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Fourier Transforms
by Eric W. HansenFourier Transforms: Principles and Applications explains transform methods and their applications to electrical systems from circuits, antennas, and signal processors--ably guiding readers from vector space concepts through the Discrete Fourier Transform (DFT), Fourier series, and Fourier transform to other related transform methods. Featuring chapter end summaries of key results, over two hundred examples and four hundred homework problems, and a Solutions Manual this book is perfect for graduate students in signal processing and communications as well as practicing engineers.
Fourier Transforms, Filtering, Probability and Random Processes: Introduction to Communication Systems (Synthesis Lectures on Communications)
by Jerry D. GibsonThis book provides backgrounds and the mathematical methods necessary to understand the basic transforms in signal processing and linear systems to prepare for in depth study of analog and digital communications systems.This tutorial presentation provides developments of Fourier series and other orthogonal series, including trigonometric and complex exponential Fourier series, least squares approximations and generalized Fourier series, and the spectral content of periodic signals. This text thoroughly covers Fourier transform pairs for continuous time signals, Fourier transform properties, and the magnitude and phase of Fourier transforms. The author includes discussions of techniques for the analysis of continuous time linear systems in the time and frequency domains with particular emphasis on the system transfer function, impulse response, system/filter bandwidth, power and energy calculations, and the time domain sampling theorem.
The Fourth Dimension: Toward a Geometry of Higher Reality
by Rudy RuckerThis is the definitive popular exploration of what the fourth dimension means, both physically and spiritually. Mathematician and science-fiction novelist Rudy Rucker takes readers on a guided tour of a higher reality that explores what the fourth dimension is and what it has meant to generations of thinkers. The exciting and challenging journey is enhanced by more than 200 illustrations and a host of puzzles and problems (with answers)."This is an invigorating book, a short but spirited slalom for the mind." -- Timothy Ferris, The New York Times Book Review "Highly readable. One is reminded of the breadth and depth of Hofstadter's Gödel, Escher, Bach." -- Science"Anyone with even a minimal interest in mathematics and fantasy will find The Fourth Dimension informative and mind-dazzling... [Rucker] plunges into spaces above three with a zest and energy that is breathtaking." -- Martin Gardner "Those who think the fourth dimension is nothing but time should be encouraged to read The Fourth Dimension, along with anyone else who feels like opening the hinges of his mind and letting in a bit of fresh air." -- John Sladek, Washington Post Book World"A mine of mathematical insights and a thoroughly satisfying read." -- Paul Davies, Nature Magazine
The Fourth Dimension Simply Explained
by Henry P. ManningTo remove the contents of an egg without puncturing its shell or to drink the liquor in a bottle without removing the cork is clearly unthinkable -- or is it? Understanding the world of Einstein and curved space requires a logical conception of the fourth dimension.This readable, informative volume provides an excellent introduction to that world, with 22 essays that employ a minimum of mathematics. Originally written for a contest sponsored by Scientific American, these essays are so well reasoned and lucidly written that they were judged to merit publication in book form. Their easily understood explanations cover such topics as how the fourth dimension may be studied, the relationship of non-Euclidean geometry to the fourth dimension, analogues to three-dimensional space, some four-dimensional absurdities and curiosities, possible measurements and forms in the fourth dimension, and extensive considerations of four-dimensional space's simpler properties.Since each essay is independently conceived, all of the writers offer fresh viewpoints and original examples. Because of this, some of the most important principles relating to the fourth dimension are viewed from several different angles at once -- an invaluable aid to visualizing these abstruse but fascinating ideas. New Introduction by Thomas F. Banchoff, Brown University. 82 figures.
