Open access peer-reviewed chapter

# Towards a Precise and Mathematical Fractalesque Architecture

Written By

John Charles Driscoll

Submitted: 27 April 2022 Reviewed: 02 June 2022 Published: 21 August 2022

DOI: 10.5772/intechopen.105677

From the Edited Volume

## Genetic Algorithms

Edited by Sebastián Ventura, José María Luna and José María Moyano

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## Abstract

This paper reviews a design process in the context of algorithmic architecture design for establishing a scale-invariant and rigorous self-similar motif(s) that can be applied generally to any design problem. An architect (author) defines a genetic algorithm (GA) using a population of design variants iterated over multiple generations. Exemplars are selected based on their fractal dimension (FD) along with the architect and fit to solve a real-world architectural problem. The algorithm is coded in Python and Ruby with an interface in SketchUp. The architect is able to modify exemplars and iterate them as many times as required in the GA until an acceptable solution is achieved. Solutions are critiqued by a jury of professional architects regarding their fractal qualities. Results show a fractal motif that is not strictly self-similar and not strictly scale-invariant. Discussion is focused here on the philosophical implications of this research in terms of better defining a fractalesque architecture. The case for a more precise and mathematical fractalesque architecture is discussed concluding that further development of the algorithmic design process is necessary to clarify the value of such a tool.

### Keywords

• architecture
• Genetic algorithm
• algorithmic design
• fractals
• fractal dimension
• box-counting dimension
• characteristic complexity
• motif
• jury
• critique

## 1. Introduction

This paper reviews a design process in the context of algorithmic architecture design for establishing a scale-invariant and rigorous self-similar motif that can be applied generally to any design problem. An architect (author) defines a genetic algorithm (GA) using a population of design variants iterated over multiple generations. Exemplars are selected based on their fractal dimension (FD) along with the architect and fit to solve a real-world architectural problem. This study diverges from some precedent in that it positions FD analysis up front from within the creative milieu of an architect’s process. This study explores the use of FD from within a computational framework aimed at establishing both a scale-invariant and self-similar pattern as the organizing principle for the building’s form and parti relative to various features of the building. These features are primarily plan, section and elevation as well as the relationship to a contextual building’s relevant elevation. The algorithm is coded in Python and Ruby with an interface in SketchUp. The framework is directed towards the creation of fractaleque architecture by incorporating a mathematically rigorous and iterative evolutionary strategy from within the creative milieu of an architect’s process. The philosophical implications of the framework are the focus of this paper. The algorithmic design process consists of 3 steps as follows:

1. A GA is used to iterate a 2-dimensional line composition for a select number of runs. The GA uses a binary representation for individual lines which can be traded with other strings through crossover and mutation and inherited by subsequent generations. From an initial population of individuals exemplars are chosen based on their FD and tournament selection as well as cloning and used for creating additional population pools. This process is repeated many times until the FD of exemplars approaches a pre-selected FD. Line compositions are extruded into 3-dimensional massing models stochastically.

2. This step is referred to as fitting. GA outputs are imported into SketchUp, a solid modeling environment, and are available for a designer’s review and selection relative to the sample problem. The designer is able to modify the exemplars chosen and use them as seeds for additional GA runs (repeating step 1). The GA and designer thereby establish a self-similar geometry that is reflected in various features of the design. Through the use of this tool the designer may potentially create a building with a similar FD throughout its scale range in terms of its orthogonal projections. 3 scale levels are used for this case study but the design process is not limited to a specific number of levels and could reflect a greater range of scales as the designer prefers but not an infinite range of scale due to practical limitations.

3. A jury of professional architects engage in a critique of the design at 3 stages during the process. The stages are referred to as “pin-ups” and consist of presentation images and renderings of both the GA outputs in the form of a timeline (Figure 1) and the designer’s modifications. Jurors critiques are based in praxis and personal design philosophy.

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## 2. Background

Fractal geometry was, from the beginning, closely related to natural inorganic and biological morphology [1]. Mandelbrot generalized a basis for what he termed “fractals” by gathering together the work of mathematicians and topologists such as Cantor, Gaston Julia, Felix Hausdorff, Jean Perrin and others who prior to computer technology had not been able to fully visualize the ramifications of such models [2]. Fractal dimension (FD) is a technique Mandelbrot adopted to describe an object’s characteristic complexity and is related to its degree of irregularity at multiple scales although it does not explicitly define the object’s geometry or whether or not it is strictly self-similar or fractal. The term characteristic complexity does however provide insight into an object’s scale invariance. For this reason it is helpful in understanding a range of irregular geometries found in nature and architecture and is considered one of half a dozen or so measures of complexity [3].

