Geometry and its Applications in Arts, Nature and Technology

Geometry and its Applications in Arts, Nature and Technology
Author: Georg Glaeser
Publisher: Springer Nature
Total Pages: 694
Release: 2020-12-18
Genre: Mathematics
ISBN: 3030613984

This book returns geometry to its natural habitats: the arts, nature and technology. Throughout the book, geometry comes alive as a tool to unlock the understanding of our world. Assuming only familiarity with high school mathematics, the book invites the reader to discover geometry through examples from biology, astronomy, architecture, design, photography, drawing, engineering and more. Lavishly illustrated with over 1200 figures, all of the geometric results are carefully derived from scratch, with topics from differential, projective and non-Euclidean geometry, as well as kinematics, introduced as the need arises. The mathematical results contained in the book range from very basic facts to recent results, and mathematical proofs are included although not necessary for comprehension. With its wide range of geometric applications, this self-contained volume demonstrates the ubiquity of geometry in our world, and may serve as a source of inspiration for architects, artists, designers, engineers, and natural scientists. This new edition has been completely revised and updated, with new topics and many new illustrations.

Natural Operations in Differential Geometry

Natural Operations in Differential Geometry
Author: Ivan Kolar
Publisher: Springer Science & Business Media
Total Pages: 440
Release: 2013-03-09
Genre: Mathematics
ISBN: 3662029502

The aim of this work is threefold: First it should be a monographical work on natural bundles and natural op erators in differential geometry. This is a field which every differential geometer has met several times, but which is not treated in detail in one place. Let us explain a little, what we mean by naturality. Exterior derivative commutes with the pullback of differential forms. In the background of this statement are the following general concepts. The vector bundle A kT* M is in fact the value of a functor, which associates a bundle over M to each manifold M and a vector bundle homomorphism over f to each local diffeomorphism f between manifolds of the same dimension. This is a simple example of the concept of a natural bundle. The fact that exterior derivative d transforms sections of A kT* M into sections of A k+1T* M for every manifold M can be expressed by saying that d is an operator from A kT* M into A k+1T* M.

The Fractal Geometry of Nature

The Fractal Geometry of Nature
Author: Benoit Mandelbrot
Publisher: Echo Point Books & Media, LLC
Total Pages: 0
Release: 2021-07-16
Genre:
ISBN: 9781648370410

Written in a style that is accessible to a wide audience, The Fractal Geometry of Nature inspired popular interest in this emerging field. Mandelbrot's unique style, and rich illustrations will inspire readers of all backgrounds.

Sub-Riemannian Geometry

Sub-Riemannian Geometry
Author: Andre Bellaiche
Publisher: Birkhäuser
Total Pages: 404
Release: 2012-12-06
Genre: Mathematics
ISBN: 3034892101

Sub-Riemannian geometry (also known as Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been a full research domain for fifteen years, with motivations and ramifications in several parts of pure and applied mathematics, namely: control theory classical mechanics Riemannian geometry (of which sub-Riemannian geometry constitutes a natural generalization, and where sub-Riemannian metrics may appear as limit cases) diffusion on manifolds analysis of hypoelliptic operators Cauchy-Riemann (or CR) geometry. Although links between these domains had been foreseen by many authors in the past, it is only in recent years that sub- Riemannian geometry has been recognized as a possible common framework for all these topics. This book provides an introduction to sub-Riemannian geometry and presents the state of the art and open problems in the field. It consists of five coherent and original articles by the leading specialists: Andr Bellache: The tangent space in sub-Riemannian geometry Mikhael Gromov: Carnot-Carathodory spaces seen from within Richard Montgomery: Survey of singular geodesics Hctor J. Sussmann: A cornucopia of four-dimensional abnormal sub-Riemannian minimizers Jean-Michel Coron: Stabilization of controllable systems.

New Horizons In Differential Geometry And Its Related Fields

New Horizons In Differential Geometry And Its Related Fields
Author: Toshiaki Adachi
Publisher: World Scientific
Total Pages: 257
Release: 2022-04-07
Genre: Mathematics
ISBN: 9811248117

This volume presents recent developments in geometric structures on Riemannian manifolds and their discretizations. With chapters written by recognized experts, these discussions focus on contact structures, Kähler structures, fiber bundle structures and Einstein metrics. It also contains works on the geometric approach on coding theory.For researchers and students, this volume forms an invaluable source to learn about these subjects that are not only in the field of differential geometry but also in other wide related areas. It promotes and deepens the study of geometric structures.

