Geometry is a dynamic branch of mathematics that also serves as a creative tool for engineers, artists, and architects. Squaring the Circle: Geometry in Art and Architecture includes all the topics necessary for a solid foundation in geometry and explores the timeless influence of geometry on art and architecture. The text offers wide-ranging exercise sets and related projects that allow students to practice and master the mathematics presented. Each chapter introduces mathematical concepts geometrically and illustrates their nontraditional applications in art and architecture throughout the centuries. Appropriate for both basic mathematics courses and cross-discipline courses in mathematics and art, Squaring the Circle requires no previous mathematics.
Every age and every culture has relied on the incorporation of mathematics in their works of architecture to imbue the built environment with meaning and order. Mathematics is also central to the production of architecture, to its methods of measurement, fabrication and analysis. This two-volume edited collection presents a detailed portrait of the ways in which two seemingly different disciplines are interconnected. Over almost 100 chapters it illustrates and examines the relationship between architecture and mathematics. Contributors of these chapters come from a wide range of disciplines and backgrounds: architects, mathematicians, historians, theoreticians, scientists and educators. Through this work, architecture may be seen and understood in a new light, by professionals as well as non-professionals. Volume I covers architecture from antiquity through Egyptian, Mayan, Greek, Roman, Medieval, Inkan, Gothic and early Renaissance eras and styles. The themes that are covered range from symbolism and proportion to measurement and structural stability. From Europe to Africa, Asia and South America, the chapters span different countries, cultures and practices.
The age of the Baroque—a time when great strides were made in science and mathematics—witnessed the construction of some of the world's most magnificent buildings. What did the work of great architects such as Bernini, Blondel, Guarini, and Wren have to do with Descartes, Galileo, Kepler, Desargues, and Newton? Here, George Hersey explores the ways in which Baroque architecture, with its dramatic shapes and playful experimentation with classical forms, reflects the scientific thinking of the time. He introduces us to a concept of geometry that encompassed much more than the science we know today, one that included geometrics (number and shape games), as well as the art of geomancy, or magic and prophecy using shapes and numbers. Hersey first concentrates on specific problems in geometry and architectural design. He then explores the affinities between musical chords and several types of architectural form. He turns to advances in optics, such as artificial lenses and magic lanterns, to show how architects incorporated light, a heavenly emanation, into their impressive domes. With ample illustrations and lucid, witty language, Hersey shows how abstract ideas were transformed into visual, tactile form—the epicycles of the cosmos, the sexual mystique surrounding the cube, and the imperfections of heavenly bodies. Some two centuries later, he finds that the geometric principles of the Baroque resonate, often unexpectedly, in the work of architects such as Frank Lloyd Wright and Le Corbusier. A discussion of these surprising links to the past rounds out this brilliant reexamination of some of the long-forgotten beliefs and practices that helped produce some of Europe's greatest masterpieces.
This work takes a close look at a broad range of 20th-century examples of design, architecture and illustration, revealing underlying geometric structures in their compositions.
Geometry Civilized is a unique combination of history and mathematics. It contains a full introduction to plane geometry and trigonometry within a fascinating historical framework that sets off the technical material. This approach to geometrical ideas gives the book its unique, readable style. The author has included a wide range of unusual and engaging geometric problems, many of which are taken from practical applications, drawn from sources ranging from ancient to modern. The study of geometry has been an important element of education in Europe since the time of the Greeks. This book helps us to understand why such emphasis has been placed on obtaining a good understanding of geometry. But the history presented here is not confined to the Western tradition. Examples drawn from other cultures, particularly Chinese and Indian, underscore the peculiarities of the geometry we have inherited from the Greeks, and thereby make Euclid's approach more accessible. Book reviews from the hardback: 'He has written a marvellous tale of how, throughout much of recorded history, geometrical thinking and civilisation have been closely intertwined. ...Definitely a book to dip into and reflect on a superior form of brainfood for the beach this summer perhaps? Heilbron's enthusiasm is contagious.' Ian Stewart, New Scientist 'The book is wonderfully illustrated. There are diagrams on almost every page, apt illustrations from older books, and well chosen photographs,together with eight colour plates. The appearance of the book is quite seductive, for which Oxford University Press should be congratulated.' Jeremy Gray, Nature 'This is a handsome book, well researched and entertainingly written. It shows how powerfully a historically informed account can communicate. If you thought Euclidean geometry was something only your great-grandparents did - try it, you will be surprised.' BJune Barrow-Green, The TIMES Higher Education Supplement
A professional guide to drawing two-dimensional geometric shapes in simple step-by-step instructions.
