Parametric Design as a Computational Thinking Approach for Mathematical Communication in Industrial Design Paul Tsuyen Wu University of Edinburgh, College of Arts, Humanities and Social Sciences Edinburgh College of Art, School of Design Research Methods MSR (Product Design) November 2014 / March 2020 E-mail: ujalatsuyen@live.com Works: issuu.com/ujalatsuyen
Abstract There is a common belief that the majority of people consider mathematics to be a rather difficult subject, owing to an array of factors related to instruction and learners’ cognitive (thinking), affective (feeling) and psychomotor (kinesthetic) attributes. If wrongly conceived, mathematics could appear as a daunting subject of esoteric theorems and tedious computations, rigorous and only accessible to a select few individuals. However, there is in fact more to mathematics than rigor and formality. While it might seem like a purely formulaic pursuit in the eyes of the majority, mathematics has itself, indeed, been described both as a creative art and as a natural science motivated by beauty and practicality. Application wise, not only does mathematics provide us with a universal language with which to describe, model, and understand phenomena in our world, it also delivers inspiration to artists and designers. In architectural design, mathematics is a powerful tool that contributes to the form development of architecture, thereby resulting in the birth of parametric design. With additive manufacturing and artificial intelligence fueling the fourth industrial revolution, computational design methods and generative design technology become prominent and needed by degrees for the reason that it is no longer about ordering of form, but rather about the ordering of code that orders form for achieving optimal design. This paper elaborates on how parametric design, treated as a math-driven approach by implementing mathematical algorithms in industrial design, transforms seemingly abstract mathematical concepts into an interactive experience for designers to achieve intuitive, emergent solutions in art and design in the digital realm. Keywords: Algorithm, generative design, parametric design, parametric equations, parametric, parametricism, mathematical programming, optimization.
1 | INTRODUCTION Aside from searching for ways to communicate mathematics in a non-rigorous manner, this research also focuses on how we can leverage digital technologies in order to design models that will approach the efficiency and the beauty of living eukaryotic systems. It is evident that nature can play a useful role in providing solutions to human problems. The biological systems of eukaryotes, for instance, offer wide array of astounding properties that designers can draw from. Botanical species in particular are composed of modular building blocks that possess nature’s algorithms such as their symmetries, spirals, branches, meanders, waves, foams, tessellations, cracks and stripes. These building blocks do not exclusively give eukaryotes a regularity of sorts; they are able to generate a wide range of variation provided as design inspiration that can potentially contribute to the development of a new design methodology, highlighting a vast array of impacts from design optimization to design innovation in terms of complexity and emergence of forms. By using algorithmic modeling platforms such as McNeel's Grasshopper and Autodesk’s Dynamo, parametric design potentially enables dissimilar, innovative and mathematical ways of generating multiple design solutions to structural form-finding problems. Such approach is also known as generative design, which is similar to parametric design in terms of design process as both involve the user gradually modulating design inputs until a desired aesthetic result is found. In a more technical manner, generative design is a formfinding process that can replicate natural world’s evolutionary approach with math and computing to provide optimal solutions to an engineering problem.[1] How was such design methodology developed though? Moreover, how does mathematics even factor into parametric design and generative design? To answer these questions, it is necessary to look into its origin first.