Architecture has been trending towards ever larger buildings. From 2012 to 2018 the average floorspace of buildings has increased by11% in the United States according to the 2018 Energy Information Administration, Commercial Buildings Energy Consumption Survey [4]. Unfortunately, As buildings grow the morphology of buildings often tends to become simplified with less detail and less articulation (images of big box stores and strip malls come to mind). Larger buildings can be paradoxically cheaper through modularization and standardization as opposed to custom – one-off – buildings. With automated strategies on the horizon however it is increasingly possible to create a more articulate and detail rich architecture that is not prohibitively expensive.

FD incorporated in general analytic tools has been useful in understanding a building’s detail in terms of characteristic complexity and have been gaining traction over the lasts decades. Michael Batty and Carl Bovil introduced FD as a serious tool in the analysis of architecture and urban form based on its underlying mathematical structure using box-counting dimension (BCD) to approximate FD (In this paper FD and BCD are used interchangeably). This tool is used to analyze 2-dimensional urban form and architectural plans and elevations of various buildings after they have been built [5, 6]. FD has since been adopted by many researchers to study the dynamics and complexity of cities and urban scaling as well as urban infrastructure [7, 8, 9, 10, 11, 12, 13, 14]. Similar tools are also being used increasingly in the analysis of the characteristic complexity of architecture. FD is often used to make correlations between styles of architecture as well as the natural environment. Such analyses are related to fractal geometry a priori with respect to the traditional application of the tool, i.e., relating to morphogenesis. For this reason, architectural styles purporting to be based in nature are often a focus such as the American School or organic movement developed by Louis Sullivan and Frank Lloyd Wright [9, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Otswald introduces a standardized calibrated model for the analysis of architecture in 2013 [24]. Lorenz et al. have offered additional analytic “proportion methods” using BCD offering a potentially more descriptive approach [25].

Algorithmic design or computational design (CD) uses computer technology to aid in generating architectural solutions and has developed rapidly over the last decades. CD approaches differ from simply using computer tools in the design process but rather use computation to create designs [26]. Generative design is a subset of CD and includes evolutionary approaches such as GAs [27, 28, 29, 30, 31, 32]. GAs are methods analogous to biological evolution consisting of autonomous and stochastic search algorithms. GAs, when used generatively, can produce complex and unpredictable outcomes [26]. GAs are well suited to complex problems such as those represented by the multi-variate form and function requirements in architecture [33]. Trends in research are towards biomimetic approaches combining evolution-based computational methods with morphogenetic processes inspired by nature, where form is generated by computer technology, incorporating the rules and constraints of fabrication [34]. GA methods have been developed to help solve a variety of architectural problems such as geometric (lattice) deformation by Wattabe and sequences of polygonal operators by McGuire [35]. Latham and Todd developed the PC mutator system at IBM UK’s Scientific Centre with individual projects as well as commercially available software [35]. Hemberg and the Emergent Design Group (EDG) at MIT developed Genr8 in 2001 which is a GA plug-in for the modeling and animation software Maya [36, 37, 38]. Galapagos is an evolutionary plug-in for Grasshopper [39, 40] which is used as a visual programming aid for architects within the BIM software Rhino. Grasshopper has been widely used for parametric modeling applications including a method for constructing islamic ornament [41] Galapagos has been used to optimize spatial adjacencies for complicated building programs [42], daylighting and shading studies [43] as well as to find novel solutions to structural problems [44]. Galapagos has also been used to generate new fractal forms for urban environments using cellular automata [45]. Chouchoulas et al. discuss Shape Grammars using GAs which have been explored with a prototype tool called Shape Evolution to design hypothetical buildings ([46], pp. 26–34). Shape grammars and parametric design has been applied to the architecture of Frank Lloyd Wright [33] Generative design, fractal geometry and stereotomic algorithms as well as the use of automated manufacturing processes have been influential to Joris Laarman and his designs for furniture and bridges [47]. Generative design incorporating FD has been developed as a method for designing efficient and robust structural forms [48, 49, 50]. Specifically related to this paper, generative design and FD has also been used to analyze and create neighborhood plans as well as individual schematic designs of residences [51].

