The Fractal Dimension of Architecture is a book that applies the mathematical concept of fractal dimension to the analysis of the architecture of buildings. It was written by Michael J. Ostwald and Josephine Vaughan, both of whom are architecture academics at the University of Newcastle (Australia); [1] it was published in 2016 by Birkhäuser, as the first volume in their Mathematics and the Built Environment book series.
The book applies the box counting method for computing fractal dimension, via the ArchImage software system, to compute a fractal dimension from architectural drawings (elevations and floor plans) of buildings, drawn at multiple levels of detail. The results of the book suggest that the results are consistent enough to allow for comparisons from one building to another, as long as the general features of the images (such as margins, line thickness, and resolution), parameters of the box counting algorithm, and statistical processing of the results are carefully controlled. [1] [2]
The first five chapters of the book introduce fractals and the fractal dimension, and explain the methodology used by the authors for this analysis, also applying the same analysis to classical fractal structures including the Apollonian gasket, Fibonacci word, Koch snowflake, Minkowski sausage, pinwheel tiling, terdragon, and Sierpiński triangle. [3] The remaining six chapters explain the authors' choice of buildings to analyze, apply their methodology to 625 drawings from 85 homes, built between 1901 and 2007, and perform a statistical analysis of the results. [1] [3] [4]
The authors use this technique to study three main hypotheses, with a fractal structure of subsidiary hypotheses depending on them. These are
The first and third hypotheses are not convincingly supported by the analysis, but the results suggest further work in these directions. The second hypothesis, on distinctive fractal descriptions of genres and movements, does not appear to be true, leading the authors to weaker replacements for it. [1] [2] [3]
The book is aimed at architects and architecture students; its mathematical content is not deep, and it does not require much mathematical background of its readers. [1] [3] Reviewer Joel Haack suggests that it could also be used for general education courses in mathematics for liberal arts undergraduates. [3]
Benoit B.Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature.
Chaos theory is an interdisciplinary scientific theory and branch of mathematics focused on underlying patterns and deterministic laws, of dynamical systems, that are highly sensitive to initial conditions, that were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.
Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.
A New Kind of Science is a book by Stephen Wolfram, published by his company Wolfram Research under the imprint Wolfram Media in 2002. It contains an empirical and systematic study of computational systems such as cellular automata. Wolfram calls these systems simple programs and argues that the scientific philosophy and methods appropriate for the study of simple programs are relevant to other fields of science.
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.
In statistics, exploratory data analysis (EDA) is an approach of analyzing data sets to summarize their main characteristics, often using statistical graphics and other data visualization methods. A statistical model can be used or not, but primarily EDA is for seeing what the data can tell us beyond the formal modeling and thereby contrasts traditional hypothesis testing. Exploratory data analysis has been promoted by John Tukey since 1970 to encourage statisticians to explore the data, and possibly formulate hypotheses that could lead to new data collection and experiments. EDA is different from initial data analysis (IDA), which focuses more narrowly on checking assumptions required for model fitting and hypothesis testing, and handling missing values and making transformations of variables as needed. EDA encompasses IDA.
In sociology, social complexity is a conceptual framework used in the analysis of society. Contemporary definitions of complexity in the sciences are found in relation to systems theory, in which a phenomenon under study has many parts and many possible arrangements of the relationships between those parts. At the same time, what is complex and what is simple is relative and may change with time.
Information-based complexity (IBC) studies optimal algorithms and computational complexity for the continuous problems that arise in physical science, economics, engineering, and mathematical finance. IBC has studied such continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration. All these problems involve functions of a real or complex variable. Since one can never obtain a closed-form solution to the problems of interest one has to settle for a numerical solution. Since a function of a real or complex variable cannot be entered into a digital computer, the solution of continuous problems involves partial information. To give a simple illustration, in the numerical approximation of an integral, only samples of the integrand at a finite number of points are available. In the numerical solution of partial differential equations the functions specifying the boundary conditions and the coefficients of the differential operator can only be sampled. Furthermore, this partial information can be expensive to obtain. Finally the information is often contaminated by noise.
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Nikos Angelos Salingaros is a mathematician and polymath known for his work on urban theory, architectural theory, complexity theory, and design philosophy. He has been a close collaborator of the architect Christopher Alexander, with whom Salingaros shares a harsh critical analysis of conventional modern architecture. Like Alexander, Salingaros has proposed an alternative theoretical approach to architecture and urbanism that is more adaptive to human needs and aspirations, and that combines rigorous scientific analysis with deep intuitive experience.
A Theory of Architecture is a book on architecture by Nikos Salingaros, published in 2006 by Umbau-Verlag, Solingen, ISBN 3-937954-07-4. Glowing cover blurbs by Kenneth G. Masden II, Duncan G. Stroik, Michael Blowhard, and Dean A. Dykstra. Preface by Prince Charles, and Foreword by Kenneth G. Masden II. This book is a re-working of previously published articles used to teach a senior architecture studio class. Four of the twelve chapters were originally written in collaboration, and the co-authors include Michael Mehaffy, Terry Mikiten, Debora Tejada, and Hing-Sing Yu.
Applied aesthetics is the application of the branch of philosophy of aesthetics to cultural constructs. In a variety of fields, artifacts are created that have both practical functionality and aesthetic affectation. In some cases, aesthetics is primary, and in others, functionality is primary. At best, the two needs are synergistic, in which "beauty" makes an artifact work better, or in which more functional artifacts are appreciated as aesthetically pleasing. This achievement of form and function, of art and science, of beauty and usefulness, is the primary goal of design, in all of its domains.
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.
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Geometry From Africa: Mathematical and Educational Explorations is a book in ethnomathematics by Paulus Gerdes. It analyzes the mathematics behind geometric designs and patterns from multiple African cultures, and suggests ways of connecting this analysis with the mathematics curriculum. It was published in 1999 by the Mathematical Association of America, in their Classroom Resource Materials book series.
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