2 | PARAMETRIZATION: ORIGIN, HISTORY AND DEFINITION 2.1 The meaning of parameter Mathematics is the architecture of structure, and architecture is the art of structure. [2] Based on the meaning of this saying we can define mathematics as a procedural tool, originated from architecture for analyzing building structures. The birth of mathematics has doubtlessly brought a major influence to the aesthetics of architecture and thereby produced the art of mathematics. Historically, architecture was part of mathematics, and in many periods of the past, the two disciplines were indistinguishable. In the ancient world, mathematicians were architects, whose constructions - the pyramids, ziggurats, temples, stadia, and irrigation projects - we marvel at today. (O'Connor and Robertson 2000) [3] As for the idea in parametrization, various scientists and mathematicians had earlier used parametric equations to define geometry that later influenced contemporary architects, and the concept of parameter can date back to Ancient Greece, a period when architects were required to be mathematicians. By looking at the word itself, it is noticeable that parameter is a loanword from the ancient Greek para: “beside, subsidiary”; and metron: “measure.” In the middle of the seventeenth century, the word parameter was introduced as a geometry term in reference to conic sections until late 1920s when it began to be extended to “measure factor which helps to identify a particular system.”[4] In analytic geometry, curves are frequently defined by parametric equations since there are countless curves that are incapable of being expressed by the fundamental functions. When encountering mathematics in real life situations, parametric equations naturally arise, particularly when dealing with motion for which the location is a
function of time.[5] For many of these scenarios, it is comparatively easier and useful to define the problem in terms of parametric equations as separate functions of time. Closely related to variable, a parameter is a link between two dependent variables related to an independent variable of a set of parametric equations for which the range of possible values determines a collection of distinct cases in a mathematical problem.[6] 2.2 The history of parameter The definition of parametric equations states two critical criteria:
example of a parametric space curve defined by a pair of explicit functions is the formulae that describe a particular circle: x(r, t) = r â‹… cos(t) y(r, t) = r â‹… sin(t) These two formulae are said to be parametric with r and t as inputs since they satisfy the criteria with respect to the aforementioned definition.
1. A parametric equation defines a set of quantities as functions of parameters. 2. The set of quantities are related to the parameters through explicit functions. We start by considering the generic quadratic function f(x) = ax 2 + bx + c (also known as second-degree polynomial function) in which the variable x is an independent variable that designates the function’s argument whereas a(nonzero), b and c are parameters that identify which particular quadratic function is being considered. As the above discussion describes, an equation is defined to be parametric as long as it is expressed in terms of parameters on any occasion.
Fig 01: The parabola of the general form ađ?‘Ľ 2 + bx + c
Consequently, the quadratic polynomial satisfies the first criterion of the definition, and forms the graph of a parabola. Another
Fig 02: As t goes from 0 to 2Ď€ the x and y make a circle
First, they express a set of quantities (in this case an x quantity and a y quantity) in terms of a number of parameters (the radius r, which controls the shape of the curve; and the angle t, which controls where along the curve the point occurs). Secound, the outcomes (x and y) are related to the parameters (a and t) through explicit functions (functions defined in an explicit manner such that its value may be directed calculated from the independent variable). This is the origin of the term parametric: a set of quantities expressed as an explicit function of a number of parameters. The influence brought by such elementary mathematical concept has made a revolutionary change in contemporary architecture, thus laying the foundations of the theory of both parametric and generative design. Despite the role of the term and its long history in mathematics, its link with architectural design can be found in the work
of several influential scholars accomplished in the twentieth century. As for the initial appearance of the term, a host of debatable questions and plausible information in that regard still remained. Professor David Jason Gerber from USC School of Architecture credits Maurice Ruiter in his doctoral thesis Parametric Practice for first using the word in a paper from 1988 entitled Parametric Design.[7] The year was also when Parametric Technology Corporation, a company founded by mathematician Samuel Geisberg in 1985, released the first commercially parametric modeling software, ProE. In spite of the aforementioned details, some argues that the word appeared first in the 1940s’ writings of architect Luigi Moretti, in which he wrote extensively regarding parametric architecture for which he defined as the study of architecture systems with the goal of “defining the relationships between the dimensions’ dependent upon the various parameters”.[8] It is quite arguable whether the term appeared first in the 40s’ or the 80s’. Despite possibly different interpretations and uses of the term between Moretti and Ruiter, the term can be found in the writings of other scholars that were done almost one hundred years prior to Luigi Moretti’s writings, such as mathematician Samuel Earnshaw in 1839, physicist John Leslie in 1821 and geologist James Dana in 1837. The paper On the Drawing of Figures of Crystals by Dana in 1837 is an example in which Dana explains the general steps of drawing a range of crystals by using language laced with parameters, variables, and ratios.[9] For instance, in step eighteen Dana asks the reader to inscribe a parametric plane on a prism by saying If the plane to be introduced were 4P2 (a permutation expression where 4P2 = 4×3 = 12) the parametric ratio of which is 4:2:1, we should in the same manner mark off 4 parts of e, 2 of e̅ and 1 of ë. (Dana 1837)[10]
