New Horizons in Geometry

Tom M. Apostol; Mamikon A. Mnatsakanian

doi:10.5948/9781614442103

New Horizons in Geometry represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors-Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order. The hundreds of full color illustrations by Mamikon are visually enticing and provide great motivation to read further and savor the wonderful results. Lengths, areas, and volumes of curves, surfaces, and solids are explored from a visually captivating perspective. It is an understatement to say that Apostol and Mamikon have breathed new life into geometry.

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New Horizons in Geometry

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

Introduction
pp xi-xii

Foreword
pp xiii-xiv

Chapter 1 - Mamikon's Sweeping Tangent Theorem
pp 1-30

Chapter 2 - Cycloids and Trochoids
pp 31-64

Chapter 3 - Cyclogons and Trochogons
pp 65-100

Chapter 4 - Circumgons and Circumsolids
pp 101-134

Chapter 5 - The Method of Punctured Containers
pp 135-168

Chapter 6 - Unwrapping Curves from Cylinders and Cones
pp 169-212

Chapter 7 - New Descriptions of Conics via Twisted Cylinders, Focal Disks, and Directors
pp 213-242

Chapter 8 - Ellipse to Hyperbola: “With This String I Thee Wed”
pp 243-266

Chapter 9 - Trammels
pp 267-294

Chapter 10 - Isoperimetric and Isoparametric Problems
pp 295-330

Chapter 11 - Arclength and Tanvolutes
pp 331-374

Chapter 12 - Centroids
pp 375-400

Chapter 13 - New Balancing Principles with Applications
pp 401-442

Chapter 14 - Sums of Squares
pp 443-472

Chapter 15 - Appendix
pp 473-500

Bibliography
pp 501-504

Index
pp 505-512

About the Authors
pp 513-513

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