Fourth Grade Math with Confidence Instructor Guide (Math with Confidence #0)
by Kate SnowTeach Fourth Grade Math with Confidence! This scripted, open-and-go program from math educator Kate Snow will give you the tools you need to teach math with confidence–even if you’ve never taught math before. Engaging, hands-on lessons will help your child develop a strong understanding of math, step by step. This scripted, open-and-go program from math educator Kate Snow will give you the tools you need to teach math with confidence–even if you’ve never taught math before. Engaging, hands-on lessons will help your child develop a strong understanding of math, step by step. · Multiplication with multi-digit numbers · Long division · Equivalent fractions · Adding and subtracting fractions and mixed numbers · Measuring angles · Multi-step word problems · Area, perimeter, and geometry · Decimals, measurement conversions, and graphs Your child will develop both strong math skills and a positive attitude toward math with fun activities like Mental Math Arcade, Fraction War, Race to 180°, and Decimal Least to Greatest. All you’ll need are this Instructor Guide, the two Student Workbooks (Part A and Part B), and simple household items (like play money, base-ten blocks, a protractor, and playing cards) to make math come alive for your child. Hands-on, incremental lessons that steadily build conceptual understanding Daily review to ensure children retain what they’ve learned and master essential skills Games and real-world activities make math fun Easy to use, with clear directions and explanatory notes Optional picture book recommendations and math enrichment activities
The Fourth Turning Is Here: What the Seasons of History Tell Us about How and When This Crisis Will End
by Neil HoweThe visionary behind the bestselling phenomenon The Fourth Turning looks once again to America&’s past to predict our future in this startling and hopeful prophecy for how our present era of civil unrest will resolve over the next ten years—and what our lives will look like once it has. Twenty-five years ago, Neil Howe and the late William Strauss dazzled the world with a provocative new theory of American history. Looking back at the last 500 years, they&’d uncovered a distinct pattern: modern history moves in cycles, each one lasting roughly eighty to one hundred years, the length of a long human life, with each cycle composed of four eras—or &“turnings&”—that always arrive in the same order and each last about twenty years. The last of these eras—the fourth turning—was always the most perilous, a period of civic upheaval and national mobilization as traumatic and transformative as the New Deal and World War II, the Civil War, or the American Revolution. Now, right on schedule, our own fourth turning has arrived. And so Neil Howe has returned with an extraordinary new prediction. What we see all around us—the polarization, the growing threat of civil conflict and global war—will culminate by the early 2030s in a climax that poses great danger and yet also holds great promise, perhaps even bringing on America&’s next golden age. Every generation alive today will play a vital role in determining how this crisis is resolved, for good or ill. Illuminating, sobering, yet ultimately empowering, The Fourth Turning Is Here takes you back into history and deep into the collective personality of each living generation to make sense of our current crisis, explore how all of us will be differently affected by the political, social, and economic challenges we&’ll face in the decade to come, and reveal how our country, our communities, and our families can best prepare to meet these challenges head-on.
Fractal Apertures in Waveguides, Conducting Screens and Cavities
by Basudeb Ghosh Sachendra N. Sinha M. V. KartikeyanThis book deals with the design and analysis of fractal apertures in waveguides, conducting screens and cavities using numerical electromagnetics and field-solvers. The aim is to obtain design solutions with improved accuracy for a wide range of applications. To achieve this goal, a few diverse problems are considered. The book is organized with adequate space dedicated for the design and analysis of fractal apertures in waveguides, conducting screens and cavities, microwave/millimeter wave applications followed by detailed case-study problems to infuse better insight and understanding of the subject. Finally, summaries and suggestions are given for future work. Fractal geometries were widely used in electromagnetics, specifically for antennas and frequency selective surfaces (FSS). The self-similarity of fractal geometry gives rise to a multiband response, whereas the space-filling nature of the fractal geometries makes it an efficient element in antenna and FSS unit cell miniaturization. Until now, no efforts were made to study the behavior of these fractal geometries for aperture coupling problems. The aperture coupling problem is an important boundary value problem in electromagnetics and used in waveguide filters and power dividers, slotted ground planes, frequency selective surfaces and metamaterials. The present book is intended to initiate a study of the characteristics of fractal apertures in waveguides, conducting screens and cavities. To perform a unified analysis of these entirely dissimilar problems, the "generalized network formulation of the aperture problems" by Mautz and Harrington was extended to multiple-aperture geometry. The authors consider the problem of coupling between two arbitrary regions coupled together via multiple apertures of arbitrary shape. MATLAB codes were developed for the problems and validated with the results available in the literature as well as through simulations on ANSOFT's HFSS.
Fractal Dimension for Fractal Structures: With Applications to Finance (SEMA SIMAI Springer Series #19)
by Manuel Fernández-Martínez Juan Luis García Guirao Miguel Ángel Sánchez-Granero Juan Evangelista Trinidad SegoviaThis book provides a generalised approach to fractal dimension theory from the standpoint of asymmetric topology by employing the concept of a fractal structure. The fractal dimension is the main invariant of a fractal set, and provides useful information regarding the irregularities it presents when examined at a suitable level of detail. New theoretical models for calculating the fractal dimension of any subset with respect to a fractal structure are posed to generalise both the Hausdorff and box-counting dimensions. Some specific results for self-similar sets are also proved. Unlike classical fractal dimensions, these new models can be used with empirical applications of fractal dimension including non-Euclidean contexts. In addition, the book applies these fractal dimensions to explore long-memory in financial markets. In particular, novel results linking both fractal dimension and the Hurst exponent are provided. As such, the book provides a number of algorithms for properly calculating the self-similarity exponent of a wide range of processes, including (fractional) Brownian motion and Lévy stable processes. The algorithms also make it possible to analyse long-memory in real stocks and international indexes.This book is addressed to those researchers interested in fractal geometry, self-similarity patterns, and computational applications involving fractal dimension and Hurst exponent.