The above quote made by Dana mainly describes the parametric relationship between three parameters of the plane (4:2:1) and their respective division of the three lines e, e̅ , and ë. Similar statements regarding how various parameters of crystal equations affect the drawing of assorted crystals can be found from the rest of Dana's paper. Today, which is 177 years after the time when Dana published his paper concerning his crystal equations, resemble those that would be used by us in the present day to develop parametrically generated architectural forms. 2.3 The definition of parametric design The debate, however, among scholars regarding whether parametric design should be perceived as a style or a method has long been discussed. Some designers adamantly consider parametric design to be a procedural approach that purely enables them to fasten the design process of their work. Mark Wigley, a New Zealand born architect argues that there is no such thing as parametricism and it is not a good description to explain what people have been doing in recent years.[11] Kas Oosterhuis, an Amsterdam born architect also argues that parametricism is not a style and it should not be regarded as such. His arguments against parametric design, which he considered a form of superficiality, could be perceived as a populist act for not respecting the underlying values of design.[12] According to the Chileanbased team of Chido Studio, parametric design is an abstract concept related to mathematical processes that allow us to, even more freely, control the precision of our designs to arrive at the most optimal result.[13] It is a computational thinking process and a methodical approach to the evaluation of generative systems. It enables the expression of parameters and rules that define the relationship between design intent and design response. Both parametric design and generative design at this point share a common attribute: utilizing algorithms to predict personal design preference. This imperceptibly makes “form follows function”
no longer the guiding principle in design; rather, the idea of “form follows programming.” Parametric design is critically considered a fast way of exploring design possibilities for its practice in various design disciplines such as architectural design, communication design, engineering design, interior design, graphic design and industrial design. One of the important aspects of parametric design is the generation of natural complex geometric patterns with declared parameters involved for designers or users to define.
3 | PARAMETRIC DESIGN AS AN APPROACH TO COMMUNICATE MATHEMATICS 3.1 Algorithm as a way of mathematical implementation In Elements of Parametric Design by Robert Woodbury, the beginning of the first chapter explains the need to possess a solid level of complexity of skills for designers to master parametric design. These skills are about combining the basic ideas of parametric systems with equally basic ideas from both mathematics and computer programming. We can argue that computer programming — a study where design is put into an executable computer program — is a field that falls within the discipline of computer science, whereas computer science — a field where one will learn about automating the algorithmic processes needed for a computer to function properly — is categorized as a branch of applied mathematics. The Dutch programmer and engineer Edsger W. Dijkstra, pioneer of computer science, published his 1979 paper On the Interplay Between Mathematics and Programming in which he addressed: Programming is one of the hardest branches of applied mathematics because it is also one of the hardest branches of engineering, and vice versa. (Dijkstra 1979)[14]
From a conceptual standpoint, this allows us to be daring in putting forward an opinion that use of computer algorithms means implementation of mathematics as there is mathematics involved in building algorithms. The two core technologies of information security used these days are public-key cryptography and digital signatures, which are indeed numerical algorithm based. Contributed by discrete mathematics employed to analyze program behavior, these technologies provide security services such as privacy, authenticity and integrity.[15] The study of algorithm design has qualitatively changed mathematics to the extent that questions regarding algorithms have become thoroughly intertwined with the mainstream of research in the fields of combinatorics and graph theory.[16] The history of the modern digital computer, interestingly enough, dates to the seventeenth century when mathematicians such as Napier (1614), Pascal (1642), Leibniz (1672) all built computing devices for the development of binary arithmetic, which is an essential part of all today’s computers that enables a computer to perform mathematical operations on binary numbers. Almost nothing was done regarding binary for a couple of centuries after works of the previously mentioned mathematicians until the birth of the world's first programmable, electronic, digital computer in 1938 by Konrad Zuse, for which the system used Boolean logic and binary floating-point numbers.[17] Pascal’s calculator, also known as the first workable arithmetic machine, is particularly important as the prototype of today’s mechanical calculators. The main ideas behind all those computing devices laid the theoretical foundation of the modern digital computer. Over and above that, discrete mathematics and logic are foundations for computer-based disciplines.[18] Woodbury made a remark in Element of Parametric Design with regard to having the difficulty to amalgamate ideas of parametric architecture with ideas from both mathematics and computing. There he explained in order to master parametric design one must be part designer, part computer scientist and part
