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24 books match the query 'Geometry'

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After Euclid

Jesse Norman

CSLI, 2006

What does it mean to have visual intuition? Can we gain geometrical knowledge by using visual reasoning? And if we can, is it because we have a faculty of intuition? In After Euclid, Jesse Norman reexamines the ancient and long-disregarded concept of visual reasoning and reasserts its potential as a formidable tool in our ability to grasp various kinds of geometrical knowledge. The first detailed philosophical case study of its kind, this text is essential reading for scholars in the fields of mathematics and philosophy.

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Architecture and Geometry in the Age of the Baroque

George L. Hersey

University of Chicago Press, 2001

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.

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Dynamics, Geometry, Number Theory

The Impact of Margulis on Modern Mathematics

Edited by David Fisher, Dmitry Kleinbock, and Gregory Soifer

University of Chicago Press, 2022

This definitive synthesis of mathematician Gregory Margulis’s research brings together leading experts to cover the breadth and diversity of disciplines Margulis’s work touches upon.

This edited collection highlights the foundations and evolution of research by widely influential Fields Medalist Gregory Margulis. Margulis is unusual in the degree to which his solutions to particular problems have opened new vistas of mathematics; his ideas were central, for example, to developments that led to the recent Fields Medals of Elon Lindenstrauss and Maryam Mirzhakhani. Dynamics, Geometry, Number Theory introduces these areas, their development, their use in current research, and the connections between them. Divided into four broad sections—“Arithmeticity, Superrigidity, Normal Subgroups”; “Discrete Subgroups”; “Expanders, Representations, Spectral Theory”; and “Homogeneous Dynamics”—the chapters have all been written by the foremost experts on each topic with a view to making them accessible both to graduate students and to experts in other parts of mathematics. This was no simple feat: Margulis’s work stands out in part because of its depth, but also because it brings together ideas from different areas of mathematics. Few can be experts in all of these fields, and this diversity of ideas can make it challenging to enter Margulis’s area of research. Dynamics, Geometry, Number Theory provides one remedy to that challenge.

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Euclid and His Twentieth Century Rivals

Diagrams in the Logic of Euclidean Geometry

Nathaniel Miller

CSLI, 2007

Twentieth-century developments in logic and mathematics have led many people to view Euclid’s proofs as inherently informal, especially due to the use of diagrams in proofs. In Euclid and His Twentieth-Century Rivals, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to mathematicians, computer scientists, and anyone interested in the use of diagrams in geometry.

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Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

Robert Bryant, Phillip Griffiths, and Daniel Grossman

University of Chicago Press, 2003

In Exterior Differential Systems, the authors present the results of their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincaré-Cartan forms. They also cover certain aspects of the theory of exterior differential systems, which provides the language and techniques for the entire study. Because it plays a central role in uncovering geometric properties of differential equations, the method of equivalence is particularly emphasized. In addition, the authors discuss conformally invariant systems at length, including results on the classification and application of symmetries and conservation laws. The book also covers the Second Variation, Euler-Lagrange PDE systems, and higher-order conservation laws.

This timely synthesis of partial differential equations and differential geometry will be of fundamental importance to both students and experienced researchers working in geometric analysis.

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Geometrical Vectors

Gabriel Weinreich

University of Chicago Press, 1998

Every advanced undergraduate and graduate student of physics must master the concepts of vectors and vector analysis. Yet most books cover this topic by merely repeating the introductory-level treatment based on a limited algebraic or analytic view of the subject.

Geometrical Vectors introduces a more sophisticated approach, which not only brings together many loose ends of the traditional treatment, but also leads directly into the practical use of vectors in general curvilinear coordinates by carefully separating those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition.

Written in an informal and personal style, Geometrical Vectors provides a handy guide for any student of vector analysis. Clear, carefully constructed line drawings illustrate key points in the text, and problem sets as well as physical examples are provided.

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Geometry and Chronometry in Philosophical Perspective

Adolf Grunbaum

University of Minnesota Press, 1968

Geometry and Chronometry in Philosophical Perspective was first published in 1968. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions.