Fractal Dimensions of Networks (Springerbriefs In Computer Science Ser.)
by Eric RosenbergCurrent interest in fractal dimensions of networks is the result of more than a century of previous research on dimensions. Fractal Dimensions of Networks ties the theory and methods for computing fractal dimensions of networks to the “classic” theory of dimensions of geometric objects.The goal of the book is to provide a unified treatment of fractal dimensions of sets and networks. Since almost all of the major concepts in fractal dimensions originated in the study of sets, the book achieves this goal by first clearly presenting, with an abundance of examples and illustrations, the theory and algorithms for sets, and then showing how the theory and algorithms have been applied to networks. Thus, the book presents the classical theory and algorithms for the box counting dimension for sets, and then presents the box counting dimension for networks. All the major fractal dimensions are studied, e.g., the correlation dimension, the information dimension, the Hausdorff dimension, the multifractal spectrum, as well as many lesser known dimensions. Algorithm descriptions are accompanied by worked examples, many applications of the methods are presented, and many exercises, ranging in difficulty from easy to research level, are included.
Fractal Flow Design: How to Design Bespoke Turbulence and Why
by Yasuhiko Sakai Christos VassilicosThis book focuses on turbulent flows generated and/or influenced by multiscale/fractal structures. It consists of six chapters which demonstrate, each one in its own way, how such structures and objects can be used to design bespoke turbulence for particular applications and also how they can be used for fundamental studies of turbulent flows.
Fractal Functions, Dimensions and Signal Analysis (Understanding Complex Systems)
by Santo Banerjee D. Easwaramoorthy A. GowrisankarThis book introduces the fractal interpolation functions (FIFs) in approximation theory to the readers and the concerned researchers in advanced level. FIFs can be used to precisely reconstruct the naturally occurring functions when compared with the classical interpolants. The book focuses on the construction of fractals in metric space through various iterated function systems. It begins by providing the Mathematical background behind the fractal interpolation functions with its graphical representations and then introduces the fractional integral and fractional derivative on fractal functions in various scenarios. Further, the existence of the fractal interpolation function with the countable iterated function system is demonstrated by taking suitable monotone and bounded sequences. It also covers the dimension of fractal functions and investigates the relationship between the fractal dimension and the fractional order of fractal interpolation functions. Moreover, this book explores the idea of fractal interpolation in the reconstruction scheme of illustrative waveforms and discusses the problems of identification of the characterizing parameters. In the application section, this research compendium addresses the signal processing and its Mathematical methodologies. A wavelet-based denoising method for the recovery of electroencephalogram (EEG) signals contaminated by nonstationary noises is presented, and the author investigates the recognition of healthy, epileptic EEG and cardiac ECG signals using multifractal measures. This book is intended for professionals in the field of Mathematics, Physics and Computer Science, helping them broaden their understanding of fractal functions and dimensions, while also providing the illustrative experimental applications for researchers in biomedicine and neuroscience.
Fractal Geometry: Mathematical Foundations And Applications
by Kenneth FalconerThe seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines. Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences. Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals Carefully explains each topic using illustrative examples and diagrams Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics Features a wide range of exercises, enabling readers to consolidate their understanding Supported by a website with solutions to exercises and additional material http://www. wileyeurope. com/fractal Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)
Fractal Geometry
by Kenneth FalconerThe seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions.Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines.Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences.Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractalsCarefully explains each topic using illustrative examples and diagramsIncludes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topicsFeatures a wide range of exercises, enabling readers to consolidate their understandingSupported by a website with solutions to exercises and additional material http://www.wileyeurope.com/fractalLeads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)
Fractal Geometry: Mathematical Foundations and Applications (Progress In Probability Ser. #70)
by Kenneth FalconerThe seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines. Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences. Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals Carefully explains each topic using illustrative examples and diagrams Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics Features a wide range of exercises, enabling readers to consolidate their understanding Supported by a website with solutions to exercises and additional material http://www.wileyeurope.com/fractal Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)
Fractal Geometry and Stochastics V
by Christoph Bandt Kenneth Falconer Martina ZähleThis book collects significant contributions from the fifth conference on Fractal Geometry and Stochastics held in Tabarz, Germany, in March 2014. The book is divided into five topical sections: geometric measure theory, self-similar fractals and recurrent structures, analysis and algebra on fractals, multifractal theory, and random constructions. Each part starts with a state-of-the-art survey followed by papers covering a specific aspect of the topic. The authors are leading world experts and present their topics comprehensibly and attractively. Both newcomers and specialists in the field will benefit from this book.