mathematician. To this point, we can understand despite being able to use creativity and design thinking skills to highlight novelty and innovation, it is highly expected for designers of today’s world to use mathematics and programming to understand parametric design inasmuch as doing so can bring some design concepts into sharp focus. 3.2 Which mathematics? However, which area of mathematics are we directly referring to since mathematics is a broad and diverse field of study? Mathematics can mean algebra, calculus, geometry, theory of computing and optimization, and more. Despite the diversity of mathematics, it is a natural affinity for us to consider geometry to be the target subject as it is the most visually intuitive manifestation of mathematics. As how Helen Castle describes it in Architectural Design (AD) that geometry plays as a liaison role, which has held sway across continent and time, and is most often seen in architectural forms such as ancient shrines, Renaissance churches, Islamic structures and contemporary organic shapes and forms.[19] With the power of computing, we can even be more engaged in integrating geometrical content into fine art. The onset of computation has, however, offered us the chance not only to reconnect architecture with geometry and pursue the possibilities of non-Euclidean geometries, but also to realize the opportunities that other branches of mathematics, such as calculus and algorithms, afford. (Castle 2011)[20] According to Wolfram Mathematica, the study of Voronoi diagrams is classified as the study of triangulation, the core of which is formed by the study of computational geometry, the branch of mathematics and computer science concerned with finding efficient algorithms for solving geometric problems.[21] The Voronoi diagram in mathematics has been favorably
taken by designers as an inspiration to achieve interesting aesthetic and better computing efficiency. Numerous problems in artificial intelligence and computer graphics deal with spatiotemporal modeling that demand efficient management of objects in different dimension. In that case, the Voronoi diagram construction algorithm can provide better results in terms of robustness and performance. As a branch of computational geometry, the Voronoi diagram is an intriguing geometric structure that is widely used in various fields, particularly in computer graphics. It helps divide a plane into several adjacent and non-overlapping space regions for solving the spatial neighbor and other queries. The below given lampshade project demonstrates a parametric system involving the Voronoi diagram generated by Grasshopper 3D, an algorithmic modeling plugin developed by David Rutten for Rhinoceros 3D that uses a visual, componentbased programming language for easier and better human-computer interaction.
Fig 03: A lampshade inspired by the Voronoi diagram
“In most plants, phyllotactic patterns have symmetry — spiral symmetry or radial symmetry,” says University of Tokyo plant physiologist Munetaka Sugiyama. In his 2012 paper Phyllotactic Towers, architect Saleh Masoumi of Verk Studio proposed an inventive scheme by combining nature and design in both “the making of things” and “systems thinking.” The tower with a phyllotaxis look-alike structure,
shows its unique “open to the sky yard” arrangement similar to the leaf arrangement of a phyllotaxis, which can be defined mathematically by the formula known as the Fibonacci sequence. Since parametric design involves making a great deal of algorithmbased complex geometries, it would certainly be useful if it were also applied in other design disciplines.
3.3 Mathematical programming as a method to bring design to the next level As engineers growingly move product design and development into the computational realm, topology optimization becomes prominent and serves as a stepping stone for generative design to get straight to the best embodiment of a design.[23] Not only is topology optimization a
Fig 04/05: Masoumi’s phyllotactic towers
In 2007, Gernot Oberfell, Jan Wertel and Matthias Bär from Platform Wertel Oberfell used a mathematical algorithm to create a computer model to reproduce the fractal growth patterns of the dragon tree’s branching structure. The coffee table, also known as the Fractal-T or the Fractal.MGX, was manufactured as a single piece, without seams and joints, using stereolithographic 3-D printing modeling technology, an additive manufacturing technique that builds up and solidifies layers of resin.[22]
Fig 07: A.I. presentation at Salone del Mobile 2019
Fig 08: The A.I. Chair is a demonstration of how generative design is empowering modern designers to be versatile, innovative and highly engineering oriented
Fig 06: 3D Fractal Table by Wertel Oberfell
foundational technology on which generative design is built but also a necessary building block for generative design. It is noteworthy that in terms of philosophy, generative design and parametric design share a similarity of approach to design thinking — both of them are principally and predominantly based in thinking algorithmically. The dissimilarity between the two, however, is their design
intent; one uses declared parameters to define a form, the other mimics nature’s evolutionary approach to design.[23] Mathematical programming, the branch of applied mathematics principally concerned with the study of maximization and minimization of mathematical functions, has long played a significant role in structural engineering design, often known as structural optimization.[24] Based on concepts from mathematical programming and classical mechanics, structural optimization gained importance first in structural engineering whereas now it has matured to the point that it can be routinely applied to a wide range of real design problems, particularly in computer science and computeraided design. The study of structural optimization can be classified into dimensional, shape and topological optimization where each of them is associated with the three phases of the design process. The theory and implementation of dimensional and shape optimization have matured significantly for some time, whereas topology optimization is relatively more complex and hence more emphasized due to their technical challenges in the aeronautical industry and potential developments for the future.