In this volume Professor Grünbaum substantially extends and comments upon his essay "Geometry, Chronometry, and Empiricism," which was first published in Volume III of the Minnesota Studies in the Philosophy of Science. Commenting on the essay when it first appeared J. J. C. Smart wrote in Mind (England): "In my opinion Adolf Grünbaum's paper ... supersedes nearly all that has been written on the logical status of physical geometry and chronometry." The full text of the essay is given here with the author's extension of it and his discussion of some of the critical comment it has evoked, particularly, a critique published by Hilary Putnam.

Adolph Grünbaum is Andrew Mellon Professor of Philosophy at the University of Pittsburgh and the current president of the Philosophy of Science Association.

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Geometry and Meaning

Dominic Widdows

CSLI, 2004

From Pythagoras's harmonic sequence to Einstein's theory of relativity, geometric models of position, proximity, ratio, and the underlying properties of physical space have provided us with powerful ideas and accurate scientific tools. Currently, similar geometric models are being applied to another type of space—the conceptual space of information and meaning, where the contributions of Pythagoras and Einstein are a part of the landscape itself. The rich geometry of conceptual space can be glimpsed, for instance, in internet documents: while the documents themselves define a structure of visual layouts and point-to-point links, search engines create an additional structure by matching keywords to nearby documents in a spatial arrangement of content. What the Geometry of Meaning provides is a much-needed exploration of computational techniques to represent meaning and of the conceptual spaces on which these representations are founded.

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Geometry in Architecture

Texas Buildings Yesterday and Today

By Clovis Heimsath

University of Texas Press, 2002

Pioneer Texas Buildings opened people's eyes when it was first published in 1968. At a time when "progress" meant tearing down the weathered houses, barns, churches, and stores built by the original settlers of Central Texas, this book taught people to see the beauty, simplicity, and order expressed in the unadorned geometric forms of early Texas buildings. It inspired the preservation and restoration of many of the remaining pioneer buildings, as well as the design of modern buildings that employ the same simple geometries.

This revised edition of Pioneer Texas Buildings juxtaposes the historic structures with works by twenty contemporary architects who are inspired by the pioneer tradition to show how seamlessly the basic geometries translate from one era to another. As in the first edition, sketches and brief commentary by Clovis Heimsath explain how squares, triangles, and circles take shape in the cubic, triangular, and cylindrical forms that comprise houses and other buildings. Then black-and-white photographs, the heart of the book, illustrate these geometric forms in historic and modern buildings. The book also includes two essays in which Heimsath discusses the factors that led him and his wife Maryann to document early Texas buildings and the results in historic preservation and timeless architectural designs that have followed from their efforts.

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Geometry of Grief

Reflections on Mathematics, Loss, and Life

Michael Frame

University of Chicago Press, 2021

In this profound and hopeful book, a mathematician and celebrated teacher shows how mathematics may help all of us—even the math-averse—to understand and cope with grief.

We all know the euphoria of intellectual epiphany—the thrill of sudden understanding. But coupled with that excitement is a sense of loss: a moment of epiphany can never be repeated. In Geometry of Grief, mathematician Michael Frame draws on a career’s worth of insight—including his work with a pioneer of fractal geometry Benoit Mandelbrot—and a gift for rendering the complex accessible as he delves into this twinning of understanding and loss. Grief, Frame reveals, can be a moment of possibility.

Frame investigates grief as a response to an irrevocable change in circumstance. This reframing allows us to see parallels between the loss of a loved one or a career and the loss of the elation of first understanding a tricky concept. From this foundation, Frame builds a geometric model of mental states. An object that is fractal, for example, has symmetry of magnification: magnify a picture of a mountain or a fern leaf—both fractal—and we see echoes of the original shape. Similarly, nested inside great loss are smaller losses. By manipulating this geometry, Frame shows us, we may be able to redirect our thinking in ways that help reduce our pain. Small‐scale losses, in essence, provide laboratories to learn how to meet large-scale losses.

