Designing with Algorithms A Mathematical Guide (Arash Soleimani) (z-library.sk, 1lib.sk, z-lib.sk)

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“Designing with Algorithms: A Mathematical Guide takes a foundational approach to computational thinking and algorithmic design. Dr. Soleimani begins with clear explanations of how algorithms work, the basic types of algorithms, and their sys- tems of notation. He then uses this framework to move from the algorithm of cheese and tomato sandwich to explanations of parametric color and mathematical transformations. Each chapter builds on the previous allowing for the foundations set in the early chapters to enable the understanding of the increasingly complex topics such as fractals and diffusion. The focus on algorithms and mathematics over software-specifc instruction allows this book to work in a variety of settings and applications. I would highly recommend this book for faculty looking to provide their students with a strong foundation in algorithmic design and for professionals looking to evolve their practice.” —Mark Ericson, Professor of Architecture, Woodbury University “In a world increasingly defned by complexity, where systems are interconnected yet often unpredictable, Designing with Algorithms brings order to the chaos. Arash Soleimani invites designers from all disciplines to explore where creativity, math- ematics, and computation converge, unlocking the power of algorithmic design. From parametric geometries to chaotic systems, this book teaches not just tech- niques, but a way of thinking that will reshape creative activity.” —Dr. Keith Evan Green, McLean Professor of Human Centered Design and Professor of Mechanical and Aerospace Engineering, Cornell University

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Designing with Algorithms Computing is revolutionizing the creative process, transforming how designers approach ideation, prototyping, fabrication, communication, and representation. Yet many lack the foundational knowledge needed to fully harness its potential. Designing with Algorithms: A Mathematical Guide bridges this gap, offering a comprehensive, step-by-step exploration of algorithmic practices applicable across diverse design felds. This guide begins with foundational theories, illuminating the role of mathemat- ics and computational logic in design thinking. It progresses to practical techniques for creating parametric 2D and 3D geometries, grid systems, patterns, forms, and fonts. The book culminates in real-world case designs, showcasing diverse applica- tions of generative algorithms across urban design, architecture, graphic design, typography, fashion, and more. Unlike software-specifc guides, this book empha- sizes process and adaptability, teaching readers to think algorithmically and develop skills transferable to any programming platform. Whether you’re a student, practitioner, or enthusiast, this book empow- ers you to future-proof your design practice and unlock the creative potential of computation. Arash Soleimani is Professor and Chair of Design Computation and Applied Computer Science—Media Arts at Woodbury University’s School of Architecture in Los Angeles, California. He holds a multidisciplinary PhD in Architectural Robotics from Clemson University’s School of Architecture, where his research intersected electrical and computer engineering, materials science and engineering, educa- tion, and human factors psychology. He also earned a Master of Architecture from the University of Nottingham, UK, with a focus on Responsive and Interactive Environments in Museums—a subject on which he published a book. His academic journey in design began with a Bachelor of Architecture from Isfahan University of Art, Iran.

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Designing with Algorithms A Mathematical Guide Arash Soleimani

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Designed cover image: © Arash Soleimani First published 2026 by Routledge 4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 605 Third Avenue, New York, NY 10158 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2026 Arash Soleimani The right of Arash Soleimani to be identifed as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifcation and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Soleimani, Arash author Title: Designing with algorithms : a mathematical guide / Arash Soleimani. Description: Abingdon, Oxon ; New York, NY : Routledge, 2026. | Includes bibliographical references and index. | Identifers: LCCN 2025014532 | ISBN 9781032704432 hardback | ISBN 9781032704395 paperback | ISBN 9781032704463 ebook Subjects: LCSH: Design--Mathematics Classifcation: LCC NK1510 .S667 2026 | DDC 744.0285/513--dc23/eng/20250625 LC record available at https://lccn.loc.gov/2025014532 ISBN: 9781032704432 (hbk) ISBN: 9781032704395 (pbk) ISBN: 9781032704463 (ebk) DOI: 10.4324/9781032704463 Typeset in Univers by KnowledgeWorks Global Ltd.