finding the best profile for a topologically fixed structure under a wide range of constraints. In contrast to the shape optimization, topological optimization is a technique developed to determine the best paths transferring structural loads, which means it seeks the optimal distribution of the material within the design domain while satisfying the specified design constraints.[25] As a mathematical technique for addressing engineering design problems, topological optimization can be programmed using different algorithms. These algorithms in general are mathematical programming algorithm and bioinspired intelligent [26] algorithm. The traditional gradient-based mathematical programming algorithm seeks to search an algorithm that gives an optimal solution, whereas the bioinspired intelligent algorithm seeks to develop an algorithm that simulates evolutionary processes by making use of bio-inspired operators. Both approaches are widely adopted in architecture and product design.
Fig 09: Three major stages of the design process
The optimization function built in Autodesk’s Fusion 360 is structural shape optimization, which performs different tasks in comparison with the topology optimization built in Autodesk’s Nastran In-CAD. Shape optimization is predominantly concerned with
Fig 10/11: Used about 112 kilometers of the biopolymer in production, ACTLAB demonstrates how 3D printing and algorithms can meet to create truly stunning architecture
The question about whether generative design will be the future of manufacturing or otherwise has been discussed extensively in recent years as industry 4.0 gains momentum in the manufacturing world. We are currently witnessing how designers and engineers integrate the merging of computing skills and computational thinking concepts into product design and development. After years of development as Project Dreamcatcher, Fusion 360 Ultimate by Autodesk made its commercial debut in 2018, in which the new topology optimization and generative design features allow designers and engineers to filter through and choose the outcomes that best meet their stated requirements. Topology optimization is a mathematical method based on finite element analysis for the development of optimization considering design parameters such as expected loads, available design space, material distribution and cost. This mathematically based technology aids designers to germinate ideas that would be unattainable by conventional methods due to its high geometric complexity resulted from the technology. With the increased usage of this technology, the need and expectation for additive manufacturing becomes a lot more tangible than they were in the past.
Fig12: Parametric form finding by embedding algorithmic design methodologies in the form finding process
We can consider parametric modeling as a generative design tool of how mathematics
is employed in design, for showing the beauty of mathematics by utilizing it creatively to solve design problems. In the days ahead, the influence of generative design on product design development is expected to grow rapidly since a vast array of design firms is poised to capture a competitive edge by investing in this advancing technology with the aid of digital fabrication.
4 | BENEFITS OF PARAMETRIC DESIGN AND DIGITAL FABRICATION Thus far, we are aware that parametric design allows complex topological form generation, which not only does it provide visual impact but also offers various design possibilities. This enables designers to make optimal selection for clients. Furthermore, parametric method has great potentials in reducing waste at the design stage, bringing structural optimization in product design and thereby helps achieve weight, material and cost reduction. Parametric method also benefits user input, which leads to the availability of customization. For the reason that each individual has unique intent needs, parametric design provides three advantages: benefiting clients for inputting their personal choice of parameters to achieve optimal satisfaction, benefiting designers for optimizing design process, benefiting business organizations for large-scale production and reducing transportation cost. With all these factors and the advance and help of additive manufacturing, parametric design is gradually proving to be the future of manufacturing and creativity. In architectural and furniture design, the most commonly seen parametric structures are the shape of interpolated contours, the Voronoi-Delaunay duality, and the metaball forms. Besides having smoother flowing shapes and very few hard edges, modeling and
fabrication of complex organic models are oftentimes a challenge for designers and makers. This is when computational approach and 3-D printing become effective. Algorithm-
strength is needed and the thinner where it is not. Meanwhile, material distribution, weight and stability are all optimized while material quantity is minimized in the bone chair.