Fractal Geometry and Stochastics VI (Progress in Probability #76)
by Uta Freiberg Ben Hambly Michael Hinz Steffen WinterThis collection of contributions originates from the well-established conference series "Fractal Geometry and Stochastics" which brings together researchers from different fields using concepts and methods from fractal geometry.Carefully selected papers from keynote and invited speakers are included, both discussing exciting new trends and results and giving a gentle introduction to some recent developments. The topics covered include Assouad dimensions and their connection to analysis, multifractal properties of functions and measures, renewal theorems in dynamics, dimensions and topology of random discrete structures, self-similar trees, p-hyperbolicity, phase transitions from continuous to discrete scale invariance, scaling limits of stochastic processes, stemi-stable distributions and fractional differential equations, and diffusion limited aggregation.Representing a rich source of ideas and a good starting point for more advanced topics in fractal geometry, the volume will appeal to both established experts and newcomers.
Fractal Geometry in Human Geography and Planning (ISTE Invoiced)
by Cécile TannierThe aim of this book is to provide an overview of the concepts and methods used in fractal analysis and modeling in order to study the spatial distribution of human settlements. The main concepts and mathematical tools are reviewed, and emphasis is placed on the practical benefits of their application and the pitfalls to be avoided when using them. The first part of the book is devoted to the basic concepts and geometric reference figures required for fractal analysis in human geography. In the second part, the principles of four major families of analysis methods are explained in detail: fractal analyses of point sequences, fractal methods for morphological delineation of urban agglomerations, multifractal analyses and cross scale signatures. The third part of the book is devoted to applications of fractal geometry in urban and regional planning.
The Fractal Geometry Of Nature
by Benoit MandelbrotExplore the wondrously complex repeating shapes of the natural world in The Fractal Geometry of Nature. Written in a style that is accessible to a wide audience, computer scientist, professor, mathematician, economist, and visionary Benoit B Mandelbrot's fascinating work has inspired popular interest in the geometry inherent in the natural world. Unlike the squares, circles, spheres, and cones of fundamental geometry, nature has rough edges and no straight lines or perfect curves. Mandelbrot observed that, even with this roughness, there still exists a kind of symmetry, which he dedicated his work to document and study. This became the basis for his development of a new kind of geometry; indeed, he coined the term "fractal."
Fractal Patterns in Nonlinear Dynamics and Applications: Patterns in Nonlinear Dynamics and Applications
by Santo Banerjee M K Hassan Sayan Mukherjee A GowrisankarMost books on fractals focus on deterministic fractals as the impact of incorporating randomness and time is almost absent. Further, most review fractals without explaining what scaling and self-similarity means. This book introduces the idea of scaling, self-similarity, scale-invariance and their role in the dimensional analysis. For the first time, fractals emphasizing mostly on stochastic fractal, and multifractals which evolves with time instead of scale-free self-similarity, are discussed. Moreover, it looks at power laws and dynamic scaling laws in some detail and provides an overview of modern statistical tools for calculating fractal dimension and multifractal spectrum.
Fractal Patterns with MATLAB (SpringerBriefs in Complexity)
by Santo Banerjee A. Gowrisankar Komandla Mahipal ReddyThis book presents the iterative beauty of fractals and fractal functions graphically with the aid of MATLAB programming. The fractal images generated using the MATLAB codes provide visual delight and highly encourage the fractal lovers for creative thinking. The book compiles five cutting-edge research chapters, each with state-of-the art fractal illustrations. It starts with the fundamental theory for the construction of fractal sets via the deterministic iteration algorithm. Incorporating the theoretical base, fractal illustrations of elementary fractal sets are provided with the explicit MATLAB code. The book gives examples of MATLAB codes to present the fractal surfaces. This book is contributed to all the research beginners as well as the professionals on the field of fractal analysis. As it covers basic fractals like Sierpinski triangle to advanced fractal functions with explicit MATLAB code, the presented fractal illustrations hopefully benefit even the non-field readers. The book is a useful course to all the research beginners on the fractal and fractal-related fields.
