Annotated Bibliography
Related to Geometry

Last Updated: August 2004

Any additions or corrections are welcomed. Send to dwh2@cornell.edu

Most of these books (and articles) are in my personal library – they are grouped into subject matter sections. The remainder I have read or consulted in connection with the writing of my books. The annotations in quotes (" ") are taken from the listed book, usually from the Preface.  When available I have included links to online versions of the book. In addition, the The University of Michigan historic books collection contains many geometry books that may be of interest – some of these are listed at the end.

Sections:

AD. Art and Design

AG. Analytic Geometry

AN. Analysis

AT. Ancient Texts

CA. Calculus

CE. Cartography, the Earth

CG. Computers and Geometry

CT. College Teaching

DC. Dissections and Constructions

DG. Differential Geometry

DS. Dimensions and Scale

EG. Expositions - Geometry

EM. Expositions - Mathematics

FO. Foundations of Geometry

FR. Fractals

GC. Geometry in Different Cultures

GS. Geometry and Science

HI. History of Mathematics

HM. History of a Mathematician

HY. Hyberbolic Geometry

IN. Inversions

LA. Linear Algebra and Geometry

LS. Learning/Students

ME. Mechanisms

MI. Minimal Surfaces

MP. Models and Polyhedra

MS. Mathematics and Social Issues

NA. Nature

PA. Projective and Affine Mathematics

PH. Philosophy of Mathematics

RN. Real Numbers

SA. Sacred Geometry

SG. Symmetry and Groups

SP. Spherical Geometry

TG. Teaching Geometry

TM. Teaching Mathematics

TP. Topology

TX. Geometry Texts

UN. The Physical Universe

University of Michigan historic books collection

 

AD. Art and Design

Albarn, Keith, Smith, Jenn Mial, Steele, Stanford, and Walker, Dinah. The Language of Pattern. New York: Harper & Row, 1974. 

                    Inspired by Islamic decorative pattern, the authors of this book, who are all designers, explore pattern step by step, beginning with simple numerical and geometrical relationships and progressing through the dimensions

 

Alexander, Christopher, Ishikawa, Sara, and Silverstein, Murray. A Pattern Language: Towns, Bulidings, Construction. New York: Oxford University Press, 1977. 

                    A pattern language for building

 

Auvil, Kenneth W. Perspective Drawing. Mountain View, CA: Mayfield Publishing, 1997. 

                   

 

Baglivo, Jenny A. and Graver, ack E. Incidence and Symmetry in Design and Architecture. New York: Cambridge University Press, 1983. 

                    "The purpose of this text is to develop mathematical topics relevant to the study of the incidence and symmetry structures of geometrical objects. A secondary purpose is to expand the reader's geometric intuition. The two fundamental mathematical topics employed in this endeavor are graph theory and the theory of transformation groups."

 

Bain, George. Celtic Arts: The Methods of Construction. London: Constable, 1977. 

                    A description of the construction of Celtic patterns and designs.

 

Blackwell, William. Geometry in Architecture. New York: John Wiley & Sons, 1984. 

                    William Blackwell offers a basic review of the fascinating relationships that exist in linear design. At the same time, he uncovers new geometric principles and new applications of geometry that may have a major influence on the state of architecture today.

 

Coxeter, H.S.M., Emmer, M., Penrose, R., and Teuber, M.L:.M.C. Escher: Art  and Science,  New York: Elseview Science Publishing Co., Inc., 1986.

                   

 

Doczi, György. The Power of Limits. Boulder, CO: Shambhala, 1981. 

                   

 

Edgerton, Samuel Y., Jr. The Heritage of Giotto's Geometry Art and Science on the eve of the Scientific Revolution. Ithaca: Cornell University Press, 1993. 

                    A historical account of the development of perspective in the art of the Italian Renaisance.

 

Edmondson, Amy C. A Fuller Explanation:The Synergetic Geometry of R. Buckminster Fuller. Boston: Birkhauser, 1987. 

                    An account of the geometry and design ideas of Fuller.

 

Elam, Kimberly. Geometry of Design: Studies in Proposition and Composition. New York: Princeton Architectural Press, 2001. 

                    "This book seeks to explain visually the principles of geometric composition and offers a wide selection of professional posters, products, and buildings that are visually analyzed by these principles."

 

Emmer, Michele:.The Visual Mind: Art and Mathematics,  Cambridge: MIT Press, 1993.

                   

 

Ernst, Bruno. The Magic Mirror of M.C. Escher. New York: Random House, 1976. 

                    Throughout the book Bruno Ernst describes in detail the conception and execution of Escher's popular prints, showing with the aid of sketches and diagrams how the artist arrived at such astonishing creations as "The Balcony" and "Print Gallery." Careful attention is also paid to the graphic techniques Escher employed so successfully."

 

Escher, M.C. The Graphic Work of M.C. New York: Hawthorn Books,Inc.,Publishers, 1960. 

                    It is a fact, however, that most people find it easier to arrive at an understanding of an image by the round-about method of letter symbols than by the direct route. So it is with a view to meeting this need that I myself have written the text.

 

Field, Judith Veronica. The Invention of Infinity: Mathematics and Art in Renaissance. Oxford: Oxford University Press, 1997. 

                    Book will look at the relations between of Renaissance art and mathematics in the period from about 1300 to about 1650.

 

Fomenko, Anatolii. Mathematical Impressions. Providence,Rhode Island: American Mathematical Society, 1991. 

                    This book contains more than 80 reproductions of works by Fomenko. In the accompanying captions, Fomenko explains the mathematical motivation behind the illustrations as well as the emotional, historical, or mythical subtexts they evoke.

 

Ghyka, Matila. The Geometry of Art and Life. New York: Dover Publications, 1977. 

                   

 

Gombrich, Ernst. The Sense of Order: A Study in the Psychology of Decorative Art. Ithaca, NY: Cornell University Press, 1978. 

                   

 

Henderson, Linda. The Fourth Dimention and Non-Euclidean Geometry in Modern Art. Princeton,NJ: Princeton University Press, 1983.  

                   

 

Hersey, George L. Architectgure and Geometry in the Age of the Baroque. Chicago: The University of Chicago Press, 2000. 

                   

 

Holt, Michael. Mathematics in Art. London: Studio Vista, 1971. 

                    This book is not an account of either specialism of the title; that I leave to the acknowledged experts. Rather it is an attempt to focus on aspects common, it seems to me, to both mathematics and the visual arts. These aspects form then an anthology of creative highlights that have caught my eye.

 

Ivins, William M., Jr. Art & Geometry: A Study In Space Intuitions. New York: Dover Publications, 1946. 

                   

 

Jacobs, Michael and Fernández, Francisco. Alhambra. New York: Rizzoli, 2000. 

                   

 

Kappraf, Jay. Connections: The Geometric Bridge between Art and Science. New York: McGraw-Hill, 1991. 

                    There is a hidden harmony in the works of man and nature. From the great pyramid of Cheops to patterns of plant growth, natural and artificial designs are all governed by precise geometric laws. Design Science is the study of these hidden laws; it is the search for the connections underlying all that is beautiful and functional.

 

King, Ross. Bruelleschi's Dome: How a Renaissance Genius Reinvented Architecture. New York: Penquin Books, 2000. 

                   

 

Linn, Charles. The Golden Mean: Mathematics and the Fine Arts. Garden City, NY: Doubleday, 1974. 

                   

 

Lord, E.A. and Wilson, C.B. The Mathematical Description of Shape and Form. New york: Halsted Press, 1986. 

                    "Thus, in this survey, we are not presenting a compendium of unrelated mathematical techniques. Instead, we have attempted to present a unified view of the mathematics of form description, emphasising underlying mathematical principles."

 

Miyazaki, Kojiv. An Adventure in Multidimensional Space. New York: John Wiley and Sons, Inc., 1983. 

                    The art and geometry of polygons, polyhedra, and polytopes.

 

Schattschneider, Doris. Visions of Symmetry : Notebooks, Periodic Drawings, and Related Work of M.C. Escher. WH Freeman & Co, 1992. 

                   

 

Schattschneider, Doris, and Emmer, Michele:.M. C. Escher's Legacy:  A Centennial Celebration,  New York: Springer-Verlag, 2003.

                    The book features 40 articles, most by presenters at the Escher Centennial Congress in Rome and Ravello in 1998 and others. There is a rich array of illustrations, both of Escher's work and of original work by the authors.  The CD Rom supplements the book with presentations of art (in color), as well as some videos, animations, and demo software.

 

Schneider, Michael S. A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperPerennial, 1994. 

                   

 

Strosberg, Eliane. Art and Science. New York: Abbeville Press, 2001. 

                   

 

Taylor, Anne. Math in Art. Hayword, CA: Activity Resources Co., Inc., 1974. 

                    This book has been developed to show children the unique relationship between art and math and to help them discover concepts in each areas, as they relate to each other.

 

Watson, Ernest W. Creative Perspective for Artists and Illustrators. Mineola, NY: Dover Publications, 1992. 

                   

 

Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. 1979: Dover, 1979. 

                   

 

de Vries, Jan Vredeman. Perspective. New York: Dover, 1968. 

                    Reproductions of engravings from the 1604/1605 edition. Warning: Some of the engravings have geometrically incorrect perspective.