5 | DISCUSSION AND CONCLUSIONS
Fig13: Forms generated by using interpolation and meatball
driven design tool such as Grasshopper 3D allows the creation of complex shapes using generative algorithms for minimizing the difficulty of high-precision fabrication. Rather than explicitly building 3D models by degrees as is done in conventional 3D modeling tools, Grasshopper performs uniquely as a visual algorithm editor tightly integrated with Rhinoceros 3D for generating 3D models. It uses a tremendous amount of vector geometry in programming to develop a 3D configuration in a systematic manner, often far more complex than could ever be attempted by hand. Parametric furniture formed by the contour interpolation method, for instance, is easy to fabricate and assemble. The most used material for making such furniture is plywood due to its ease of fabrication of curved surfaces and its high uniform strength. Jens Dyvik’s work, at this point, can be good examples. A chair generated by the Voronoi patterns is another example of showing weight and material reduction. The most used materials for making Voronoi structures are polycrystalline alloys, cellular foams, geo materials, trabecular bone, and other materials that benefit weight and material reduction. The famous bone chair designed by Joris Laarman is another generative form of a piece of furniture that uses an algorithm to translate the complexity of human bone and tree growth into a chair. The thicker parts of the bone chair are where the
Overall, as a scientific method, parametric design evidently provides a fair number of advantages in product design. Whether it be parametric, computational, algorithmic or generative design, all of which have the principle of being a process of creating an algorithm, script, or application to create designs. It is a process that consummately takes advantage of the computer’s processing power to create designs that would almost be impossible to do without one. Computers have been traditionally used for the visual design and drafting process, now they are being used to augment the design process (see Beijing National Stadium by Herzog & de Meuron for further reference). In all above-mentioned studies, we conclude that parametric design, particularly in the generative design realm, is changing the face of architecture and product design as it allows designers to control intellectual forces (e.g., computers) many times more powerful than human minds to design and construct products that could not otherwise exist. With the recent democratization of manufacturing, especially under the advance of personal digital fabrication, designers must begin viewing computers as portals to greater exploration rather than viewing them as merely drawing tools. It is more important to know that we have mathematical laws as the language of nature that can be implemented via algorithms to provide fast and visualized solutions to design. We also have generative design as a methodical tool that can mimic nature’s approach to design. Generative design is currently addressing two ideas: optimization of mechanical properties and the creation of an
interesting aesthetic. Technologies such as Autodesk Within, Nastran and Fusion 360 follow the optimization track. As for the matter of aesthetic, Autodesk’s Dynamo Studio and McNeel’s Grasshopper 3D fall into this category and we have witnessed the creation employed in a number of architectural and furniture design projects. By means of algorithms, we are able to use computer for its high computing power to produce complex geometries, and at the same time, allowing less material consumption to achieve cost and time reduction.
[03] Nikos Angelos Salingaros, Michael W. Mehaffy, “A Theory of Architecture”, UMBAU-VERLAG Harald Püschel, pp.134, 2006
We clearly can see trails of mathematics in parametric design. In comparison with manual modeling, parametric modeling gives us the ability to program every aspect of the design modeling process, which allows the user to control each data set to make design changes based on other parameters such as distance or space. In contrast to parametric modeling, manual modeling builds models by drawing curves in sketches and building bodies from these sketches using modeling commands. Parametric modeling unequivocally overcomes the limitations of what manual modeling has brought to us. Combining with rapid prototyping technology, parametric modeling technique shows more advantages in terms of prototyping. Through parametric design, not only can we explore mathematics, we can also be more engaged in understanding the beauty and values of mathematics in a visually practical manner.