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Contents Foreword x About the Author xii Acknowledgments xiii Organization of the Book xv Reader’s Guide xvii 1 Algorithmic Thinking: Redefning Design Through Computation 1 Theories and Concepts in Algorithmic Design 2 Algorithmic Design Thinking: The Designer’s New Tool 7 Logics and Applications of Scripting in Design 8 The Why and How of Algorithmic Design 9 2 Parametric Geometries 11 Why Parametric Geometry? 12 Curves 14 Triangles 15 Quadrilaterals 18 Polygons with n Sides 20 3 Transformative Geometries 24 Superimposed Composites 26 Overlay 35 Morphing Geometries: From Static to Dynamic Forms 45 Singular Transformation 45 Collective Transformation 49 4 Rule-Based Grid and Pattern Systems 52 Metric-Based Grids: The Invisible Skeleton of Design 53 Modulation: The Art of Control in Design 66 Attractors: Magnetic Fields of Form 75 Point Attractors 78 Line/Curve Attractors 79

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■ Contents Area/Shape Attractors 79 Dynamic Attractors 80 Parametric Colors: Painting with Algorithms 87 5 Transformative Patterns 98 Superimposed Patterns: Unveiling Layers of Complexity 101 Morphing Patterns: The Evolution of Design Over Time 105 The Element of Time in Morphing Patterns 106 Singular Transformation 110 Collective Transformation 111 6 Multiverse of Mathematical Forms 120 N-Dimensionality: Beyond the Familiar Space 120 Polytope Theory: The Architecture of Multidimensional Spaces 124 Fusion Polytopes: Blending Complexity into Harmony 126 Fractal Geometries: The Infnite Beauty of Recursion 129 Iterated Function Systems (IFS): Creating Through Repetition 129 Escape Time Algorithms: Mapping the Infnite 142 Exploring the Depths of 3D Fractals: Beyond the Surface 149 Chaos Theory: Dancing on the Edge of Order 157 Strange Attractors: The Chaotic Sculptors 157 7 Case Designs 169 Algorithmic Typography: Letters as Dynamic Forms 169 Parametric Typography 170 Procedural Typography 173 Responsive Typography 176 Islamic Patterns: Where Algorithm Meets Spirituality 179 Girih Patterns 180 Muqarnas: Geometry, Structure, and Light 182 Zellige: The Art of Mathematical Precision 186 Calligraphic Integration 189 Islamic Patterns in Higher Dimensions 190 Transformative Textiles: Weaving New Realities 192 Plain Weave 193 Twill Weave 195 Multi-Objective Beam Optimization: Structural Programming 197 Longitudinal Section Optimization Using Moment and Shear Diagrams 198 Cross-Section Optimization Using Defection Models 199 Cross-Section Optimization with Integrated Perforation Systems 201 3D Optimization Using Parametric Curving 202 Urban Fabrics: Designing the City of Tomorrow 203 Historical Fabrics 204 Hybrid Fabrics 206 viii □

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Contents ■ Modernist Fabrics 209 Organic Fabrics 213 Simulation: The Art of Data Visualization 217 Light Simulation 218 Sound Simulation 221 Water Simulation 224 Wind Simulation 227 Human Activity Simulation 230 Conclusion: The Future of Algorithmic Design 234 Index 236 ix □

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Foreword This century’s standard practice for design creation has been drawing and modeling using computer-driven methods. The lowest design production level is keyboard-and- mouse entry of points, lines, and planes. Researchers refer to this method of design production as digital design. High-level design production is accomplished through scripting and computer programming. Here, a designer creates drawings and models using a rule-based generative method encoded within a script or program. We refer to high-level computing as automated design or design computation. Low-level design is slow yet fexible, allowing designers to model signifcant variations. A good design model is composed of a wide variety of 3D objects in detail across all dimensions and locations in the model. High-level design offers a fast way to create drawings and models. Unfortunately, design automation is not as fexible as digital design. Dr. Arash Soleimani’s text, Designing with Algorithms, is a proper start to the enterprise of computational design. Soleimani brings a notable perspective to this feld, grounded in a cross-disciplinary PhD in Architectural Robotics and a practice that spans design, computation, and media arts. The text begins with conversations on the theory of computation and the many ways it can be included in design. His book concludes with algorithms and expressions of the hundreds of ways design algorithms can be applied to real-world design problems. These methods of production include parametric examples such as rules, heuristics, and patterns. Another application shows functional examples of algorithms used to create urban patterns, illustrating key methods of com- putation, fabrication, and working with functional models. The text presents various possibilities, from the visual to the physical and, in many ways, the spiritual. Designers make numerous decisions. An architect considers the form and details of a building. The lead designer formulates ideas on paper and as models in collaboration with clients, engineers, bankers, and other designers. Various factors infuence the fnal product throughout the design process. New questions regard- ing the future of decision-making in the design feld are emerging. These concepts and inquiries relate to the role of AI. Utilizing computation as a decision-maker is complex. AI, as a factor in design decision-making, shifts the conversation. Digital design is keystroke 2D drawing and 3D modeling used to represent our ideas. Practically, it is the lowest level of data entry. Design students, interns, and emerg- ing professionals learn digital design in school. Keystroke modeling is an arduous chal- lenge of learning how to create geometry. Learning geometric modeling in school is even more challenging for anyone unfamiliar with design. A novice designer with basic skills in 3D modeling understands the benefts of scripting and algorithmic design. Generative systems make design fun. They are also fast ways to develop very complex geometry.