 

 

AG. Analytic Geometry

 

Hahn, Liang -shin. Complex Numbers & Geometry. Washington DC: Mathematical Association of America, 1994. 

                    The purpose of the book is to demonstrate that these two subjects can be blended together beautifully, resulting in easy proofs and natural generalizations of many theorems in plane geometry.

 

Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton: Princeton University Press, 1999. 

                    "This book is intended for all those mathematicians, engineers, and physicists who have to know, or who want to know, more about the modern theory of quaternions. Primarily, as the title page suggests, it is an exposition of the quaternion and its primary application as a rotation operator."  Included are applications of spherical geometry.

 

Postnikov, M. Lectures in Geometry Semester I Analytic Geometry. Moscow: MIR publishers, 1982. 

                    The subject matter is presented on the basis of vector axiomatics of geometry with special emphasis on logical sequence in introduction of the basic geometrical concepts.

 

Schwerdtfeger, Hans. Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry. New York: Dover Publications, Inc., 1979. 

                    This book uses complex numbers to analyze inversions in cricles and then their relationship to hyperbolic geometry.

 

Smogorzhevsky, A.S. The Method of Coordinates. Moscow: Mir Publishers, 1984. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

 

AN. Analysis

 

Bishop, Errett and Douglas,Bridges. Constructive Analysis. New York: Springer-Verlag, 1985. 

                    The main book on constructive analysis.

 

Bressoud, David. A Radical Approach to Real Analysis. Washington, DC: Mathematical Association of America, 1994. 

                   

 

Goldblatt, Robert. Lectures on the Hyperreals. New York: Springer, 1998. 

                   

 

Hairer, E. and Wanner, G. Analysis by Its History. New York: Springer, 1996. 

                   

 

Rudin,Walter. Principles of Mathematical Analysis. New York: McGraw Hill, 1964. 

                    For many years a standard text in analysis.

 

Strichartz, Robert S. The Way of Analysis. Boston: Jones and Bartlett Publishers, 1995. 

                    The presentation of the material in this book is often informal. A lot of space is given to motivation and a discussion of proof strategies." This is only recent analysis book that I know of that is direct and honest about Archimedean Axiom.

 

 

AT. Ancient Texts

 

AL-Khawarizmi, Muhammad Ibn Musa. Al-Jabr wa-l-Muqabala. Baghdad: House of Wisdom, 825. 

                    Traslated in English in Karpinski, L.C., ed., Robert of Chester's Latin Translation of Al'Khowarizmi's Algebra, New York: Macmillan, 1915.

 

Perga, Apollonius of. On Cutting Off a Ratio. Fairfield: The Golden Hind Press, 1987. 

                    An Attempt to Recover the Original Argumentation through a Critical Translation of the Two Extant Medieval Arabic Manuscript.

 

Perga, Apollonius of. Treatise on Conic Sections. New York: Dover, 1961. 

                    This is the standard work on conic sections from the Greek world.

 

Baudhayana. Sulbasutram. Bombay: Ram Swarup Sharma, 1968. 

                    This is translated from the Sanskrit manual for the construction of alters. The beginning of the book contains a discussion the geometry needed for the construction of the altars  this beginning section is apparently the oldest surviving geometry textbook.

 

Berggren, J. Lennart and Jones, Alexander. Ptolemy's /it Geography : an annotated translation of the theoretical chapters. Princeton, NJ: Princeton University Press, 2000. 

                   

 

Berggren, J.L. and Thomas, R.S.D. Euclid's Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy. New York: Garland Publishing, 1996. 

                    Contains the only accessible English translation of Euclid's Phaenomena. This work is, alas, out of print, but a brief, and more easily obtained account of its comments can be found in: Berggren, J.L., and Thomas, R.S.D.,                                           "Mathematical Astronomy in the Fourth Century B.C. as found in Euclid's Phaenomena", Physis, Vol XXIX (1992), 7-33.

 

Bonasoni, Paolo. Algebra Geometrica. Annapolis: The Golden Hind Press, 1985. 

                    being the only known work of this nearly forgotten Renaissance mathematician (excepting a still unpublished treatise on the division of circles).

 

Cardano, Girolamo. The Great Art or the Rules of Algebra. Cambridge: MIT Press, 1968. 

                    This is the book that first describes algebraic algorithms for solving most cubic equations.

 

Coxeter, H.S.M. and Greitzer, S.L. Geometry Revisited. New York: The L.W. Singer Company, 1967. 

                    "Using whatever means will best suit our purposes, let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused."

 

Descartes, Rene. The Geometry of Rene Descartes. New York: Dover Publications,Inc., 1954. 

                    This the book in which Descartes develops the use of what we now call Cartesian coordinates for the study of curves.

 

Euclid: Phaenomena. Euclidis opera omnia. Menge H, (eds). anonymous Lipsiae, B.G. Teubneri, 1883,

                   

 

Euclid:Optics, . Journal of the Optical Society of America 35, no. 5 (1945), 357-372.

                    This is a translation of Euclid's work that contains the elements of what we now call perjective geometry.

 

Euclid. Elements. New York: Dover, 1956. 

                    This is edition of Eulid's Elements to which one is usually referred. Heath has added a large collection of very useful historical and philosophical notes.

 

Euclid. Elements. London: Dent & Sons, 1933. 

                    Todhunter's translation of Euclid.

 

Euclid. Elements. Green Lion Press, 2002. 

                    Thomas L. Heath translation, edited by Dana Densmore, all in one volume without Heath extensive notes

 

Galilei, Galileo: Trattato della Sphaera (1586-87). Galilei Opere. Favaro A, (eds). anonymous Florence, G. Barbera, 1953,

                   

 

Guthrie, Kenneth. The Pythagorean Sourcebook and Library. Grand Rapids: Phanes Press, 1987. 

                    An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy.

 

Heath, T.L. Euclid: The Thirteen Books of the Elements. New York: Dover, 1956. 

                    This is edition of Eulid's Elements to which one is usually referred. Heath has added a large collection of very useful historical and philosophical notes. His notes are more extensive than Euclid's text.

 

Karpinski, L.C.:.Robert of Chester's Latin Translation of Al'Khowarizmi's Algebra,  New York: Macmillan, 1915.

                   

 

Khayyam, Omar: a paper  (no title), . Scripta Mathematica 26 (1963), 323-337.

                    In this paper Khayyam discusses algebra in relation to geometry.

In this paper Khayyam discusses algebra in relation to geometry.

 

Khayyam, Omar. Risâla fî sharh mâ ashkala min musâdarât Kitâb 'Uglîdis. Alexandria, Egypt: Al Maaref, 1958. 

                   

 

Khayyam, Omar. Algebra. New York: Columbia Teachers College, 1931. 

                    In this book Khayyam gives geometric techniques for solving cubic equations.

 

Plato. The Collected Dialogues. Princeton,NJ: Bollinger, 1961. 

                    Plato discusses mathematical ideas in many of his dialogues.

 

Plotinus. The Enneads. Burdette, NY: Larson Publications, 1992. 

                   

 

Proclus. Proclus: A Commentary on the First Book of Euclid's Elements. Princeton: Princeton University Press, 1970. 

                    These commentaries by Proclus (Greek, 410-485) are a source of much of our information about the thinking of mathematicians toward the end of the Greek era.

 

Saccheri, Girolamo. Euclides Vindicatus. New York: Chelsea pub. Co., 1986. 

                    In this book Girolamo Sacchri set forth in 1733, for the first time ever, what amounts to the axiom systems of non-Euclidean geometry." It is not mentioned in this volume that Saccheri borrowed many ideas from Khayyam's Risâla fî sharh mâ ashkala min musâdarât Kitâb 'Uglîdis.

 

Smyrna, Theon of. Mathematics Useful for Understanding Plato. San Diego: Wizards Bookshelf, 1978. 

                    This work appears to have been a text book intended for students who were beginning a study of the works of Plato. In its original form there were five sections: 1) Arithmetic 2) Plane Geometry 3) Stereometry (solid geometry) 4) Music 5) Astronomy. Sections 2 and 3 on Geometry have been lost while the others remain in their entirety and are presented here." The section on Astronomy contains discussions of the shape of space.

 

Thomas, Ivor:.Selections Illustrating the History of Greek Mathematics,  Cambridge, MA: Harvard University Press, 1951.

                    A collection of primary sources.

 

 

CA. Calculus

 

Amdahl, Kenn and Loats, Jim. Calculus For Cats. Broomfield, CO: Clearwater Publishing, 2001. 

                   

 

Berlinski, David. a tour of the calculus. New York: Pantheon Books, 1995. 

                   

 

Cohen, David W.: Henle, James M. Conversational Calculus. Reading, MA: Addison-Wesley, 1997. 

                   

 

Devlin, Keith. An Electronic Companion to Calculus. Cogito Learning Media, Inc, 1997. 

                   

 

Simmons, George F. Calculus Gems. New York: McGraw-Hill, Inc., 1992. 

                   

 

Grabiner, Judith V. The Origins of Cauchy's Rigorous Calculus. Cambridge: MIT Press, 1981. 

                   

 

Spivak, Michael. The Hitchhiker's Guide to Calculus: A Calculus Course Companion. Houston: Polished Pebble Press, 1995. 

                   

 

 

CE. Cartography, the Earth

 

Bagrow, L. A History of Cartography. Cabridge, MA: Harvard University Press, 1964. 