[06] The Editors of Encyclopaedia Britannica, “Parameter”, Mathematics and Statistics,
[04] Douglas Harper , “Parameter”, Online Etymology Dictionary, www.etymonline.com/word/parameter, 2001-2020
[05] Andrew D. Loveless, “Math 124: Calculus Lecture Note on Parametric Equation”, University of Washington Department of Mathematics, 2017
www.britannica.com/topic/parameter
[07][08][09][10] Daniel Davis, “Modelled on Software Engineering: Flexible Parametric Models in the Practice of Architecture”, Encyclopædia Britannica, Inc. pp.18-19 [11] Mark Wigley, “Instead of saying I want the banana”, What is parametricism? Vol. 2, Issue 4. www.co-l-on.com/2/Colon%20Volume%202%20 Issue %204.pdf, 2017. [12] Kas Oosterhuis, “Parametric design is not a style”, 019, http://www.oosterhuis.nl/?cat=49 [13] Koenraad Willems, “Create the Future: what is Parametric Design?”, Cloudalize, www.cloudalize.com/blog/create-the-future-what-isparametric-design/, 2019
Books, websites, articles
[14] Dijkstra E.W., “On the Interplay Between Mathematics and Programming”, 1979
[01] Autodesk: “GENERATIVE DESIGN”,
[15] Ninghui Li, “CS526: Information Security,
REFERENCES
www.autodesk.com/solutions/generative-design, 2020
[02] Leilei Chu, Tianjun Wang, Suiyang Chen, “Contemporary Mathematics and Cultures of Computing”, Beijing Institute of Technology Press, pp.40
topic 6”, Purdue University, 2014
[16] Timothy Gowers, June Barrow-Green, “The Princeton Companion to Mathematics”, The Princeton Companion to Mathematics, 2010, pp.875
[17] Jerry M. Lodder, “Binary Arithmetic: From Leibniz to von Neumann”, The Cyberspace Handbook By Jason Whittaker, New Mexico State University, 2011, pp.75
[18] Paul A. FreibergerMichael R. Swaine, “Pascaline”, ] ENCYCLOPÆDIA BRITANNICA, www.britannica.com/technology/Pascaline
[19][20] Helen Castle, Editorial, “MATHE+MATICS OF SPACE” , Architectural Design, Profile No. 212, July-Augusr 2011, pp.5
[21] Eric W. Weisstein, "Voronoi Diagram", MathWorld: A Wolfram Web Resource, mathworld.wolfram.com/VoronoiDiagram.html
[22] Materialise, 3D Printing Innovators, https://www.materialise.com/en/mgx/collection/fractalmgx
[23] AMFG, “Generative Design and 3D Printing: The Manufacturing of Tomorrow”, 2018, www.amfg.ai/2018/10/25/generative-design-3dprinting-the-manufacturing-of-tomorrow/?cnreloaded=1&cn-reloaded=1
[24] Xiao Lai, Zheng He, “Generalized parametric structural design: Framework and perspective”, Dalian University of Technology, 2019
[25] Yukang Sui, Xirong Peng, “Modeling, solving and application for topology optimization of continuum structures”, Elsevier Inc., 2018, pp. Preface
[26] Aykut Kentli, “Topology Optimization Applications on Engineering Structures”, www.intechopen.com/books/truss-and-frames-recentadvances-and-new-perspectives/topology-optimizationapplications-on-engineering-structures, 2019 Pictures, photographs Fig 01: www.onlinemathlearning.com/imagefiles/xquadraticfunction.png.pagespeed.ic.nSV1dtTw7f.webp
Fig 02: www.mathsisfun.com/algebra/images/circleparametric.svg Fig 03: Personal project done in school during the master’s program Fig 04: www.solaripedia.com/images/large/6201.jpg Fig 05: www.solaripedia.com/images/large/6172.jpg Fig 06: s3files.core77.com/blog/images/fractal_table.jpg Fig 07: covetedition.com/wpcontent/uploads/2019/04/7-Kartell-The-First-ChairCreated-by-Artificial-Intelligence-620x520.jpg
Fig 08: media.madeindesign.com/nuxeo/products/8/c/stackablearmchair-a-i-metallicgrey_madeindesign_330885_original.jpg
Fig 09: www.researchgate.net/publication/282419796/figure/fig 13/AS:325713233236000@1454667494965/Threelevels-of-structural-optimization.png
Fig 10: www.actlab.net/uploads/2/6/2/9/26291501/figure-5_orig.jpg
Fig 11: www.actlab.net/uploads/2/6/2/9/26291501/figure17_orig.jpg
Fig 12: Personal project done in postgraduate school during the first semester: a project aimed for the 2014-2015 RSA design competition Fig 13: https://i.ytimg.com/vi/LS0nLTAsqQM/maxresdefault .jpg