Interweaving original illustrations, clear introductions to advanced topics in geometry, and wisdom gleaned from his own experience with illness and others’ remarkable responses to devastating loss, Frame’s poetic book is a journey through the beautiful complexities of mathematics and life. With both human sympathy and geometrical elegance, it helps us to see how a geometry of grief can open a pathway for bold action.

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Geometry of Nonpositively Curved Manifolds

Patrick B. Eberlein

University of Chicago Press, 1997

Starting from the foundations, the author presents an almost entirely
self-contained treatment of differentiable spaces of nonpositive
curvature, focusing on the symmetric spaces in which every geodesic lies
in a flat Euclidean space of dimension at least two. The book builds to
a discussion of the Mostow Rigidity Theorem and its generalizations, and
concludes by exploring the relationship in nonpositively curved spaces
between geometric and algebraic properties of the fundamental group.

This introduction to the geometry of symmetric spaces of non-compact
type will serve as an excellent guide for graduate students new to the
material, and will also be a useful reference text for mathematicians
already familiar with the subject.

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Geometry, Rigidity, and Group Actions

Edited by Benson Farb and David Fisher

University of Chicago Press, 2011

The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others.

The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.

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Grammar, Geometry, and Brain

Jens Erik Fenstad

CSLI, 2010

This original study considers the effects of language and meaning on the brain. Jens Erik Fenstad—an expert in the fields of recursion theory, nonstandard analysis, and natural language semantics—combines current formal semantics with a geometric structure in order to trace how common nouns, properties, natural kinds, and attractors link with brain dynamics.

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Group Actions in Ergodic Theory, Geometry, and Topology

Selected Papers

Robert J. Zimmer

University of Chicago Press, 2019

Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology.

In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.

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Inversions

I. Ya. Bakel'man

University of Chicago Press, 1975

In this book, I. Ya. Bakel'man introduces inversion transformations in the Euclidean plane and discusses the interrelationships among more general mathematical concepts. The author begins by defining and giving examples of the concept of a transformation in the Euclidean plane, and then explains the "point of infinity" and the "stereographic projection" of the sphere onto the plane. With this preparation, the student is capable of applying the theory of inversions to classical construction problems in the plane.

The author also discusses the theory of pencils of circles, and he uses the acquired techniques in a proof of Ptolemy's theorem. In the final chapter, the idea of a group is introduced with applications of group theory to geometry. The author demonstrates the group-theoretic basis for the distinction between Euclidean and Lobachevskian geometry.

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Liberty's Grid

A Founding Father, a Mathematical Dreamland, and the Shaping of America

Amir Alexander

University of Chicago Press, 2024

The surprising history behind a ubiquitous facet of the United States: the gridded landscape.

Seen from an airplane, much of the United States appears to be a gridded land of startling uniformity. Perpendicular streets and rectangular fields, all precisely measured and perfectly aligned, turn both urban and rural America into a checkerboard landscape that stretches from horizon to horizon. In evidence throughout the country, but especially the West, the pattern is a hallmark of American life. One might consider it an administrative convenience—an easy way to divide land and lay down streets—but it is not. The colossal grid carved into the North American continent, argues historian and writer Amir Alexander, is a plan redolent with philosophical and political meaning.

In 1784 Thomas Jefferson presented Congress with an audacious scheme to reshape the territory of the young United States. All western lands, he proposed, would be inscribed with a single rectilinear grid, transforming the natural landscape into a mathematical one. Following Isaac Newton and John Locke, he viewed mathematical space as a blank slate on which anything is possible and where new Americans, acting freely, could find liberty. And if the real America, with its diverse landscapes and rich human history, did not match his vision, then it must be made to match it.

From the halls of Congress to the open prairies, and from the fight against George III to the Trail of Tears, Liberty’s Grid tells the story of the battle between grid makers and their opponents. When Congress endorsed Jefferson’s plan, it set off a struggle over American space that has not subsided. Transcendentalists, urban reformers, and conservationists saw the grid not as a place of possibility but as an artificial imposition that crushed the human spirit. Today, the ideas Jefferson associated with the grid still echo through political rhetoric about the country’s founding, and competing visions for the nation are visible from Manhattan avenues and Kansan pastures to Yosemite’s cliffs and suburbia’s cul-de-sacs. An engrossing read, Liberty’s Grid offers a powerful look at the ideological conflict written on the landscape.

