. The design language explored in Building for Life contributes a taxonomy of biophilic design conditions that overlap with the spatial concepts of fractal geometry.
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8. Tim F Flannery, Here on earth: a natural history of the planet, (New York: Atlantic Monthly Press, 2010). Here on Earth: A Natural History of the Planet (2010) by Tim F. Flannery claims that humans progress isn’t possible without love and understanding of the planet we live on. Here on Earth studies a variety of animals that bring unique elements into the architecture of our natural environments form and function. Super organisms are studied in a chapter dedicated to better understand the interconnected qualities of nature. The concept of the super organism is of interest to this thesis because it embodies the interdependency a fractal has between the different iterations of its self.
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Research Essay Images fig 1. Benoit B. Mandelbrot, The Fractal Geometry of Nature (San Francisco: W.H. Freeman, 1982). Page c10.
fig 9. Benoit B. Mandelbrot, The Fractal Geometry of Nature (San Francisco: W.H. Freeman, 1982). Page 144.
fig 2. Benoit B. Mandelbrot, The Fractal Geometry of Nature (San Francisco: W.H. Freeman, 1982). Page 164
fig 10. Michael Batty, and Paul Longley. Fractal cities: a geometry of form and function, (London: Academic Press, 1994). Page 236.
fig 3. Ron Eglash, African Fractals: Modern Computing and Indigenous Design (New Brunswick, N.J. : Rutgers University Press, 1999). Page 69.
fig 11. www.raremaps.com
fig 4. Ron Eglash, African Fractals: Modern Computing and Indigenous Design (New Brunswick, N.J. : Rutgers University Press, 1999). Page 119. fig 5. Ron Eglash, African Fractals: Modern Computing and Indigenous Design (New Brunswick, N.J. : Rutgers University Press, 1999). Page 119. fig 6. Ron Eglash, African Fractals: Modern Computing and Indigenous Design (New Brunswick, N.J. : Rutgers University Press, 1999). Page 27. fig 7. Stephen R. Kellert, Building for life designing and understanding the humannature connection. (Washington, DC: Island Press, 2005). Page 154. fig 8. Michael Batty, and Paul Longley. Fractal cities: a geometry of form and function, (London: Academic Press, 1994). Page 96.
fig 12. Benoit B. Mandelbrot, The Fractal Geometry of Nature (San Francisco: W.H. Freeman, 1982). Page 65.
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