                   

 

Monmonier, Mark. Drawing the Line: Tales of Mapes and Cartocontroversy. New York: Henry Holt and Company, 1995. 

                   

 

Monmonier, Mark. How to Lie with Maps. University of Chicago Press, 1996. 

                   

 

Pottmann, Helmut: "Rational curves and surfaces with rational offsets", Computer Aided Geometric Design, 12 (1995), 175-192.

                   

 

Smith, James R. Introduction to Geodesy (The History and Concepts of Modern Geodesy). John Wiley Interscience, 1997. 

                   

 

Snyder, John P. Flattening the Earth: Two Thousand Years of Map Projections. Chicago: University of Chicago Press, 1993. 

                    A history and mathematical description of numerous map projections of the sphere.

 

Sobel, Dava. Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time. New York: Penquin Books, 1995. 

                    A account of the struggles to develop a method for determining the longitude of ships at sea.

 

Tomilin, Anatoly. How People Discovered the Shape of the Earth. Moscow: Raduga Publishers, 1984. 

                    A childrens book with nice colored illustrations.

 

 

CG. Computers and Geometry

 

Vision Geometry, Contemporary Mathematics.119: Washington DC: American Mmathematical Society, 1989.

                    "Computer vision is concerned with obtaining descriptive information about a scene by computer analysis of images of the scene."

 

The Geometer's Sketchpad: Dynamic Geometry for the 21st Century, Key Curriculum Press

                    A program running on Windows or Mac platforms which allows you to construct geometric drawing with points, lines, and circles and then to dynamically vary constituent parts.

 

Angel, Edward. Interactive Computer Graphics: A Top-Down Approach with OpenGL. Addison-Wesley, 1999. 

                   

 

King, James. Geometry Through the Circle with The Geometer's Sketchpad. Key Curriculum Press, 1994. 

                   

 

King, James, and Schattschneider, Doris (editors): Geometry Turned On: Dynamic Software in Learning, Teaching and Research, MAA Notes.41:, 1997.

                    A book of 26 papers about aspects of dynamic software for geometry

 

Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton: Princeton University Press, 1999. 

                    "This book is intended for all those mathematicians, engineers, and physicists who have to know, or who want to know, more about the modern theory of quaternions. Primarily, as the title page suggests, it is an exposition of the quaternion and its primary application as a rotation operator."  Included are applications of spherical geometry.

 

Litchfield, Dan, Goldenheim, Dave, and Dietrich, Charles H.: "Euclid, Fibonacci, and Sketchpad", Math Horizons, Feb 1997, 9-10.

                   

 

Lord, E.A. and Wilson, C.B. The Mathematical Description of Shape and Form. New york: Halsted Press, 1986. 

                    "Thus, in this survey, we are not presenting a compendium of unrelated mathematical techniques. Instead, we have attempted to present a unified view of the mathematics of form description, emphasising underlying mathematical principles."

 

Mortenson, Michael E. Geometric Modeling. New York: John Wiley and Sons, 1997. 

                    "[This text] offers the reader a comprehensive look at the indispensable core concepts of geometric modeling, describing and comparing all the important mathematical structures for modeling curves, surfaces, and solids, and showing how to shape and assemble those elements into more complex models."

 

Pottmann, Helmut: "Rational curves and surfaces with rational offsets", Computer Aided Geometric Design, 12 (1995), 175-192.

                   

 

Prenowitz, Walter and Jordan, Meyer. Basic Concepts of Geometry. New York: Ardsley house Publishers, 1989. 

                   

 

Richter-Gerbert, Jürgen, and Kortenkamp, Ulrich H.:Cinderella: The Interactive Geometry Software, Heidelberg: Springer-Verlag  (1999)

                    A Java based dynamic geometry software.

 

Rovenski, V.Y. Geometry of Curves and Surfaces with MAPLE. Boston: Birkhäuser,

                    This concise text on geometry with computer modeling presents some elementary methods for analytical modeling and visualization on curves and surfaces.

 

Taylor, Jean: Computing Optimal Geometries, Providence: American Mathematical Society  (1991)

                    "This videotape testifies to the influence of computing and computer graphics in mathematical research. The material on the videotape was presented in a Special Session on Computing Optimal Geometries, held at the Joint Mathematics Meetings in San Francisco in January, 1991."

 

Taylor, Walter F. The Geometry of Computer Graphics. Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1992. 

                    "This book is a direct presentation of elementary analytic and projective geometry, as modeled by vectors and matrices and as applied to computer graphics."

 

 

CT. College Teaching

 

Case, Bette Anne (editors): You're the Professor, What Next?, Ideas and Resources For Preparing College Teachers,  Washington DC.: The Mathematical Association of America, 1994.

                   

 

Ewing, John (editors): Towards Excellence, Leading Dotoral Mathematics Department In The 21st Century,  Washington DC.: American Mathematical Society, 1999.

                   

 

Fisher, Naomi D., Keynes, Harvey B., and Wagreich, Philip D (editors): Mathematicians and Education Reform, Issues in Mathematics Education.3: Providence, Rhode Island: American Mathematical Society, 1993.

                   

 

Sciences, The ConferenceBoardoftheMathematical. The Mathematical Education of Teachers. Providence, RI: American Mathematical Society, 2001. 

                   

 

 

DC. Dissections and Constructions

 

Beskin, N.M. Dividing a Segment in a Given Ratio. Moscow: Mir Publishers, 1975. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Boltjansky, V. and Gohberg, I. Results and Problems in Combinatorial Geometry. Cambridge: Cambridge University Press, 1985. 

                   

 

Boltyanski, Vladimir G. Hilbert's Third Problem. New York: John Wiley & Sons, 1978. 

                    A discussion of dissections on the plane, sphere, and hyperbolic spaces.

 

Boltyanski, Vladimir and Soifer, Alexander. Geometric Etudes in Combinatorial Mathematics. Colorada Springs, CO: Center for Excellence in Mathematical Education, 1991. 

                   

 

Boltyanskii, Vladimir G. The Decomposition of Figures into Smaller Parts. Chicago: University of Chicago Press, 1980. 

                   

 

Dudley, Underwood. A budget of Trisections. New York: Springer-Verlag, 1987. 

                    "This book is about angle trisections and the people who attempt them. Its purposes are to collect many trisections in one place, inform about trisectors, amuse the reader, and, perhjaps most importantly, to reduce the number of trisectors."

 

Eves, Howard. A Survey of Geometry. Boston: Allyn & Bacon, 1963. 

                    A textbook that contains an extensive coverage of the dissection theory of polygons.

 

Frederickson, Greg. Dissections: Plane and Fancy. New York: Cambridge University Press, 1997. 

                    This book is a collection of interesting dissection puzzles, old and new, and is an instructive manual on the art and science of geometric dissections.

 

Frederickson, Greg. Hinged Dissections: Swinging & Twisting . Cambridge, UK: Cambridge University Press, 2002. 

                    The book explores all manner of dissections whose pieces are hinged together, along with techniques that allow you to design them.  It is a nice sequel to it Dissections: Plane & Fancy.

 

Ho, Chung -Wu: "Decomposition of a Polygon into Triangles", Mathematical Gazette, 60 (1976), 132-134.

                    This article contains a proof that all planar polygons can be dissected into triangles and discusses the many mistakes made by other (many well-known ones) authors in their "proofs" of the same result.

 

Lindgren, Harry. Geometric Dissections. Princeton, NJ: D. Van Nostrand Company, 1964. 

 

                   

Lindgren, Harry. Recreational Problems in Geometric Dissection and How to Solve Them. New York: Dover, 1972. 

 

 

Martin, George E. Geometric Constructions. New York: Springer, 1998. 

                   

 

Sah, C.H. Hilbert's Third Problem: Scissors Congruence. London: Pitman, 1979. 

                    A detailed discussion of the three dimensional dissections.

 

Soifer, Alexander. How Does One Cut a Triangle? Colorado Springs, CO: Center for Excellence in Mathematical Education, 1990. 

                   

 

 

DG. Differential Geometry

 

Berger, M. and Gostiaux, B. Differential Geometry: Manifolds, Curves, and Surfaces. New York: Springer-Verlag, 1988.  

                   

 

Bishop, Richard L. and Goldberg, Samuel I. Tensor Analysis on Manifolds. New York: Dover Publications, 1980. 

                    The subject is treated as a continuation of advanced calculus. The standards of rigor and logical completeness are high throughout the text, and many excellent problems are presented

 

Bloch, Ethan D. A First Course in Geometric Topology and Differential Geometry. Boston: Birkhauser, 1997. 

                    Contains the topological classification and differential geometry of surfaces.

 

Casey, James. Exploring Curvature. Wiesbaden: Vieweg, 1996. 

                    A truly delightful book full of "experiments" to physically explore curvature of curves and surfaces.

 

Dodson, C. T. J. and Poston, T. Tensor Geometry. London: Pitman, 1979. 

                    A very readable but technical text using linear (affine) algebra to study the local intrinsic geometry of spaces leading up to and including the geometry of the theory of relativity.

 

Dubrovin, B.A., Fomenko, A.T., and Novikov, S.P. Modern Geometry: Methods and Applications(Part I. The Geometry of Surfaces, Transformation Groups, and Fields). New York: Springer-Verlag, 1984. 