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Spacetime And Geometry

The Alfred Schild Lectures

Edited by Richard A. Matzner and L. C. Shepley

University of Texas Press, 1982

These reports, at the forefront of relativity theory when they were written, in particular the geometrical aspects of spacetime theory, were the result of the Alfred Schild Memorial Lecture Series presented at the University of Texas at Austin beginning in 1977. Each article is a self-contained summary of an important area of contemporary gravitational physics, while the book as a whole provides an overview of a wide variety of the problems of general relativity and gravitation.

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Surfaces and Superposition

Field Notes on some Geometrical Excavations

Ernest W. Adams

CSLI, 2001

Buildings appear to rest on top of the earth's surface, yet the surface is actually permeated by the buildings' foundations-out of view. If a foundation's blueprints are unavailable, as in archaeology, excavation would be needed to discover what actually supports a specific building. Analogously, the fields of geometry and topology have easily observable concepts resting on the surface of theoretical underpinnings that have not been completely discovered, unearthed or understood. Moreover, geometrical and topological principles of superposition provide insight into probing the connections between accessible superstructures and their hidden underpinnings. This book develops and applies these insights broadly, from physics to mathematics to philosophy. Even analogies and abstractions can now be seen as foundational superpositions.

This book examines the dimensionality of surfaces, how superpositions can make stable frameworks, and gives a quasi-Leibnizian account of the relative `spaces' that are defined by these frameworks. Concluding chapters deal with problems concerning the spatio-temporal frameworks of physical theories and implications for theories of visual geometry. The numerous illustrations, while surprisingly simple, are satisfyingly clear.

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The Decomposition of Figures Into Smaller Parts

Vladimir Grigor'evich Boltyanskii and Izrail' Tsudikovich Gokhberg

University of Chicago Press, 1980

In contrast to the vast literature on Euclidean geometry as a whole, little has been published on the relatively recent developments in the field of combinatorial geometry. Boltyanskii and Gohberg's book investigates this area, which has undergone particularly rapid growth in the last thirty years. By restricting themselves to two dimensions, the authors make the book uniquely accessible to interested high school students while maintaining a high level of rigor. They discuss a variety of problems on figures of constant width, convex figures, coverings, and illumination. The book offers a thorough exposition of the problem of cutting figures into smaller pieces. The central theorem gives the minimum number of pieces into which a figure can be divided so that all the pieces are of smaller diameter than the original figure. This theorem, which serves as a basis for the rest of the material, is proved for both the Euclidean plane and Minkowski's plane.

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The Geometry of Modernism

The Vorticist Idiom in Lewis, Pound, H.D., and Yeats

By Miranda B. Hickman

University of Texas Press, 2006

Addressing both the literature and the visual arts of Anglo-American modernism, The Geometry of Modernism recovers a crucial development of modernism's early years that until now has received little sustained critical attention: the distinctive idiom composed of geometric forms and metaphors generated within the early modernist movement of Vorticism, formed in London in 1914. Focusing on the work of Wyndham Lewis, leader of the Vorticist movement, as well as Ezra Pound, H.D., and William Butler Yeats, Hickman examines the complex of motives out of which Lewis initially forged the geometric lexicon of Vorticism—and then how Pound, H.D., and Yeats later responded to it and the values that it encoded, enlisting both the geometric vocabulary and its attendant assumptions and ideals, in transmuted form, in their later modernist work.

Placing the genesis and appropriation of the geometric idiom in historical context, Hickman explores how despite its brevity as a movement, Vorticism in fact exerted considerable impact on modernist work of the years between the wars, in that its geometric idiom enabled modernist writers to articulate their responses to both personal and political crises of the 1930s and 1940s. Informed by extensive archival research as well as treatment of several of the least-known texts of the modernist milieu, The Geometry of Modernism clarifies and enriches the legacy of this vital period.

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