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## 3. Methodology

The case study was an architectural project in Ithaca, NY as defined by a request for proposals (RFP) issued by the city in the Spring of 2018 (https://www.cityofithaca.org/DocumentCenter/View/7614/Green-St-Garage-Redevelopment-IURA-RFP-Revised-5718). The project program consisted of a mixed use residential and retail building in an urban zone along Green Street to the south and with access to a pedestrian commons to the north. The case study focused on the schematic design phase with a level of completion considered to be adequate for an RFP. The design process consisted of a GA using BCD as the fitness criteria. The GA was written in Python and Ruby with a user interface plug-in for the solid modeling software SketchUp.

The GA started with a number of 2D line compositions representing individuals. Lines were placed horizontally and vertically on a page in random locations initially. Individual line compositions consisted of a set number of horizontal and vertical lines perpendicular or parallel to each other and the edge of the page. The lines are allowed to go from edge to edge or stop at another line. Each composition was measured with BCD to determine its scalar fitness value. The GA used tournament selection to search exemplar individuals whose value most closely approximated a target BCD. The target was 1.562 which was the BCD of the facade of the contextual building used in this study (Figure 2). Tournament selection consisted of randomly selecting 3 individuals from the pool with replacement and choosing the one whose value most closely matched the target. Tournaments were iterated until a new generation of individuals was created. New individuals were subject to random rates of mutation. Mutations consisted of altering the locations of lines an amount determined stochastically from a Poisson distribution. The full code for this research is available at https://github.com/JohnCDriscoll/Fractalesque-Architecture/new/main?readme=1.

Figure 1 shows a sample of a timeline which is out-put by the GA with the designs progressing from earlier designs on the left to later designs on the right. Exemplar compositions for each generation are shown along the upper-most row for each column. The images below the exemplar show the same line composition but with a number of rectangles defined by the lines stochastically selected as “masses”(shown in red). Masses are allowed to overlap and any number of masses may be set in the corresponding parameter. The row of images below show the masses extruded stochastically to an elevation along the Z axis. The graphs at the bottom display the exemplar FD for each run and the mean over successive generations and r-squared of individual designs. The FD is a measure of only the 2D compositions. Figure 1 is an arbitrary example culled from preliminary GA runs. An important leverage point offered by a GA is the sheer number of potential variants a designer may peruse and develop. For this example the FD for exemplars highlighted are 1.683 and 1.699. As the series progresses to the right, the BCD increases. The GA presented here is understood as a proof-of-concept model and as such was explored briefly in terms of its capabilities. For instance, the runs executed were limited to 10 which was adequate to find a successful composition. Longer runs were executed with a target FD of 2 (maximum FD) as an experiment. A higher FD was achieved using 500 runs but the lines tended to bunch up along the edges of the page. This issue was not resolved in this research. Developing the GA further would focus on this issue as well as include more complex architectural elements other than simple 2D line compositions. This potential is discussed more in the Next Steps section below. As a proof-of-concept study these limitations were not considered significant. The GA developed in this research was intended to scale up in future versions and is presented here to demonstrate that an under-the-hood approach was valuable to the architect for a variety of reasons. A GA mirrors some aspects of the design process such as iteration and evolution and thereby allows for a more integrated approach where the architect is in control of the algorithm to a degree and enabled to visualize options from inception through realization. This is especially valuable in terms of fractal geometry which shares some of the attributes of the algorithm.

After the GA determined an exemplar composition it was out-pouted to a solid modeling environment where it was used to generate the design for the building. The designer made alterations and these designs were flatted to 2D compositions and used as seeds for the GA a second time. This back and forth happened 2 times in total. After this the designer modeled the building without the use of the GA. The completed design was measured for FD. Milestones during the process were presented to jurors as traditional architectural renderings. These milestones were referred to as pin-ups. There were 3 pin-ups in total. Comments were collected asynchronously and salient issues addressed and, in some cases, used to adjust the GA and fitting steps for subsequent GA and designer iterations leading to the final scheme.