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Foreword ■ Powerful algorithms, such as fractals used to generate trees and plants compu- tationally, are game-changing for landscape design. Fractal programming makes the generation and visualization of biological constructs possible. A possible future is the creation of algorithmic systems that generate building elements in shapes and forms similar to those in nature. Complex geometrical models of building elements are challenging to generate by keystroke commands. Double-curvature surface models, curved edges, and corners elevate the challenge. Modeling objects com- posed of small honeycombs, cellular forms, and interlocking mechanical descrip- tions is hard. They are examples of a complex future. Soleimani’s text demonstrates how a designer can generate systems such as frac- tals, iterated functional systems (IFS), and recursion. Designing with Algorithms presents an option for generating, seeing, and making our ideas through the creation of scripts. The beneft is speed. The designer can iterate through many drawings, renderings, ani- mations, and information for machining in seconds. Jenny Sabin’s research group is an excellent example of the potential of generative systems. They have applied generative systems to produce PolyForm, an interactive digitally fabricated outdoor pavilion com- posed of four large crystalline forms built of waterjet-cut sheet material. Computational methods involve decomposing the form into an assembly of planar elements. Soleimani demonstrates the relationship between math and visual design with drawings. He illustrates how to transform symbolic logic spatially with procedures. The results are large shapes and forms composed of many repeating symbols, in particular with regular polygons. His work with polytopes moves beyond three- dimensional space, encouraging designers to explore higher-dimensional constructs. He also shows how to compute shapes frst with fxed points, edges, and planes. Throughout the book are demonstration models of complex operations. Soleimani’s examples of Mandelbrot patterns, Julia sets, and AI-generated fractals illustrate the incredible and unimaginable forms that scripted algorithms can produce. Computers can manipulate large amounts of geometry very well. We should not be fooled into believing that fast generative systems are innovative or helpful. They cannot understand or reason through the physical world. Computer programs and AI systems are weak at inferring and discovering a future. Computers can build at some level of anticipation with existing information. There exist examples of software used to generate city communication. AI works well with text because it manages a fxed number of characters, words, and symbols. On the other hand, the spatial world of design is endless. There are many ways to add meaning to points, lines, and planes within a design problem. We do not know how to program robots to see, move, or manipulate objects without much input from the user. A possible future for design computation is creating a specifc visual language. Developing algorithms that meet aesthetic goals is also a concern. The future of design may look more like symbol-making, manipulation, and math. These elements will solve problems related to labor, con- structability, materials reuse, and energy. The impact will be a better world. Soleimani’s work is a welcome guide for this future. His book provides not only the mathematical tools but also the philosophical grounding to shape the next gen- eration of algorithmic designers. It is both a technical resource and a visionary text. —Dr. Lawrence (Larry) Sass, Professor and Chair, Computation Group MIT School of Architecture and Planning xi □

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About the Author An educator, architect, and computational designer, Dr. Arash Soleimani is Professor and Chair of MS Architecture, Design Computation, and Applied Computer Science—Media Arts in the School of Architecture at Woodbury University in Los Angeles, California. Dr. Soleimani holds a multidisciplinary PhD in Architectural Robotics from Clemson University’s School of Architecture, where his research bridged elec- trical and computer engineering, materials science and engineering, education, and human factors psychology. He also earned a Master of Architecture from the University of Nottingham, UK, with a focus on Responsive and Interactive Environments in Museums, a subject on which he published a book. His academic foundation in design began with a Bachelor of Architecture from Isfahan University of Art, Iran. Dr. Soleimani has received multiple awards and research grants for his contribu- tions to design computation, with his work showcased at international conferences and community-driven design initiatives. Before joining Woodbury University, he taught at Kennesaw State University, the University of North Carolina at Charlotte, and Clemson University. As an educator, he continues to push the boundaries of design innovation, integrating computational methodologies and emerging tech- nologies to shape the next generation of designers.