Let your creativity flow by drawing amazing geometric shapes! Whether you're a beginning artist looking for a way to get in touch with your creative side, a harried executive looking for a meditative hobby to reduce stress, or are already an experienced artist, learning to draw geometrically can expand your horizons, giving you a new angle on how you view your world. Even if the thought of drawing daunts you, anyone can draw simple shapes like triangles, circles, squares, and hexagons. This kit will show you how to put those shapes together in surprising ways and patterns. This kit includes: - Hypotroid Spirograph Stencil Ruler for creating amazing spiral drawings - Two 8x6 geometric drawing stencils featuring a wide variety of geometric shapes - 112 page drawing tablet with illustraitions and prompts to guide you on your journey of drawing geometric - Plus empty pages for your own amazing creations!
Smartgeometry (SG) is a key influence on the architectural community who explore creative computational methods for the design of buildings. An informal international network of practitioners and researchers, the group meets annually to experiment with new technologies and collaborate to develop digital design techniques. When SG was founded in 2001 by London-based architects and friends Hugh Whitehead (Foster + Partners), J Parrish (AECOM) and Lars Hesselgren (PLP), there was little in the way of parametric tools for architecture. SG was founded to encourage the development, discussion and experimentation of digital design techniques driven by design intent rather than on construction specifications. SG calls for a re-consideration of the design process, where the creation of computational mechanisms become an integral part of designing – not a task done prior to or separate from the process. In the early years of the workshops this need for new ways of design thinking led to the development of Bentley´s GenerativeComponents software. In recent years, the ecology of these design environments has diversified to include multiple software platforms, as well as innovative fabrication techniques and interactive environments. SG has grown accordingly from a handful of experts to an international network of designers who are defining the future of design. Founded by digital pioneers, it creates the algorithmic designers of the future. Inside Smartgeometry can be seen as a retroactive manifesto for SG, examining and contextualising the work of the SG community: the digital spaces, prototypes and buildings designed using bespoke tools created in response to architectural ideas. From interactive crowd-sourcing tools to responsive agent-based systems to complex digitally fabricated structures, it explores more than a decade of advances that have been influential for architecture. Through 23 original texts including reflections by the founders, and key contributors such as Robert Aish, Martin Bechthold, Mark Burry, Chris Williams and Robert Woodbury, the book offers a critical state of the art of computational design for architecture. Many international design and engineering firms have participated in SG and the book includes chapters by practitioners from offices such as CASE, Design2Production, Foster + Partners, Grimshaw, Populous and SOM.
A new interdisciplinary perspective on architecture and geometry--from Athenian Ruins to the Chrysler building
An introduction to geometry without measurements.
The first edition of Connections was chosen by the National Association of Publishers (USA) as the best book in OC Mathematics, Chemistry, and Astronomy OCo Professional and ReferenceOCO in 1991. It has been a comprehensive reference in design science, bringing together in a single volume material from the areas of proportion in architecture and design, tilings and patterns, polyhedra, and symmetry. The book presents both theory and practice and has more than 750 illustrations. It is suitable for research in a variety of fields and as an aid to teaching a course in the mathematics of design. It has been influential in stimulating the burgeoning interest in the relationship between mathematics and design. In the second edition there are five new sections, supplementary, as well as a new preface describing the advances in design science since the publication of the first edition. Contents: Proportion in Architecture; Similarity; The Golden Mean; Graphs; Tilings with Polygons; Two-Dimensional Networks and Lattices; Polyhedra: Platonic Solids; Transformation of the Platonic Solids I; Transformation of the Platonic Solids II; Polyhedra: Space Filling; Isometries and Mirrors; Symmetry of the Plane. Readership: Polytechnic students, architects, designers, mathematicians and general readers."
The three most basic shapes -- squares, triangles and circles -- are all around us, from the natural world to the one we've engineered. Full of fascinating facts about these shapes and their 3D counterparts, Shapes in Math, Science and Nature introduces young readers to the basics of geometry and reveals its applications at home, school and everywhere in between. Puzzles and activities add to the fun factor.
