This remarkable book endures as a true masterpiece of mathematical exposition. The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. Geometry and the Imagination is full of interesting facts, many of which you wish you had known before. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is ``Projective Configurations''. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the pantheon of great mathematics books.

This book offers an archeology of the undeveloped potential of mathematics for critical theory. As Max Horkheimer and Theodor W. Adorno first conceived of the critical project in the 1930s, critical theory steadfastly opposed the mathematization of thought. Mathematics flattened thought into a dangerous positivism that led reason to the barbarism of World War II. The Mathematical Imagination challenges this narrative, showing how for other German-Jewish thinkers, such as Gershom Scholem, Franz Rosenzweig, and Siegfried Kracauer, mathematics offered metaphors to negotiate the crises of modernity during the Weimar Republic. Influential theories of poetry, messianism, and cultural critique, Handelman shows, borrowed from the philosophy of mathematics, infinitesimal calculus, and geometry in order to refashion cultural and aesthetic discourse. Drawn to the austerity and muteness of mathematics, these friends and forerunners of the Frankfurt School found in mathematical approaches to negativity strategies to capture the marginalized experiences and perspectives of Jews in Germany. Their vocabulary, in which theory could be both mathematical and critical, is missing from the intellectual history of critical theory, whether in the work of second generation critical theorists such as Jürgen Habermas or in contemporary critiques of technology. The Mathematical Imagination shows how Scholem, Rosenzweig, and Kracauer’s engagement with mathematics uncovers a more capacious vision of the critical project, one with tools that can help us intervene in our digital and increasingly mathematical present. The Mathematical Imagination is available from the publisher on an open-access basis.

"This book considers conditions of applicability of mathematics to the study of natural phenomena. The possibility of such an application is one of the fundamental assumptions underlying the enormous theoretical and practical success of modern science. Addressing problems of matter, substance, infinity, number, structure of cognitive faculties, imagination, and of construction of mathematical object, Dmitri Nikulin examines mathematical (geometrical) objects in their relation to geometrical or intelligible matter and to imagination. The author explores questions in the history of philosophy and science, particularly in late antiquity and early modernity. The focus is on key thinkers Plotinus and Descartes (with the occasional appearance of Plato, Aristotle, Euclid, Proclus, Newton and others), in whom the fundamental presuppositions of ripe antiquity and of early modernity find their definite expression."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved

Geometry is a central subject in Steiner-Waldorf schools, weaving into different subject areas throughout the 12 years. Geometry helps children explore both the outward world, and the inner human world. It helps them develop spacial harmony, and their analytical thinking.This comprehensive book has sections on Pre-Geometry, First Lessons, Pentagon and Pentagram, The Four Rules of Arithmetic, The Five Regular Solids, The Conic Sections, and Projective Geometry.It will be a particularly valuable resource for teachers of Years 6 to 8, and into High School.

With wit and clarity, the authors progress from simple arithmetic to calculus and non-Euclidean geometry. Their subjects: geometry, plane and fancy; puzzles that made mathematical history; tantalizing paradoxes; more. Includes 169 figures.

Mathematics for the Imagination provides an accessible and entertaining investigation into mathematical problems in the world around us. From world navigation, family trees, and calendars to patterns, tessellations, and number tricks, this informative and fun new book helps you to understand the maths behind real-life questions and rediscover your arithmetical mind. This is a follow-up to the popular Mathematics for the Curious, Peter Higgins's first investigation into real-life mathematical problems. A highly involving book which encourages the reader to enter into the spirit of mathematical exploration.

This book collects the papers of the conference held in Berlin, Germany, 27-29 August 2012, on 'Space, Geometry and the Imagination from Antiquity to the Modern Age'. The conference was a joint effort by the Max Planck Institute for the History of Science (Berlin) and the Centro die Ricerca Matematica Ennio De Giorgi (Pisa).

An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students. Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists--who include mathematicians and scientists--examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery. Classroom-tested activities and problem solving Accessible problems that move beyond regular art school curriculum Multiple solutions of varying difficulty and applicability Appropriate for students of all mathematics and art levels Original and exclusive essays by contemporary artists Forthcoming: Instructor's manual (available only to teachers)

Within cognitive science, two approaches currently dominate the problem of modeling representations. The symbolic approach views cognition as computation involving symbolic manipulation. Connectionism, a special case of associationism, models associations using artificial neuron networks. Peter Gärdenfors offers his theory of conceptual representations as a bridge between the symbolic and connectionist approaches. Symbolic representation is particularly weak at modeling concept learning, which is paramount for understanding many cognitive phenomena. Concept learning is closely tied to the notion of similarity, which is also poorly served by the symbolic approach. Gärdenfors's theory of conceptual spaces presents a framework for representing information on the conceptual level. A conceptual space is built up from geometrical structures based on a number of quality dimensions. The main applications of the theory are on the constructive side of cognitive science: as a constructive model the theory can be applied to the development of artificial systems capable of solving cognitive tasks. Gärdenfors also shows how conceptual spaces can serve as an explanatory framework for a number of empirical theories, in particular those concerning concept formation, induction, and semantics. His aim is to present a coherent research program that can be used as a basis for more detailed investigations.