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## 4. Results

Results established a preliminary composition of 30 lines with a FD of 1.477. This result was somewhat lower than the target 1.562 but was significantly higher than random compositions which were in the 1.200 range. The composition was acceptable in terms of its design elements and potential constructibility. The lower FD was not considered problematic as a small scale element as will be discussed next. Figure 3 shows a 2D image which was then extruded by the GA and modified by the architect. (Figure 4). This block became a module conceived as a masonry unit as well as the motif and was subsequently assigned scale-dependent functions and also established a general organizational strategy. The motif occurred at primarily 3 distinct scale ranges in the building termed micro, mezzo and macro. The micro level consisted of the modular element (Figure 4). The mezzo represented larger building systems such as facade elements and fenestration assemblies (Figure 5). The macro level was at the scale of the building overall as represented in elevation and plan including the site plan (Figures 69) The BCD of the various elements and scale ranges were coordinated within the architectural context of the site by approximating a similar BCD as the primary adjacent building at the macro level (Figure 2). The following description refers to Figures 69. Figures are shown as dual images with the left image representing the building element and the right image showing superimposed elements of composition which are discussed next.

The motif is composed of two intersecting lines (shown in red) with a third line (shown in blue) which stops at one of the intersecting lines. The main intersection of the red lines creates a visual focal point or center of interest. This point is then reinforced with a rectangle (shown in green) further establishing the focus. The intersection of the red and blue lines creates a secondary center of interest which is further established with the addition of the yellow rectangle (Figure 10). The micro level (Figures 4 and 10) represented a masonry unit and as such was simplified from the original GA output. The BCD reduced from 1.48 to 1.26. This was a significant reduction and not preferable. However, the requirements for forming the block outweighed the reduction in BCD in this instance.

The motif was again expressed at the mezzo level in various ways. A clear expression of the motif is shown in Figure 5 where the proportions of the original block were preserved. The function of the element changes. Now, the primary focal point serves to denote one of the main entrances into the building leading to a primary circulation component while the secondary focal point serves to denote the entrance and storefront for a separate retail space. The red and blue lines outline the masses of the building and provide axes for the building to expand along. The mezzo scale was also included fenestration assemblies (Figure 2). Here the BCD was modified in a back and forth between architect and algorithm to increase the BCD back to the original GA output which was 1.48.

The macro scale was considered in terms of the entire south facade of the building as well as the site plan. The facade was measured and had a BCD of 1.59. This was higher than the original output but did compare favorably to the adjacent building which served as the contextual element for the building. This building had a south facade BCD of 1.56 which was considered a contextual relationship. Again, the elements of composition highlighted certain features of the building. These were mainly circulation routes and established a hierarchical arrangement of space which highlighted programmatic features of the building. The motif here serves to discern commercial from residential zones as well as demarcating the main access to the pedestrian commons located to the north. of the site (Figure 6). The site plan (Figure 7) shows this relationship more clearly where now the motif is superimposed on the ground plane. The access rout to the commons is well defined as well as the pedestrian strip between the building and the contextual building to the north. The green rectangles used to enhance the focal point are important aspects of the program, namely a park pavilion and the residential tower. The blue line is an important secondary feature of the building denoting a vehicular access point necessary for loading and delivery as well as egress and fire truck requirements. The next two images (Figures 8 and 9) are isometrics and show the same elements of composition and their 3-dimensional expression. Figure 11 is a perspective rendering of the building and its relationship to the block. A key component of the building’s program was to allow for views from the 11 story building to the north (contextual building) while also helping to balance the various proportions of the city block which consisted of smaller 4 and 5 story buildings. The inter-relations of scale included the building’s components as well as the immediate urban fabric (Figure 2).

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## 5. Discussion

As mentioned in the background section, previous studies have often discussed fractals in architecture within a context of an architectural tradition ascribing to nature based principles, e.g., the American School or organic architecture as espoused by Louis Sullivan and Frank Lloyd Wright [5, 15, 17, 18, 20, 52, 53]. This discussion continues in this tradition in focusing on the philosophical implications from a similar context of organic architecture with regard to the algorithmic method presented above.

James Walter Schildroth who was an apprentice of Frank Lloyd Wright’s (see http://www.schildrotharchitect.net/autobiography.html) as well as a juror for this study writes in response to Harris’ claims concerning the fractal ontology of Frank Lloyd Wright’s design for the Palmer house [20]:

“I really don’t see fractals in the Palmer House. Of course there are equilateral triangles. The unit is made with equilateral triangles. The plan is made by relating all parts of the plan as it is made to the unit system. The design is not made by repeatingself-similar triangles. The unit system gives the whole unity. I think this unity approaches the unity in all of the natural world. What we call beauty.”