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Acknowledgments Writing this book has been a journey—one that has been shaped by countless indi- viduals who have inspired, challenged, and supported me along the way. Algorithmic design is about connections, about patterns that emerge through iteration and collaboration. In many ways, this book is a refection of those very principles—an accumulation of ideas, conversations, and shared moments with those who have infuenced my thinking and practice. First and foremost, I extend my deepest gratitude to Dr. Keith Evan Green, my PhD advisor, whose mentorship and intellectual generosity have been instrumen- tal in shaping my path. Keith, your guidance not only sharpened my approach to research but also instilled in me the importance of interdisciplinary exploration. Your ability to weave together architecture, computation, and human-centered design has left an indelible mark on my work. Thank you for your continued support and mentorship. A special thank-you to Seba Alabdullatif, a gifted multimedia designer who played an essential role in shaping the graphics of this book. Seba was my student at Woodbury University, where she earned her Bachelor of Architecture with a minor in Applied Computer Science–Media Arts, before continuing her studies at New York University, earning a master’s degree in Integrated Design and Media. From academia to practice, Seba has been with me on this journey, and her creative vision, technical expertise, and dedication have elevated this book’s visual narrative. I am grateful for her talent and commitment to this work. To my advisors, mentors, and peers, who have fueled my curiosity and chal- lenged my perspectives, I am deeply grateful. Your insights, critiques, and encour- agement have been essential in refning the ideas presented in this book. The academic and design communities I have been fortunate to be a part of—both past and present—have continuously reminded me of the power of collaboration and intellectual exchange. To my colleagues, both at Woodbury University and beyond, thank you for the rich discussions, shared projects, and the camaraderie that makes this profession so rewarding. Your dedication to education and research is a constant source of motivation. To my students—past, present, and future—this book is, in many ways, for you. Teaching has always been a reciprocal process, and I have learned just as much from your questions and ideas as I hope you have learned from my guidance. Seeing you embrace computational thinking, challenge conventions, and develop

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■ Acknowledgments your own voices as designers has been one of the most fulflling aspects of my career. Your creativity and ambition are a reminder of why this feld continues to evolve and inspire. To my publishers and the entire editorial team, thank you for believing in this work and helping me shape it into its fnal form. Your patience, feedback, and atten- tion to detail have been invaluable. Finally, to my family and loved ones, thank you for your unwavering support, patience, and understanding throughout this process. Writing a book is as much a personal endeavor as it is an intellectual one, and your encouragement has been my foundation. You remind me that while algorithms can shape forms, it is human connection that gives them meaning. This book stands as a testament to the many voices, minds, and experiences that have infuenced me over the years. To everyone who has been part of this journey—thank you. xiv □

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Organization of the Book Designing with Algorithms is structured to take readers on a journey through the world of algorithmic design, from foundational theories to advanced applications. The book is divided into seven chapters, each of which focuses on a different aspect of algorithmic design. Chapter 1 – Algorithmic Thinking: Redefning Design Through Computation This chapter sets the foundation by exploring the theoretical underpinnings of algorithmic design. It delves into how mathematics forms the backbone of our uni- verse and how these principles can be harnessed in design thinking. By looking at pivotal works such as Max Tegmark’s Our Mathematical Universe and Greg Lynn’s Animate Form, the chapter builds a compelling argument for integrating algorithmic logic into creative practices. The focus shifts to scripting and its applications, intro- ducing the reader to generative, evolutionary, predictive, and analytical algorithms. These are not just tools but new methodologies for design, redefning the creative process in both artistic and functional domains. Chapter 2 – Parametric Geometries Parametric geometry is explored here as a dynamic system that transforms static design into fuid, evolving forms. The chapter starts with basic geometric shapes—curves, triangles, and polygons—and extends into complex multisided fgures, all of which are described through parametric equations. By incorporating time and other variables, parametric design pushes the boundaries of traditional geometry, enabling a multitude of possibilities. The chapter explains how these parametric forms can shift from the paper-and-pencil stage into algorithm-driven creativity, making geometry an active participant in the design process rather than a static element. Chapter 3 – Transformative Geometries Building on the principles introduced in Chapter 2, this chapter focuses on transforming basic geometric shapes into more complex structures using time- based methods. Techniques such as shape morphing and superimposition are intro- duced to show how simple forms evolve into intricate geometric composites. The emphasis here is on the dynamic interaction of time, space, and topology, dem- onstrating how design can move from static states to fuid, evolving forms that respond to external forces and user inputs. Chapter 4 – Rule-Based Grid and Pattern Systems Grids and patterns have long been used to bring order to design, but this chap- ter takes them into a new dimension—parametric systems. Through rule-based