                    A well-written graduate text.

 

Gauss, C.F. Modern Differential Geometry of Curves and Surfaces. Hawlett, NY: Raven Press, 1965. 

                    A translation into English of Gauss' early papers on surfaces.

 

Gibson, C.G. Elementary Geomery of Differentiable Curves: An Undergraduate Introduction. Cambridge, UK: Cambridge University Press, 2001. 

                   

 

Gray, A. Modern Differential Geometry of Curves and Surfaces. CRC, 1993. 

                    This is a very extensive book based on computations using Mathematica©.

 

Guggenheimer, Heinrich. Differential Geometry. New York: McGraw Hill, 1963. 

                   

 

Henderson, David W. Differential Geometry: A Geometric Introduction. Upper Saddle River, NJ: Prentice Hall, 1998. 

                    In this book we will study a foundation for differential geometry based not on analytic formalisms but rather on these underlying geometric intuitions.

 

Hicks, Noel J. Notes of Differential Geometry. New York: Van Nostrand Reinhold Company, 1971. 

                    The first three chapters of this book provide a short course on classical differential geometry and could be used at the junior level with a little outside reading in linear algebra and advanced calculus.

 

Koenderink, Jan J. Solid Shape. Cambridge: M.I.T. Press, 1990. 

                    Written for engineers and applied mathematicians, this is a discussion of the extrinsic properties of three-dimensional shapes. There are connections with applications and a nice section "Your way into the literature."

 

Kreyszig, Erwin. Mathematical Expositions No. 11: Differential Geometry. Toronto: University of Toronto Press, 1959. 

                    This book provides an introduction to the differential geometry of curves and surfaces in three-dimensional Euclidean space... In the theory of surfaces we make full use of the tensor calculus, which is developed as needed.

 

Laugwitz, Detlef. Differential and Riemannian Geometry. New York: Academic Press, 1965. 

                    This textbook is intended to be an introduction to classical differential geometry as well as to the tensor calculus and to Riemannian geometry.

 

McCleary, John. Geometry from a Differential Viewpoint. Cambridge, UK: Cambridge University Press, 1994. 

                    The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of ... the hyperbolic plane.

 

Millman, R.S. and Parker, G.D. Elements of Differential Geometry. Englewood Cliffs, NJ: Prentice-Hall, 1977. 

                    A well-written text, which uses linear algebra extensively to treat the formalisms of extrinsic differential geometry.

 

Morgan, Frank. Riemannian Geometry: A Beginner's Guide. Boston: Jones and Bartlett, 1993. 

                    An accessible guide to Riemannian geometry including a chapter on the theory of relativity and the calculation of the precession in the orbit of Mercury.

 

Oprea, John. Introduction to Differential Geometry and Its Applications. Upper Saddle River: Prentice Hall, 1997. 

                   

 

Penrose, Roger: The Geometry of the Universe. Mathematics Today. Steen L, (eds). New York, Springer-Verlag, 1978,

                    An expository discussion of the geometry of the universe.

An expository discussion of the geometry of the universe.

 

Prakash. Differential Geometry: An Integrated Approach. New Delhi: Tata McGraw-Hill Publishing Company Limited, 1981. 

                   

 

Rovenski, V.Y. Geometry of Curves and Surfaces with MAPLE. Boston: Birkhäuser,

                    This concise text on geometry with computer modeling presents some elementary methods for analytical modeling and visualization on curves and surfaces.

 

Santander, M.: "The Chinese South-Seeking chariot: A simple mechanical device for visualizing curvature and parallel transport", American Journal of Physics, 60 (9) (1992), 782-787.

                    An old mechanical device, the Chinese South-Seeking chariot, presumably designed to work on a flat plane, is shown to perform parallel transport on arbitrary surfaces. Its use affords experimental demonstration and even numerical checking (within a reasonable accuracy) of all the features of curvature and parallel transport of vectors in a two-dimensional surface.

 

Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Wilmington, DE: Publish or Perish, 1979. 

                    In five(!) volumes Spivak relates the subject back to the original sources. Volume V contains an extensive bibliography (up to 1979).

 

Stahl, Saul. The Poincaré Half-Plane. Boston: Jones and Bartlett Publishers, 1993. 

                    This text is an analytic introduction to some of the ideas of intrinsic differential geometry starting from the Calculus.

 

Thurston, William. Three-Dimensional Geometry and Topology, Vol. 1. Princeton, NJ: Princeton University Press, 1997. 

                    This is a detailed excursion through the geometry and topology of two- and three-manifolds. "The style of exposition in this book is intended to encourage the reader to pause, to look around and to explore.

 

Weeks, Jeffrey. The Shape of Space. New York: Marcel Dekker, 1985. 

                    An elementary but deep discussion of the geometry on different two- and three-dimensional spaces.

 

Weeks, Jeffrey. Shape of Space. New York: Marcel Dekker, 2002. 

                   

 

do Carmo, Manfredo Perdigão. Riemannian Geometry. Boston: Birkhäuser, 1992. 

                    The object of this book is to familiarize the reader with the basic language of and some fundamental theorems in Riemannian Geometry.

 

 

DS. Dimensions and Scale

 

Abbott, Edwin A. Flatland. New York: Dover Publications, Inc., 1952. 

                    A fantasy about two-dimensional beings in a plane encountering the third dimension.

 

Banchoff, Thomas and Wermer, John. Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions. New York: Springer-Verlag, 1983. 

                    This book treats a number of themes that center on the notion of dimensions, tracing the different ways in which mathematicians and others have met them in their work.

 

Burger, Dionys. Sphereland. New York: Thomas Y. Crowell Co., 1965. 

                    A sequel to Abbott's Flatland.

 

Henderson, Linda. The Fourth Dimention and Non-Euclidean Geometry in Modern Art. Princeton,NJ: Princeton University Press, 1983. 

                   

 

Kohl, Judith and Kohl, Herbert. The View from the Oak: The Private Worlds of Other Creatures. New York: Sierra Club Books/Charles Scribner's Sons, 1977. 

                    This delightful books describes the various experiential worlds of different creatures and is a good illustration of intrinsic ways of thinking. Included are differing dimensions and scales of these worlds.

 

Morrison, Phillip and Morrison, Phylis. Powers of Ten: About the Relative Size of Things in the Universe. New York: Scientific American Books, Inc., 1982. 

                    A beautiful book (and a video with the same title) that starts with a square meter on earth and then zooms out and in by powers of ten describing and illustrating at each power of ten what can be seen until it reaches (by zooming out) vast stretches of empty space in the universe or (by zooming in) the empty space within elementary particles.

 

Rucker, Rudy. Geometry, Relativity and the Fourth Dimension. New York: Dover, 1977. 

                    [The author's] goal has been to present an intuitive picture of the curved space-time we call home.

 

Rucker, Rudy. The Fourth Dimension. Boston: Houghton Mifflin Co., 1984. 

                    A history and description of various ways that people have considered the fourth dimension.

 

Sommerville, D.M.Y. An Introduction to the Geometry of N Dimensions. New York: Dover, 1958. 

                   

 

 

EG. Expositions – Geometry

 

Artmann, Benno. Euclid-The Creation of Mathematics. New York: Springer, 1999. 

                    Here the present book takes a clear position: The Elements are read, interpreted, and commented upon from the point of view of modern mathematics.

 

Beskin, N.M. Dividing a Segment in a Given Ratio. Moscow: Mir Publishers, 1975. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Beskin, N.M. Images of Geometric Solids. Moscow: Mir Publishers, 1985. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Blatner, David. The Joy of PI. New York: Walker Publishing Company, 1997. 

                   

 

Boltjansky, V. and Gohberg, I. Results and Problems in Combinatorial Geometry. Cambridge: Cambridge University Press, 1985. 

                   

 

Boltyanski, Vladimir and Soifer, Alexander. Geometric Etudes in Combinatorial Mathematics. Colorada Springs, CO: Center for Excellence in Mathematical Education, 1991. 

                   

 

Carroll, Lewis. Euclid and His Modern Rivals. New York: Dover Publications, Inc., 1973. 

                    Yes! Lewis Carroll of Alice in Wonderland fame was a geometer. This book is written as a drama; Carroll has Euclid defending himself against modern critics.

 

Coxeter, H.S.M. and Greitzer, S.L. Geometry Revisited. New York: The L.W. Singer Company, 1967. 

                    "Using whatever means will best suit our purposes, let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused."

 

Darley, George. Geometrical Companion: in which the Elements of Abstract Geometry are Familiarised, Illustrated, and Rendered Practically Useful to the Various Purposes of Life. London: Taylor and Walton, 1841. 

                   

 

Edwards, A.W.F. Cogwheels of the Mind: The Story of Venn Diagrams. Baltimore, MD: Johns Hopkins University Press, 2004. 

                    geometric and historical aspects of Venn diagrams, including Venn diagrams on the sphere.

 

Fetisov, A.I. Proof in Geometry. Moscow: Mir Publishers, 1978. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Gaffney, Matthew P. and Steen, Lynn Arthur. Annotated Bibliography of Expository Writing in the Mathematical Sciences. Washington, DC: M.A.A., 1976. 

                   

 

Gorini, Catherine A. (editors): Geometry at Work: Papers in Applied Geometry, MAA Notes.Number 53: Washington, DC: Mathematical Association of America, 2000.