A valid criticism from the jury was that the unifying motif is formal and did not originate from some functional requirement or relation to site but was established a priori and thereby applied functional requirements to the form, i.e., function followed form. This aspect of the process was built into the algorithm at the outset by limiting the fitness criterion to FD in selecting forms rather than a host of criteria such as: client’s needs, budget, program, topography, solar attitude, materiality, tectonic systems, adjacencies, meaning, beauty, etc. This was done to simplify representation in the GA and generalize the system towards a parsimonious tool capable of solving a host of design problems. As the results show, the motif undergoes modification at different scales and as different functions are applied to it. The primary scale ranges employ the motif in different ways that are not exact replicas of each other although they are proportional as will be discussed. The building is therefore not self-similar in a strict sense. The scales range is limited to 3 levels although there is overlap and compositional unity throughout the building. Therefore, the building is clearly not scale-invariant, at least in terms of the repeating motif. The building may be considered as being composed of natural fractal structures in a material sense but not in so far as they were designed within the architectural concept. The building can not be said to be a literal representation in the mathematical or natural sense of fractal geometry in light of these two criteria not being met. The question of whether the building represents fractal architecture is more difficult to assess because of the ambiguity around this definition [53]. The following discussion does not argue that the building is fractal, fractalesque or fractal-like [54] but rather raises issues pertinent to this question.

One issue raised in this work and elsewhere in the literature is whether a higher FD is correlated with more quality design. Some suggest there is a “magic number” that is more esthetically pleasing or proportionally harmonious ranging from 1.3 to 1.52 [25]. The jury was mixed on this point with no clear indication that compositions with a specific FD were more compelling visually or had inherent merit architecturally when divorced from programmatic and functional requirements. Ken Kroeger comments, “ To address the question about increased FD and improvement of the compositional quality – I don’t think it specifically improves the composition. It simply is a variation of line work. Without having any performance/outcome values, I would bet that if you ask 50 different people which one they preferred, you would get that many answers.”

This study however did not employ FD as an esthetic device or assume that one value of FD improved one composition over another but incorporated it from within an architect’s design process to gauge and thereby relate the various elements within an overall organizational strategy. FD was considered on multiple levels from ornamentation on the facade (masonry blocks) to the distribution of spaces and circulation systems to the overall parti of the building and relationship to the larger urban fabric.

FD indicates an object’s characteristic complexity but does not reveal the specific fractal shape. For instance, different fractals may have the same FD if they share the same characteristic complexity. This limitation however was not problematic in this study because the self-similar motif was defined by the GA and architect not by its FD alone. The GA can be thought of as a computational tool that takes over when the architect can no longer process. Discovering this edge and broadening it is a challenge for the architect. For this reason, FD used here does not present a conflagration of irregular objects with fractal objects. FD is not used to define self-similar geometries or indicate a higher quality design per se but is used compositionally to direct the eye.

A higher characteristic complexity in one element over another is useful in directing the eye [55]. In this study eye movement was an organizational strategy a priory and shown to be a convincing device in differentiating and highlighting the programatic elements as well as creating visual effect. The compositional elements of the motif established focal points and as such direct the eye towards key features of the building. The building as a whole is unified through the repetitive use of the motif and the dynamic quality established in the ensemble. The eye is lead from one place to another at various scale ranges in the building to discover the whole reflected in the part and vice versa. A juror, Bret Holverstott, comments, “I think I understand why our aesthetics seems to dictate that the fractal dimension increase as the scale increases. It is because the overall massing of a building should strive for a composition that is compelling if seen from a distance, not overly simple as in a homogenous skyscraper. I am reminded of how gothic cathedrals evolved to utilize alternating bays in order to keep your eye from immediately slipping to the end of the hall; good skyscrapers also break up the composition into something that allows your eye to linger on elements instead of slipping up to the top of the building.”