Fractal Signatures in the Dynamics of an Epidemiology: An Analysis of COVID-19 Transmission
by Santo Banerjee A. GowrisankarThe recent Covid-19 pandemic threw the world into complete chaos with its rapid and devastating spread. Scientists are still trying to obtain a better understanding of the patterns of COVID-19 and trying to get a deeper understanding of mutant strains and their pathogenicity by performing genomic sequences of more samples. Fractal-based analysis provides its unique forecasting policy to reduce the spread of COVID-19, and in general, of any outbreaks. The book presents fractal and multifractal models of COVID-19 and reviews the impact of the pandemic including epidemiology, genome organization, transmission cycle, and control strategies based on mathematical models towards developing an immune intervention. Also, it covers non-clinical aspects such as economic development with graphical illustrations, meeting the needs of onlookers outside the sector who desire additional information on the epidemic. The fractal signatures describe the fractal textures in the patterns of Corona virus. Studies on the epidemiology of Covid-19 in relation with the fractals and fractal functions serve to exhibit its irregular chaotic nature. Moreover, the book with its wide coverage on the Hurst exponent analysis and the fractal dimension estimation, greatly aids in measuring the epidemiology.
Fractal Solutions for Understanding Complex Systems in Earth Sciences
by V. P. DimriThis book deals with fractals in understanding problems encountered in earth science, and their solutions. It starts with an analysis of two classes of methods (homogeneous fractals random models, and homogeneous source distributions or "one point" distributions) widely diffused in the geophysical community, especially for studying potential fields and their related source distributions. Subsequently, the use of fractals in potential fields is described by scaling spectral methods for estimation of curie depth. The book also presents an update of the use of the fractal concepts in geological understanding of faults and their significance in geological modelling of hydrocarbon reservoirs. Geophysical well log data provide a unique description of the subsurface lithology; here, the Detrended Fluctuation Analysis technique is presented in case studies located off the west-coast of India. Another important topic is the fractal model of continuum percolation which quantitatively reproduce the flow path geometry by applying the Poiseuille's equation. The pattern of fracture heterogeneity in reservoir scale of natural geological formations can be viewed as spatially distributed self-similar tree structures; here, the authors present simple analytical models based on the medium structural characteristics to explain the flow in natural fractures. The Fractal Differential Adjacent Segregation (F-DAS) is an unconventional approach for fractal dimension estimation using a box count method. The present analysis provides a better understanding of variability of the system (adsorbents - adsorbate interactions). Towards the end of book, the authors discuss multi-fractal scaling properties of seismograms in order to quantify the complexity associated with high-frequency seismic signals. Finally, the book presents a review on fractal methods applied to fire point processes and satellite time-continuous signals that are sensitive to fire occurrences.
Fractal Teletraffic Modeling and Delay Bounds in Computer Communications
by Ming LiBy deploying time series analysis, Fourier transform, functional analysis, min-plus convolution, and fractional order systems and noise, this book proposes fractal traffic modeling and computations of delay bounds, aiming to improve the quality of service in computer communication networks. As opposed to traditional studies of teletraffic delay bounds, the author proposes a novel fractional noise, the generalized fractional Gaussian noise (gfGn) approach, and introduces a new fractional noise, generalized Cauchy (GC) process for traffic modeling. Researchers and graduates in computer science, applied statistics, and applied mathematics will find this book beneficial. Ming Li, PhD, is a professor at Ocean College, Zhejiang University, and the East China Normal University. He has been an active contributor for many years to the fields of computer communications, applied mathematics and statistics, particularly network traffic modeling, fractal time series, and fractional oscillations. He has authored more than 200 articles and 5 monographs on the subjects. He was identified as the Most Cited Chinese Researcher by Elsevier in 2014–2020. Professor Li was recognized as a top 100,000 scholar in all fields in 2019–2020 and a top 2% scholar in the field of Numerical and Computational Mathematics in 2021 by Prof. John P. A. Ioannidis, Stanford University.
Fractal Time: Why a Watched Kettle Never Boils (Studies of Nonlinear Phenomena in Life Science, Volume 14)
by Susie VrobelThis book provides an interdisciplinary introduction to the notion of fractal time, starting from scratch with a philosophical and perceptual puzzle.
Fractal Worlds: Grown, Built, and Imagined
by Michael Frame Amelia UrryFractal geometry is a uniquely fascinating area of mathematics, exhibited in a range of shapes that exist in the natural world, from a simple broccoli floret to a majestic mountain range. In this essential primer, mathematician Michael Frame—a close collaborator with Benoit Mandelbrot, the founder of fractal geometry—and poet Amelia Urry explore the amazing world of fractals as they appear in nature, art, medicine, and technology. Frame and Urry offer new insights into such familiar topics as measuring fractal complexity by dimension and the life and work of Mandelbrot. In addition, they delve into less-known areas: fractals with memory, the Mandelbrot set in four dimensions, fractals in literature, and more. An inviting introduction to an enthralling subject, this comprehensive volume is ideal for learning and teaching.