An illustrated guide to harmonics--the sacred geometry principles that underlie the natural world--and its practical applications • Demonstrates how the vesica piscis is a matrix from which ideas and forms emanate, connecting cosmic time cycles, measures of space, and musical tones • Provides harmonic analyses of ancient sculpture, architecture, the solar system, the Earth-Moon relationship, and the structure of water and waves • Explains how to apply sacred geometry to create building floor plans, pottery figures, gardens, and sacred ceremonial spaces We are in the midst of a revival of an ancient way of looking at the world--an approach that enabled great civilizations of the past to bring forth inventions of great beauty and power. This school of thought--harmonics--envisioned the natural world and the solar system as an interlocking matrix of harmonious numbers, perfectly woven into the creative fabric of life and the surrounding universe. Exploring the art and science of harmonics, John Oscar Lieben shows how to create harmonious forms using the ancient tools of number, geometry, and musical tone--an approach that resonates with nature’s own ways of creation. He demonstrates many practical applications that result from the study of harmonics, providing analyses of ancient sculpture and architecture, as well as original examples of building floor plans, pottery figures based on planetary proportions, gardens based on harmonic principles, and ceremonial spaces that honor cosmic harmonies and sacred geometric relationships. Showing how harmonics can also be applied to the mysteries of time and space, the author demonstrates how the vesica piscis and many other variations of the vesica shape reveal numerical synchronicities and correspondences that connect cosmic time cycles, measures of space, and musical tones. The author applies harmonics and the “vesica construction” matrix to illustrate many of nature’s wonders, including the Earth-Moon relationship, the interactions of the Golden Number and the musical scale, and how the Flower of Life symbol connects the universal field with the pattern of raindrops falling on a pond. Offering an approach to sacred geometry that pairs the mystical with the practical, the cosmic with the earthly, the author reveals how the art and science of harmonics should be required study for both the artist and the seeker of eternal truths as well as the scientist who seeks an entrance into the sacred foundations of nature.
Whicher explores the concepts of polarity and movement in modern projective geometry as a discipline of thought that transcends the limited and rigid space and forms of Euclid, and the corresponding material forces conceived in classical mechanics. Rudolf Steiner underlined the importance of projective geometry as "a method of training the imaginative faculties of thinking, so that they become an instrument of cognition no less conscious and exact than mathematical reasoning." This seminal approach allows for precise scientific understanding of the concept of creative fields of formative (etheric) forces at work in nature--in plants, animals and in the human being.
This classic survey of Italian Baroque art & architecture focuses on the arts in every center between Venice & Sicily in the early, high, & late Baroque periods. Wittkower's text is now accompanied by a critical introduction & substantial new bibliography. This edition-now published in three volumes-will also include color illustrations for the first time.
In the early 1960s Italian design legend Bruno Munari published his visual case studies on shapes: Circle, Square, and, a decade later, Triangle. Using examples from ancient Greece and Egypt, as well as works by Buckminster Fuller, Le Corbusier, and Alvar Aalto, Munari invests the three shapes with specific qualities: the circle relates to the divine, the square signifies safety and enclosure, and the triangle provides a key connective form for designers. One of the great designers of the twentieth century, Munari contributed to the fields of painting, sculpture, design, and photography while teaching throughout his seventy-year career. After World War II he began to focus on book design, creating children's books known for their simplicity and playfulness.
Malcolm Stewart has discovered a remarkable geometrical device. The 'starcut diagram', as he has called it, is at first glance a simple way of dividing the area of a square. After extensive research, however, he found that it has extraordinary mathematical properties, suggesting that it may be no less than the source of the number system used when ancient humanity first built cities.He shows that the starcut diagram underlies many significant patterns and proportions across the world: in China, the shaman's dance; in Egypt, the Great Pyramid; in Europe, a Raphael fresco; in Asia, the Vedic Fire Altar, and many others. This book is an intellectual adventure, written for a general reader without specialist knowledge. Illustrated with around 180 photographs, drawings and diagrams, it tells the story of many fresh discoveries, bringing sacred geometry to life in an original and inspiring way.
A hugely enjoyable, brilliantly researched explanation of the basic principles of maths.