                   

Gorini, Catherine. Facts on File Geometry Handbook. New York: Facts On File, 2003. 

                   

 

Hansen, Vagn Lundsgaard. Shadows of the circle: conic sections, optimal figures and non-Euclidean geometry. River Edge, NJ: World Scientific, 1998. 

                   

 

Hargittai, István and Hargittai, Magdolna. Symmetry: A Unifying Concept. Bolinas, CA: Shelter Publications, 1994. 

                    "The single, most important purpose of this book is to help you notice the world around you, to train your eye and mind to see new patterns and make new connections."

 

Hilbert, David and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea Publishing Co., 1983. 

                    They state "it is our purpose to give a presentation of geometry, as it stands today [1932], in its visual, intuitive aspects." It includes an introduction to differential geometry, symmetry, and patterns (they call it "crystallographic groups"), and the geometry of spheres and other surfaces. Hilbert is the most famous mathematician of the first part of this century.

 

Juster, Norton. The Dot and the Line: A Romance in Lower Mathematics. New York: Random House, 1963. 

                    A mathematical fable.

 

Kaplan, Robert and Kaplan, Ellen. The Art of the Infinite: The Pleasures of Mathematics. Oxford: Oxford University Press, 2003. 

                   

 

Kutepov, A. and Rubanov, A. Problems in Geometry. Moscow: MIR publishers, 1978. 

                    "The book contains a collection of 1351 problems (with answers) in plane and solid geometry for technical schools and colleges."

 

Lang, Serge. The Beauty of Doing Mathematics: Three Public Dialogues. New York: Springer-Verlag, 1985. 

                    Expository work by a famous mathematician.

 

Lyubich, Yu.I. and Shor, L.A. The Kinematic Method in Geometrical Problems. Moscow: Mir Publishers, 1980. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Lyusternik, L.A. The Shortest Lines. Moscow: Mir Publishers, 1983. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Markushevich, A.I. Areas and Logarithms. Moscow: Mir Publishers, 1981. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Markushevich, A.I. Complex Numbers and Conformal Mappings. Moscow: Mir Publishers, 1982. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Markushevich, A.I. Remarkable Curves. Moscow: Mir Publishers,

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Mlodinow, Leonard. Euclid's Window, The Story of Geometry from Parallel Lines to Hyperspace. New York: The Free Press, 2001. 

                   

 

Nelsen, Roger B. Proofs Without Words: Exercises in Visual Thinking. Washington, D.C.: MAA, 1993. 

                   

 

Nelson, Roger B. Proofs Without Words II: More Exercises in Visual Thinking. Washington, DC: The Mathematical Association of America, 2000. 

                   

 

Nikulin, V. V. and Shafarevich, I. R. Geometries and Groups. New York: Springer-Verlag, 1987. 

                    "This book is devoted to the theory of geometries which are locally Euclidean, in the sense they are identical to the geometry of the Euclidean plane or Euclidean 3-space... The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries."

 

Polster, Burkard. A Geometrical Picture Book. New York: Springer, 1998. 

                   

 

Pritchard, Chris. The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching. Cambridge University Press and MAA, 2002. 

                    The book is an expanded collection of 57 articles published in Mathematical Gazette and Mathematics in School — two journals of The Mathematical Association, a British organization for teachers of mathematics — over about one hundred years.

 

Rosenfeld, B.A. and Sergeeva, N.D. Stereographic Projection. Moscow: Mir Publishers, 1977. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Sharygin, I. F. Problems in Solid Geometry. Moscow: MIR publishers, 1986. 

                   

 

Smogorzhevsky, A.S. Lobachevskian Geometry. Moscow: Mir Publishers, 1982. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Smogorzhevsky, A.S. The Method of Coordinates. Moscow: Mir Publishers, 1984. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Smogorzhevsky, A.S. The Ruler in Geometrical Constructions. New York: Blaisdell Publishing COmpany, 1961. 

                   

 

Soifer, Alexander. How Does One Cut a Triangle? Colorado Springs, CO: Center for Excellence in Mathematical Education, 1990. 

                   

 

Sved, Marta. Journey into Geometries. Washinton, DC: Mathematical Association of America, 1991. 

                    "This book, though not a text, is first and foremost about geometry. It is neither comprehensive, not can it claim to go very deep into the chosen topics, yet hopefully, it may initiate a spark to light the way into further progress. The central topic in this book is non-Euclidean geometry. The approach to it is made via the Poincare model, ..."

 

Tanton, James. Solve This, Math Activities For Students And Clubs. Washington DC.: The Mathematical Association of America, 2001. 

                   

 

Valens, Evans G. The Number of Things: Pythagoras, Geometry and Humming Strings. New York: E.P. Dutton and Company, 1964. 

                    This is a book about ideas and is not a textbook. Valens leads the reader through dissections, golden mean, relations between geometry and music, conic sections, etc.

 

Vasilyev, N. and Gutenmacher, V. Straight Lines and Curves. Moscow: Mir Publishers, 1985. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Walser, Hans. The Golden Section. Washington, DC: The Mathematical Association of America, 2001. 

                   

 

Wells, David. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. 

                     "Entire books have been written about tessellations alone, or topological curiosities, or geometric extremal properties, beside the wealth of classical geometry. This is my selection from that cornucopia."

 

 

EM. Expositions – Mathematics

 

Dewdney, A.K. A Mathematical Mystery Tour: Discovering the Truth and Beauty of the Cosmos. New York City: John Wiley & Sons, 1999. 

                    "The mathematical odyssey herein explores two key questions about mathematics and its relationship to reality: Why is mathematics so amazingly successful in describing the structure of physical reality? Is mathematics created, or is it discovered?" Chapter 4 of this book is about mapping the spheres, and whole book is written in a language accessible to general audience, not only mathematicians.

 

Farmer, David W. and Stanford, Theodore B. Knots and Surfaces, A Guide to Disovering Mathematics. Washington DC: American Mathematical Society, 1996. 

                   

 

GIlbert, George T. and Hatcher, Rhonda L. Mathematics Beyond the Numbers. New York: John Wiley & Sons, 2000. 

                   

 

Gaffney, Matthew P. and Steen, Lynn Arthur. Annotated Bibliography of Expository Writing in the Mathematical Sciences. Washington, DC: M.A.A., 1976. 

                   

 

Gamow, George. One, Two, Three ... Infinity. New York: Bantam Books, 1961. 

                    A well-written journey through mathematical ideas.

 

Gerdes, Paulus. Geometrical Recreations of Africa. Maputo, Mozambique: African Mathematical Union and Higher Pedagogical Institute's Faculty of Science, 1991. 

                   

 

Guillen, Michail. Bridges to Infinity: The Human Side of Mathematics. Los Angeles: Jeremy P. Tarcher, 1983. 

                   

 

Hilton, Peter, Holton, Derek, and Pedersen, Jean. Mathematical Vistas, From a room with many windows. New York: Springer, 2002. 

                   

 

Honsberger, Ross. Mathematical Gems. Washington, DC: M.A.A., 1973. 

                    Expository stories about mathematics.

 

Honsberger, Ross. Mathematical Gems II. Washington, DC: M.A.A., 1976. 

                    Expository stories about mathematics.

 

Honsberger, Ross. Mathematical Morsels. Washington, DC: M.A.A., 1978. 

                    Expository stories about mathematics.

 

Honsberger, Ross. Mathematical Plums. Washington, DC: M.A.A., 1979. 

                    Expository stories about mathematics.

 

Honsberger, Ross. Mathematical Chestnuts from Around the World. Washingon, DC: The Mathematical Association of America, 2001. 

                   

 

Ifrah, Georges. From One to Zero: A Universal History of Numbers. New York: Penguin Books, 1987. 

                   

 

Johnson, Art. Famous Problems and their Mathematicians. Englewood, CO: Teacher Ideas Press, 1999. 

                   

 

Krantz, Steven G. Mathematical Apocrypha: Stories and Anecdotes of Mathematicians and the Mathematical. Washinton, DC: The Mathematical Association of America, 2002. 

                   

 

Lacskovich, Miklós. Conjecture and Proof. Washington, DC: The Mathematical Association of America, 2001. 

                   

 

Lang, Serge. The Beauty of Doing Mathematics: Three Public Dialogues. New York: Springer-Verlag, 1985. 

                    Expository work by a famous mathematician.

 

Lieber, Lillian R. The Education of T.C. Mits (The Celebrated Man in the Street). New York: W.W. Norton, 1972. 

                    A mathematical fantasy.

 

Pappas, Theoni. Mathematical Scandals. San Carlos, CA: Wide World Publishing/Tetra, 1997. 

                   

 

Parks, Harold, Musser, Gary, Burton, Robert, and Siebler, William. Mathematics in Life, Society, and the World. Upper Saddle River, NJ: Prentice-Hall, 1997. 

                   

 

Peterson, Ivars. Mathematical Treks: From Surreal Numbers to Magic Circles. Washington, DC: The Mathematical Association of America, 2002. 

                   

 

Pickover, Clifford A. The Mathematics of OZ: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, 2002. 

                   

 

Péter, Rozsa. Playing with Infinity. New York: Dover Publishing, Inc., 1961. 

                    "Mathematical explorations and excursions."