In this study, the motif is seen to overlap or nest within itself as it changed scale. For instance, the primary focal point in Figure 5 becomes the secondary focal point in Figures 6 and 7. Such nesting characteristics define a proportional relationship between self-similar objects and are not, as Schildroth observes, simply repeating shapes at various scales. This type of expression is challenging to understand in terms of strict affine transformations, such as we see in geometric fractals like the Koch curve or Sierpinski triangle, but is germane to the overlapping richness in detail we see in architecture. Christopher Alexander discusses the multiplicity of readings and overlapping characteristics of architecture and urban planning in, A City is not a Tree [56] and demonstrates the ambiguity of mapping such characteristics. Therefore, a more refined definition of fractalesque architecture may deviate from a purely mathematical description in this regard as a description of natural fractals does, i.e., fractals in architecture are not strictly self-similar as they do not continue to iterate past a certain point and often translate in various ways or vary depending on their scale or compositional strategy, i.e., overlapping, layering, figure-ground reversal, etc. Self-similarity as explored here is emblematic of a theme which provides a sense of unity to the project yet importantly the theme undergoes development and variation. Although the motif is repeated in a finite way literally (3 levels) the self-similarity is extended phenomenologically in both directions, e.g., at a smaller scale in terms of the blocks materiality and at a larger scale in terns of the parti and master plan relating to the contextual urban fabric. In this way the building is a representation of fractal structure in two senses, both as a physical instantiation of a kind of limited self-similarity as well as a metaphysical metaphor representing a fractal structure extending beyond the building’s spatial limits.

Perhaps in light of this discussion the use of FD in assessing and designing buildings is more provocative heuristically as a general tool that can be applied to many styles and design intentions. The juror Tom Mlynarski comments, I think the most compelling thing about your work is this incorporation of fitness criteria rooted in FD. It seems FD can be used as a fitness criterion for all sorts of a different buildings with different styles and different programs and you can use the same criteria to rate them all. That is something fresh. You are no longer bound to rather simple fitness criteria like plan efficiency or exposure or whatever. So this is the most substantial part of your work.” Mlynarski goes on to say, … FD seems interesting in how it can be used to compare buildings that are very different in style and program. It’s also interesting as a tool for judging options of the same design as you demonstrated here.”

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## 6. Next steps

As mentioned, an important aspect of this work was the parsimony of the GA/FD tool. Indeed, a GA was selected as an appropriate tool for its understandability and simplicity, i.e., it is autonomous and does not require training data. Similar to applying Occam’s razor to the theory of evolution, GAs reflect a similar elegance and parsimony: essentially stochasticity within a constrained environment. However, GAs can become more complicated when multi-variate fitness functions are piled on. This study has endeavored to keep the fitness criteria simple – essentially one scalar metric – while still aiming to capture the complexity in architecture. For this reason it is not desirable to complicate FD analysis needlessly, however, an extension into a volumetric measure is a next step. 3-dimensional BCD tools exist in other fields and have been proposed for fractal analysis in architecture [25, 57, 58, 59, 60, 61]. Incorporating such a tool is necessary at this stage.

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## 7. Conclusion

This paper reviewed an algorithmic design model in the context of architecture incorporating a GA using FD as its fitness criterion and discussed its broader implications relative to characteristic complexity and a more refined definition of fractalesque architecture. The GA is presented here to demonstrate that an under-the-hood approach was valuable to the architect for a variety of reasons. A GA mirrors some aspects of the design process such as iteration and evolution and thereby allows for a more integrated approach where the architect is in control of the algorithm to a degree. This approach enabled the architect to visualize options from inception within the algorithm through realization in model space. This is especially valuable in terms of fractal geometry which shares some of the attributes of a GA such as iteration and variation within a set of constraints. The design process developed in this research established a limited self-similar motif which was employed as an organizing principle to unify the building at multiple scales in the design. This research is novel in the sense that the organizing principles incorporate the algorithm as well as the traditional modes of design. The finished product expresses its making not only in terms of material, structure and craft but in terms of its code. The motif was not merely the pattern on the facade but included the organization of algorithmic content as well as space, structure and master plan. The philosophical implications of this research suggest that a fractalesque architecture might be better conceptualized as both a partial instantiation of fractal geometry as well as in the metaphorical sense of a fractal which phenomenologically scales beyond the physical structure into its digital code and physical context. Such a design methodology required a facility on the part of the architect to incorporate the algorithm within a larger vocabulary. In essence internalizing it within the design process not unlike a material of sorts – in addition to brick and mortar – that the architect may sculpt to create a more mathematically rigorous self-similar motif with a consistent FD. The tool and process developed proved partially successful at approaching this benchmark. Further developing the GA/FD tool in the third dimension may improve the results.

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Written By

John Charles Driscoll

Submitted: 27 April 2022 Reviewed: 02 June 2022 Published: 21 August 2022