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■ Organization of the Book methods, designers can now manipulate grid structures to generate not just order, but dynamic, adaptive systems that respond to changing parameters. The introduc- tion of attractors and color parametrics provides an entirely new level of control over the visual and spatial outcomes of the design, showing how patterns can be transformed in real time based on user interactions or external stimuli. Chapter 5 – Transformative Patterns Patterns, much like geometries, can undergo transformation. This chapter delves into the evolution of patterns through techniques such as overlapping and morphing, where complex new patterns emerge from simple origins. Drawing inspiration from biological processes such as morphogenesis, this chapter con- nects the dots between natural patterns and computational design, providing the reader with a deep understanding of how algorithmic approaches can mimic the organic growth found in nature. Chapter 6 – Multiverse of Mathematical Forms This chapter is a deep dive into multidimensional mathematical forms, shaped by fractal geometry, chaos theory, and escape time algorithms. Beginning with n-dimensionality and polytope theory, the chapter takes readers on a journey beyond the familiar, exploring complex mathematical spaces. The fusion polytope concept is introduced as a new frontier in geometry, merging multiple forms into unifed structures. The chapter then shifts to fractals and chaotic systems, exploring how their infnite complexity can inform design practices. Strange attractors and 3D fractals bring the reader into the realm of dynamic, nonlinear transformations that open up new possibilities for form-making. Chapter 7 – Case Designs This fnal chapter presents real-world examples and case designs that apply the algorithmic principles discussed throughout the book. From algorithmic typography and Islamic geometric patterns to the design of transformative textiles and urban fabrics, each case demonstrates the practical applications of these advanced math- ematical and computational techniques. The examples highlight how algorithmic design can lead to innovative, functional, and aesthetically compelling solutions in various felds—from graphic design to architecture and urban planning. Each chapter is structured to progressively advance the reader’s knowledge, transitioning from foundational concepts to complex applications. Despite this pro- gression, the chapters are designed to stand alone, enabling readers to delve into any topic independently. At the conclusion of Chapters 2 through 6, a comprehen- sive assignment is provided, allowing readers to evaluate their grasp of the material covered in each chapter. xvi □

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Reader’s Guide This book is designed for a broad audience, including students, architects, design- ers, and anyone interested in the intersection of mathematics, computation, and design. Whether you are new to algorithmic design or an experienced practitioner, this book offers insights and techniques that will help you integrate algorithms into your design process. For readers new to the feld, I recommend starting with the early chapters, which provide a comprehensive introduction to the key concepts and strategies of algorithmic design. These chapters will help you build a strong foundation in the mathematical and computational principles that underlie the design processes explored in later chapters. For more advanced readers, the later chapters offer a deep dive into specifc techniques, examples, and case designs, demonstrating how algorithmic design can be applied in a wide range of contexts. These chapters are particularly useful for designers looking to apply algorithmic thinking to their own projects.

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DOI: 10.4324/9781032704463-1 1 Algorithmic Thinking Redefning Design Through Computation Design felds are undergoing a profound transformation as they embrace the poten- tial of algorithmic design. The shift from traditional hand-drawing or even computer- aided design (CAD) to algorithmically driven processes represents a fundamental change in how designers approach problem-solving. This transformation is not sim- ply technological but philosophical, marking a new paradigm in which creativity, mathematics, and computation converge. In this era, designers no longer work within static frameworks but instead engage with dynamic systems capable of evolving, learning, and optimizing. Algorithmic design, in essence, is the practice of harnessing the power of algo- rithms to create, automate, generate, evaluate, and optimize design solutions. These algorithms act as both tools and collaborators in the design process, explor- ing vast solution spaces that would be impossible to navigate manually. At the heart of this shift is a reconsideration of the role of the designer. Traditionally, designers have relied on intuition and experience to create forms and structures. While these qualities remain essential, they are now augmented by algorithms capable of processing vast amounts of data and generating solutions that are not only effcient but also innovative and unexpected. This new approach to design requires a different kind of thinking—one that is grounded in both the logic of computation and the aesthetics of form-making. Algorithmic design invites us to see the world through the lens of mathemat- ics and parameters. Max Tegmark’s Our Mathematical Universe offers a compel- ling vision of a world where reality itself is a mathematical structure. According to Tegmark, everything—from the smallest particles to the largest galaxies—can be described using mathematical equations. This idea resonates deeply with the principles of algorithmic design, where form and function are derived from math- ematical relationships. By understanding design as a mathematical process, we can begin to see patterns and structures in new ways. The algorithms that drive design processes today are rooted in this mathematical understanding, allowing designers to explore forms that are not only aesthetically pleasing but also highly effcient. Whether we are designing a city, a building, a chair, a poster, or a font, the same mathematical principles apply, enabling us to create designs that are optimized for both perfor- mance and beauty. 1 □