 

Restivo, Sal, Paul, Jean, Bendegem, Van, and Fischer, Roland (editors): MathWorlds,  New York: State University of New York Press, 1993.

                   

 

Rota, Gian -Carlo. Indiscrete Thoughts. Boston: Birkhäuser, 1997. 

                   

 

Sadovski, L.E. and Sadovskii, A.L. Mathematics and Sports. Washington DC.: American Mathematical Society, 1993. 

                   

 

Sawyer, W.W. Prelude to Mathematics. New York: Dover, 1982. 

                    "An account of some of the more stimulating and surprising branches of mathematics, introduced by an analysis of the mathematical mind, and the aims of the mathematician."

 

Shenitzer, Abe, and Stillwell, John (editors): Mathematical Evolutions, Spectrum Series Washington, DC: The Mathematical Association of America, 2002.

                   

 

Steen, Lynn Arthur (editors): Mathematics Today: Twelve Informal Essays,  New York: Springer-Verlag, 1978.

                   

 

Steen, Lynn Arthur (editors): Mathematics Tomorrow,  New York: Springer-Verlag, 1981.

                    Expository essays

 

Stein, Sherman. Strength in Numbers: Discovering the Joy and Power of Mathematics in Everyday Life. New York: John Wiley & Sons, 1996. 

                   

 

Stewart, Ian. The Problems of Mathematics. Oxford: Oxford University Press, 1987. 

                   

 

Stewart, Ian, and Jaworski, John (editors): Seven years of manifold: 1968-1980,  Nantwich, Cheshire, UK: Shiva Publishing Limited, 1987.

                    A collection of articles from a mathematics magazine published at Universtiy of Warwick, England

 

Tanton, James. Solve This: Math Activities for Students and Clubs. Washington, DC: Mathematical Association of America, 2001. 

                   

 

Uspensky, V.A. Gödel's Incompleteness Theorem. Moscow: Mir Publishers, 1987. 

                   

 

Vaderlind, Paul, Guy, RIchard, and Larson, Loren. The Inquisitive Proiblem Solver. Washington, DC: The Mathematical Association of America, 2002. 

                   

 

Wells, D.G. Recreations in Logic. New York: Dover Publications, 1979. 

                   

 

Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. London: Penquin Books, 1986. 

                   

 

Wells, David. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991. 

                    "Entire books have been written about tessellations alone, or topological curiosities, or geometric extremal properties, beside the wealth of classical geometry. This is my selection from that cornucopia."

 

Wells, David. You Are a Mathematician: A Witty and Wise Introduction to the Joy of Numbers. New York: John Wiley & Sons, 1995. 

                   

 

Wells, David. The Penquin Book of Curious and Interesting Mathematics. London: Penquin Books, 1997. 

                   

 

                   

FO. Foundations of Geometry

 

Forder, H.G. The Foundations of Geometry. Cambridge, UK: Cambridge University Press, 1927. 

                   

 

Frankland, William Barrett. Theories of Parallelism: An Historical Critique. The Cornell Library Historical Mathematics Monographs, 1910.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00100002&seq=7

                   

 

Hilbert, David. Foundation of Geometry (Grundlagen der Geometrie). LaSalle, IL: Open Court Press, 1971. 

                   

 

 

FR. Fractals

 

Edgar, Gerald A. Measure, Topology, and Fractal Geometry. New York: Springer Verlag New York Inc., 1990. 

                    This is a mathematics book. It is not about how fractals come up in nature; that is the topic of Mandelbrot's book. It is not about how to draw fractals on your computer... Complete proofs of the main results will be presented, whenever that can reasonably be done.

 

Eglash, Ron. African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press, 1999. 

                   

 

Frame, M.L., and Mandelbrot, B.B. (editors): Fractgals, Graphics, and Mathematics Education, MAA Notes Series.58: Washington, DC: The Mathematical Association of America, 2002.

                   

 

Lauwerier, Hans. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, 1991. 

                   

 

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman and Company, 1983. 

                    The book that started the popularity of fractal geometry.

 

Mumford, David, Series, Caroline, and Wright, David. Indra's Pearls: The Vision of Felix Klein. Cambridge, UK: Cambridge University Press, 2002. 

                   

 

 

GC. Geometry in Different Cultures

 

Albarn, Keith, Smith, Jenn Mial, Steele, Stanford, and Walker, Dinah. The Language of Pattern. New York: Harper & Row, 1974. 

                    Inspired by Islamic decorative pattern, the authors of this book, who are all designers, explore pattern step by step, beginning with simple numerical and geometrical relationships and progressing through the dimensions

 

Ascher, Marcia. Ethnomathematics: A Multicultural View of Mathematical Ideas. Pacific Grove, CA: Brooks/Cole, 1991. 

                    A mostly anthropological look at the mathematics indigenous to several ancient cultures.

 

Bain, George. Celtic Arts: The Methods of Construction. London: Constable, 1977. 

                    A description of the construction of Celtic patterns and designs.

 

Datta. The Science of the Sulba. Calcutta: University of Calcutta, 1932. 

                    A discussion of the mathematics in the Sulba Sutra and traditional Hindu society.

 

Eglash, Ron. African Fractals: Modern Computing and Indigenous Design. New Brunswick: Rutgers University Press, 1999. 

                   

 

Fukagawa, Hidetosi and Pedoe, Dan. Japanese Temple Geometry Problems: San Gaku. Winnipeg: The Charles Babbage Research Centre, 1989. 

                   

 

Gerdes, Paulus. Geometrical Recreations of Africa. Maputo, Mozambique: African Mathematical Union and Higher Pedagogical Institute's Faculty of Science, 1991. 

                    Gerdes describes and discusses the mathematical aspects of the central African sona sand drawings.

 

Gerdes, Paulus. The African Pythagoras: A Study in Culture and Mathematics Education. Maputo, Mozambique: Instituto Superior Pedagógico, 1994. 

                    Gerdes makes the case that the ideas behind what we call the Pythagorean Theorem could possibly have had their origins in Sub-Saharan Africa and argues for the africanization of the teaching on mathematics in Africa.

 

Gerdes, Paulus. Ethnomathematics and Education in Africa. Stockhoms Universitet, 1995. 

                   

 

Gerdes, Paulus. Women, Art and Geometry in Southern Africa. Trenton: Africa World Press, Inc., 1998. 

                    The main objective of the book Women, Art and Geometry in Southern Africa is to call attention to some mathematical aspects and ideas incorporated in the patterns invented by women in Southern Africa.

 

Gerdes, Paulus. Geometry From Africa: Mathematical and Educational Explorations. Washington: Mathematical Association of America, 1999. 

                    ... we learn of the diversity, richness, and pleasure of mathematical ideas found in Sub-Saharan Africa. Form a careful reading and working through this delightful book, one will find a fresh approach to mathematical inquiry as well as encounter a subtle challenge to Eurocentric discourses concerning the when, where, who, and why of mathematics.

 

Holme, Audun. Geometry, Our Cultural Heritage. New York: Springer, 2002. 

                   

 

Høyrup, Jens. In Measure, Number, and Weight: Studies in Mathematics and Culture. Albany: State University of New York Press, 1994. 

                   

 

Jean, Geoerges. Signs, Symbols and ciphers: Decoding the Message. London: Thames & Hudson, 1998. 

                   

 

Kline, Morris. Mathematics in Western Culture. New York: Oxford University Press, 1961. 

                   

 

Yan, Li and Shiran, Du. Chinese Mathematics: A Concise History. Oxford: Clarendon, 1987. 

                   

 

Mikami, Yoshio. The Development of Mathematics in China and Japan. New York: Chelsea, 1974. 

                   

 

Mohen, Jean -Pierre. Standing Stones: Stonehenge, Carnac and the World of Megaliths. London: Thames & Hudson, 1999. 

                    "... this book considers the special significance -- religious and cultural, architectural and scientific -- of these enigmatic Neolithic stone structures ..."

 

Nasr, Seyyed Hossein. Islamic Science: An Illustrated Study. World of Islam Festival Publishing, 1976. 

                   

 

Neihardt, John G. Black Elk Speaks: Being the Life Story of a Holy Man of the Oglala Sioux. Lincoln, NE: University of Nebraska Press, 1961. 

                    Contains descriptions of geometric ideas in Oglala Sioux culture

 

Pinxten, R., Dooren, Ingrid van, and Harvey, Frank. The Anthropology of Space. Philadelphia: University of Pennsylvania Press, 1983. 

                    Concepts of geometry and space in the Navajo culture.

 

Selin, Helaine, and D'Ambrosio, Ubiratan (editors): Mathematics Across Cultures: The History of Non-Western Mathematics,  Boston: Kluwer Academic Publishers, 2000.

                    "Every culture has mathematics. That is not to say that every culture has forms of [enumeration and calculation] ... But enumeration and calculation are only parts of mathematics; a broader definition that includes 'the study of measurements, forms, patterns, variability and change' encompasses the mathematical systems of many non-Western cultures." 

 

Zaslavsky, Claudia. Africa Counts. Boston: Prindle, Weber, and Schmidt, Inc., 1973. 

                    A presentation of the mathematics in African cultures.