Perceiving Geometry

Perceiving Geometry
Author: Catherine Q. Howe
Publisher: Springer Science & Business Media
Total Pages: 127
Release: 2005-11-18
Genre: Medical
ISBN: 0387254889

During the last few centuries, natural philosophers, and more recently vision scientists, have recognized that a fundamental problem in biological vision is that the sources underlying visual stimuli are unknowable in any direct sense, because of the inherent ambiguity of the stimuli that impinge on sensory receptors. The light that reaches the eye from any scene conflates the contributions of reflectance, illumination, transmittance, and subsidiary factors that affect these primary physical parameters. Spatial properties such as the size, distance and orientation of physical objects are also conflated in light stimuli. As a result, the provenance of light reaching the eye at any moment is uncertain. This quandary is referred to as the inverse optics problem. This book considers the evidence that the human visual system solves this problem by incorporating past human experience of what retinal images have typically corresponded to in the real world.

From Differential Geometry to Non-commutative Geometry and Topology

From Differential Geometry to Non-commutative Geometry and Topology
Author: Neculai S. Teleman
Publisher: Springer Nature
Total Pages: 398
Release: 2019-11-10
Genre: Mathematics
ISBN: 3030284336

This book aims to provide a friendly introduction to non-commutative geometry. It studies index theory from a classical differential geometry perspective up to the point where classical differential geometry methods become insufficient. It then presents non-commutative geometry as a natural continuation of classical differential geometry. It thereby aims to provide a natural link between classical differential geometry and non-commutative geometry. The book shows that the index formula is a topological statement, and ends with non-commutative topology.

World of Patterns

World of Patterns
Author: Rens Bod
Publisher: JHU Press
Total Pages: 396
Release: 2022-05-10
Genre: History
ISBN: 1421443457

A comprehensive account of the methods of knowledge production throughout human history and across the globe. The idea that the world can be understood through patterns and the principles that govern them is one of the most important human insights—it may also be our greatest survival strategy. Our search for patterns and principles began 40,000 years ago, when striped patterns were engraved on mammoths' bones to keep track of the moon's phases. What routes did human knowledge take to grow from these humble beginnings through many detours and dead ends into modern understandings of nature and culture? In this work of unprecedented scope, Rens Bod removes the Western natural sciences from their often-central role to bring us the first global history of human knowledge. Having sketched the history of the humanities in his ground-breaking A New History of the Humanities, Bod now adopts a broader perspective, stepping beyond classical antiquity back to the Stone Age to answer the question: Where did our knowledge of the world today begin and how did it develop? Drawing on developments from all five continents of the inhabited world, World of Patterns offers startling connections. Focusing on a dozen fields—ranging from astronomy, philology, medicine, law, and mathematics to history, botany, and musicology—Bod examines to what degree their progressions can be considered interwoven and to what degree we can speak of global trends. In this pioneering work, Bod aims to fulfill what he sees as the historian's responsibility: to grant access to history's goldmine of ideas. Bod discusses how inoculation was invented in China rather than Europe; how many of the fundamental aspects of modern mathematics and astronomy were first discovered by the Indian Kerala school; and how the study of law provided fundamental models for astronomy and linguistics from Roman to Ottoman times. The book flies across continents and eras. The result is an enlightening symphony, a stirring chorus of human inquisitiveness extending through the ages.

The Geometrical Beauty of Plants

The Geometrical Beauty of Plants
Author: Johan Gielis
Publisher: Springer
Total Pages: 235
Release: 2017-06-01
Genre: Science
ISBN: 9462391513

This book focuses on the origin of the Gielis curves, surfaces and transformations in the plant sciences. It is shown how these transformations, as a generalization of the Pythagorean Theorem, play an essential role in plant morphology and development. New insights show how plants can be understood as developing mathematical equations, which opens the possibility of directly solving analytically any boundary value problems (stress, diffusion, vibration...) . The book illustrates how form, development and evolution of plants unveil as a musical symphony. The reader will gain insight in how the methods are applicable in many divers scientific and technological fields.