 

 

GS. Geometry and Science

 

Abraham, Ralph H. and Shaw, Christopher D. Dynamics: The Geometry of Behaviour (in 4 volumes). Santa Cruz, CA: Aerial Press,

                    Dynamics is emerging as an important conceptual scheme, unifying the sciences -- physical, biological, and social -- in a common geometric model... All four [volumes] are written for a general audience, without the assumption of extensive training in math or the sciences. Visual presentation, the hallmark of the Visual Mathematics Library, makes the ideas accessible.

 

Abraham, Ralph H. and Shaw, Christopher D. Dynamics: A Visual Introduction. Plenum Publishing Corporation, 1987. 

                    A dynamical system is one whose state may be represented as a point in a space, where each point is assigned a vector specifying the evolution. The basic ideas of the mathematical theory of dynamical systems are presented here visually, with a minimum of discussion, using examples in low dimensions... While working together on the illustrations for a book, we discovered that we could explain mathematical ideas visually, within an easy and pleasant working partnership.

 

Blay, Michel. Reasoning with the Infinite: From the Closed World to the Mathematical Universe. Chicago and London: The University of Chicago Press, 1993. 

                    "The present essay has the object of the explaining in the development of mathematical physics on the basis of the actual process of geometrization, the difficulties that arose from trying to take the infinite into account, as well as the techniques developed to resolve or to avoid them, insofar as the neglect of the question of meaning grew out of these very difficulties.

 

Kline, Morris. Mathematics and the Physical World. New York: Thomas Y. Crowell Company, 1959. 

                    To display the role of mathematics in the study of nature is the purpose of this book. Subordinate, but by no means incidental, objectives may also be fulfilled. We may see mathematics in the process of being born... The precise manner in which mathematics produces answers to physical problems... how and why mathematics has become the essence of scientific theories.

 

Schneider, Michael S. A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperPerennial, 1994. 

                   

 

Schutz, Bernard. Geometrical methods of mathematical physics. New York: Cambridge University Press, 1993. 

                    This book aims to introduce the beginning or working physicist to a wide range of analytic tools which have their origin in differential geometry and which have recently found increasing use in theoretical physics.

 

Shaw, Robert. The Dripping Faucet as a Model Chaotic System. Santa Cruz: Aerial Press, Inc., 1984. 

                    Water drops falling from an orifice present a system which is both easily accessible to experiment and common in everyday life. As the flow rate is varied, many features of the phenomenology of nonlinear systems can be seen, including chaotic transitions, familiar and unfamiliar bifurcation sequences, hysteresis, and multiple basins of attraction.

 

Sumners, De Witt L. (editors): New Scientific Applications of Geometry and Topology, Proceedings of Symposia in Applied Mathematics.45: Providence, Rhode Island: American Mathematical Association, 1992.

                    "Recently, some of the methods and results of geometry and topology have found new utility in both wet-lab and theoretical science. Conversely, science is influencing mathematics, from posing questions which call for construction of mathematical models to the importation of theoretical methods of attack on long-standing problems of mathematical interest."

 

 

HI. History of Mathematics

 

Beckmann, Peter. A History of Pi. Boulder, CO: The Golem Press, 1970. 

                    A well-written enjoyable book about all aspects of pi.

A well-written enjoyable book about all aspects of pi.

 

Berggren, J.L. Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag, 1986. 

                    Describes many examples that are difficult to find elsewhere of the mathematical contributions from Medieval Islam.

 

Boi, Luciano. Le Probléme Mathématique de l'Espace: Une Quéte de l'Intelligible. Berlin: Springer, 1995. 

                    A historical account of non-Euclidean spaces with many interesting photos (for example, paper models of hyperbolic space constructed by Beltrami.

 

Bold, Benjamin. Famous Problems of Geometry and How to Solve Them. New York: Dover Publications, Inc., 1969. 

                   

 

Bonola, Roberto. Non-Euclidean Geomtry: A critical and Historic Study of its Developments, and "The theory of Parallels" by Nicholas Lobachevski with a supplement containing "The Science of Absolute Space" by John Bolyai. New York: Dover, 1995. 

                    Bonola's Non-Euclidean Geometry is an elementary historical and ciritcal study of the development of that subject

 

Calinger, Ronald. Classics of Mathematics. Englewood Cliffs, NJ: Prentice Hall, 1995. 

                    Mostly a collection of original sources in Western mathematics.

 

Calinger, Ronald. A Contextual History of Mathematics: to Euler. Upper Saddle River, NJ: Prentice Hall, 1999. 

                   

 

Carroll, Lewis. Euclid and His Modern Rivals. New York: Dover Publications, Inc., 1973. 

                    Yes! Lewis Carroll of Alice in Wonderland fame was a geometer. This book is written as a drama; Carroll has Euclid defending himself against modern critics.

 

Cohen, Patricia Cline. A Calculating People: The Spread of Numeracy in Early America. Chicago: The University of Chicago Press, 1982. 

                   

 

Cooke, Roger. The History of Mathematics: A Brief Course. New York: John Wiley & Sons, 1997. 

                   

 

Eves, Howard. Great Moments in Mathematics (after 1650), Dolciani Mathematical Expositions. Washington, DC: M.A.A. --AB-- This small book contains 20 lectures: 2 on non-Euclid geometry and one on Klien's "Erlanger Program" which set out to delineate various, 1981. 

                   

 

Fauvel, John and Gray, Jeremy. The History of Mathematics: A Reader. London: Macmillan Press, 1987. 

                    The selection of readings has been made for students of the Open University course MA290 Topics in the History of Mathematics ...

 

Fauvel, John, and van Maanen, Jan (editors): History in Mathematics Education: The ICMI Study,  Dordrecht: Kluwer Academic publishers, 2000.

                   

 

Field, Judith Veronica. The Invention of Infinity: Mathematics and Art in Renaissance. Oxford: Oxford University Press, 1997. 

                    Book will look at the relations between of Renaissance art and mathematics in the period from about 1300 to about 1650.

 

Frankland, William Barrett. Theories of Parallelism: An Historical Critique. The Cornell Library Historical Mathematics Monographs, 1910.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00100002&seq=7

                   

 

Goldman, Jay R. The Queen of Mathematics: A Historically Motivated Guide to Number Theory. Wellesley, MA: AK Peters, 1998. 

                   

 

Gray, Jeremy. Ideas of Space: Euclidean, Non-Euclidean and Relativistic. Oxford: Oxford University Press, 1989. 

                    A mostly historical account of Euclidean, non-Euclidean and relativistic geometry. "I shall discuss Greek and modern geometry, in particular what came to be known as the problem of parallels, that 'blot on geometry' as Saville called it in 1621."

 

Guedj, Denis. Numbers: The Universal Language. New York: Harry N. Abrams, Inc., 1997. 

                    A beautifully illustrated history of numbers from cave drawing to the present day.

 

Heath, T.L. Euclid: The Thirteen Books of the Elements. New York: Dover, 1956. 

                    This is edition of Eulid's Elements to which one is usually referred. Heath has added a large collection of very useful historical and philosophical notes. His notes are more extensive than Euclid's text.

 

Heilbron, J.L. Geometry Civilized: History, Culture, and Technique. Oxford: Clarendon Press, 2000. 

                    For many centuries, geometry was part of high culture as well as an instrument of practical utility.

 

Joseph, George. The Crest of the Peacock. New York: I.B. Tauris, 1991. 

                    A non-Eurocentric view of the history of mathematics.

 

Kaplan, Robert. The Nothing That Is: A Natural History of Zero. New York: Oxford University Press, 2000. 

                    "Look at zero and yoou see nothing, but look through it and you see the world."

 

Katz, Victor (editors): Using History to Teach Mathematics: An International Perspective, MAA Notes.#51: Washington, D.C.: Mathematical Association of America, 2000.

                   

 

Katz, Victor J. A History of Mathematics: An Introduction. Reading, MA: Addison-Wesley Longman, 1998. 

                    "... designed for junior or senior mathematics majors who intend to teach in college or high school and thus concentrates on the history of those topics typically covered in an undergraduate curriculum or in elementary or high school.

 

Kline, Morris. Mathematics in Western Culture. New York: Oxford University Press, 1961. 

                   

 

Kline, Morris. Mathematical Thought from Ancient to Modern Times. Oxford: Oxford University Press, 1972. 

                    A complete Eurocentric history of mathematical ideas including differential geometry (mostly the analytic side).

 

Laubenbacher, Reinhard and Pengelley, David. Mathematical Expeditions: Chronicles by the Explorers. New York: Springer, 1999. 

                    Contains a 53-page chapter on "Geometry: The Parallel Postulate".

 

Laugwitz, Detlef: ""Infinitely Small Quantities in Cauchy's Textbooks,"", Historia Mathematica, 14 (1987), 258-274.

                   

 

Lewinter, Marty and Widulski, William. The Saga of Mathematics: A Brief History. Upper Saddle River, NJ: Prentice-Hall, 2002. 

                   

 

Maor, Eli. e: The Story of a Number. Princeton, NJ: Princeton University Press, 1994. 

                   

 

Maor, Eli. Trigonometric Delights. Princeton, NJ: Princeton Unversity Press, 1998. 

                   

 

Monastyrsky, Michael. Modern Mathematics in the Light of the Fields Medals. Wellesley, MA: AK Peters, 1998. 

                   

 

Newell, Virginia K. (editors): Black Mathematicians and Their Works,  Ardmore, PA: Dorrance, 1980.

                   

 

Richards, Joan. Mathematical Visions. Boston: Academic Press, 1988. 

                    The pursuit of geometry in Victorian England.

 

Rosenfeld, B.A. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. New York: Springer-Verlag, 1989. 

                    A extensive history of non-Euclidean geometry based on original sources.

 

Schmandt-Besserat, Denise. Before Writing. Austin: University of Texas Press, 1992. 

                    This lavishly illustrated book develops the theory that human writing developed from counting devices.

 

Seidenberg, A.: ""The Ritual Origin of Geometry,"", Archive for the History of the Exact Sciences, 1 (1961), 488-527.

                    In this article Seidenberg makes the case that much geometry originated from the needs of various religious rituals.

 

Selin, Helaine, and D'Ambrosio, Ubiratan (editors): Mathematics Across Cultures: The History of Non-Western Mathematics,  Boston: Kluwer Academic Publishers, 2000.

                    "Every culture has mathematics. That is not to say that every culture has forms of [enumeration and calculation] ... But enumeration and calculation are only parts of mathematics; a broader definition that includes 'the study of measurements, forms, patterns, variability and change' encompasses the mathematical systems of many non-Western cultures." 

 

Singh, Simon. Fermat's Enigma: The Quest to Solve the World's Greatest Mathematical Problem. New York: Walker and Company, 1997. 

                    Description of the history of proving Fermat's Last Theorem and methods used by Andrew Wiles, but also there is some nice geometry in it.

 

Smeltzer, Donald. Man and Number. New York: Emerson Books, 1958. 

                    History and cultural aspects of mathematics

 

Smith, David Eugene. A Source Book in Mathematics. New York: Dover Publications, Inc., 1959. 

                   

 

Stillwell, John. Mathematics and Its History. New York: Springer-Verlag, 1989. 

                    "This book aims to give a unified view of undergraduate mathematics by approaching the subject though its history."

 

Struik, D.J. (editors): A source Book in Mathematics 1200-1800,  Princeton, NJ: Princeton University Press, 1986.

                   

 

Suzuki, Jeff. A History of Mathematics. Uppeer Saddle River, NJ: Prentice Hall, 2002. 

                   

 

Swetz, Frank J. Capitalism & Arithmetic: The New Math of the 15th Century. La Salle, IL: Open Court, 1987. 

                   

 

Swetz, Frank, Fauvel, John, and Bekken, Otto (editors): Learn From The Masters,  Washington DC.: The Mathematical Association of America, 1995.

                   

 

Thomas, Ivor (editors): Selections Illustrating the History of Greek Mathematics,  Cambridge, MA: Harvard University Press, 1951.

                    A collection of primary sources.

 

Toth, I. "Non-Euclidean Geometry before Euclid", Scientific American. 1969. 

                    Discusses the evidence of non-Euclidean geometry before Euclid.

 

Valens, Evans G. The Number of Things: Pythagoras, Geometry and Humming Strings. New York: E.P. Dutton and Company, 1964. 

                    This is a book about ideas and is not a textbook. Valens leads the reader through dissections, golden mean, relations between geometry and music, conic sections, etc.

 

Williams, Trevor I. A History of Invention: From Stone Axes to Silicon Chips. New York: Checkmark  Books, 2000. 

                   

 

van der Waerden, B.L. Science Awakening I: Egyptian, Babylonian, and Greek Mathematics. Princeton Junction, NJ: The Scholar's Bookshelf, 1975. 

                    "It is the intention to make this book scientific, but at the same time accessible to any one who has learned some mathematics in school and in college, and who is interested in the history of mathematics."

 

 

HM. History of a Mathematician

 

Artmann, Benno. Euclid-The Creation of Mathematics. New York: Springer, 1999. 

                    Here the present book takes a clear position: The Elements are read, interpreted, and commented upon from the point of view of modern mathematics.

 

Batterson, Steve. Stephen Smale: The Mathematician Who Broke the Dimension Barrier. American Mathematical Society, 2000. 

                   

 

Bühler, W.K. Gauss: A Biographical Study. New York: Springer-Verlag, 1981. 

                    This biography contains many quotations and lengthy passages from Gauss's writings.

 

Descartes, Rene. The Geometry of Rene Descartes. New York: Dover Publications,Inc., 1954. 

                    This the book in which Descartes develops the use of what we now call Cartesian coordinates for the study of curves.

 

Feferman, Solomon, Dawson, John W., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., and van Heijenoort, Jean (editors): Kurt Gödel: Collecdted Works, Volume I, Publications 1929-1936,  New York: Oxford University Press, 1986.

                   

 

Field, Judith Veronica and Gray, Jerome J. The Geometrical Work of Girard Desargues. New York: Springer-Verlag, 1987. 

                   

 

Flannery, Sarah. In Code: A Young Woman's Mathmatical Journey. Chapel Hill, NC: Algonquin Books of Chapel Hill, 2001. 

                   

 

Halmos, Paul R. I Want To Be a Mathematician: An Automathography In Three Parts. Washington, D.C.: MAA, Springer-Verlag, 1985. 

                   

 

Hardy, G.H. A Mathematician's Apology. Cambridge, UK: Cambridge University Press, 1967. 

                   

 

Heath, T.L. Mathematics in Aristotle. Oxford: Clarendon Press, 1949. 

                    Discusses the mathematical contributions of Aristotle.

 

Hoffman, Paul. The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth. London: Fourth Estate, 1998. 

                   

 

Kiss, Elemér. Mathematical Gems From the Boylai Chests: János Bolyai's discoveries in Number Theory and Algebra as recently deciphered from his manuscripts. Budapest: Akadémiai Kiadó & TypoTEX, 1999. 

                   

 

Marchisotto, Elena Anne and Smith, James T. The Legacy of Mario Pieri in Arithmetic and Geometry. Boston: Birkhauser, 2004. 

                   

 

Newell, Virginia K. (editors): Black Mathematicians and Their Works,  Ardmore, PA: Dorrance, 1980.

                   

 

Riemann, Bernard. Gesammelte Mathematische Werke. Leipzig: B.G. Teubner, 1902. 

                   

 

Rosenblatt, Murray (editors): ERRETT BISHOP: Reflections on Him and His Research, Contermporary Mathematics.39: Providence, RI: American Mathematical Society, 1985.

                   

 

Shasha, Dennis and Lazere, Cathy. Out Of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. New York: Copernicus, 1998. 

                   

 

Sobel, Dava. Galileo's Daughter: A Historical Memoir of Science, Faith, and Love. New York: Penquin Putman, 1999. 

                    A story of Galileo's life and works as chronicled in his correspondences with his daughter

 

Yaglom, I.M. Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century. Boston: Birkhäuser, 1988. 

                   

 

 

HY. Hyperbolic Geometry

 

Batterson, Steve. Stephen Smale: The Mathematician Who Broke the Dimension Barrier. American Mathematical Society, 2000. 

                   

 

Boi, Luciano. Le Probléme Mathématique de l'Espace: Une Quéte de l'Intelligible. Berlin: Springer, 1995. 

                    A historical account of non-Euclidean spaces with many interesting photos (for example, paper models of hyperbolic space constructed by Beltrami.

 

Bonola, Roberto. Non-Euclidean Geomtry: A critical and Historic Study of its Developments, and "The theory of Parallels" by Nicholas Lobachevski with a supplement containing "The Science of Absolute Space" by John Bolyai. New York: Dover, 1995. 

                    Bonola's Non-Euclidean Geometry is an elementary historical and ciritcal study of the development of that subject

 

Coxeter, H.S.M. Non-Euclidean Geometry. Toronto: University of Toronto Press, 1965. 

                   

 

Efimov, N. V.: "Generation of singularities on surfaces of negative curvature [Russian]", Mat. Sb. (N.S.), 106 (1964), 286-320.

                    Efimov proves that it is impossible to have a C2 isometric embedding of the hyperbolic plane onto a closed subset of Euclidean 3-space.

 

Fenchel, Werner. Elementary Geometry in Hyperbolic Space. Berlin: Walter de Gruyter, 1989. 

                   

 

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History. New York: Freeman, 1980. 

                    This is a very readable textbook that includes some philosophical discussions.

 

Hartshorne, Robin: "Non-Euclidean III.36", American Mathematical Monthly, 110 (2003), 495-502.

                    Power of a point on sphere and hyperbolic plane

 

Hilbert, David: "Über Flächen von konstanter gausscher Krümmung, Transactions of the A.M.S",  (1901), 87-99.

                    Hilbert proves here that the hyperbolic plane does not have a real analytic (or C4) isometric embedding onto a closed subset of Euclidean 3-space.

 

Kuiper, Nicolas: "On c1-isometric embeddings ii, Nederl. Akad. Wetensch. Proc. Ser. A",  (1955), 683-689.

                    Kuiper shows that there is a C1 isometric embedding of the hyperbolic plane onto a closed subset of Euclidean 3-space.

 

Milnor, Tilla: "Efimov's theorem about complete immersed surfaces of negative curvature,Advances in Math",  8 (1972), 474-543.

                    Milnor clarifies for English-reading audiences Efimov's result in [NE: Efimov].

 

Moise, Edwin E. Elementary Geometry from an Advanced Standpoint. Reading, MA: Addison-Wesley Publishing, 1990. 

                   

 

Nikulin, V. V. and Shafarevich, I. R. Geometries and Groups. New York: Springer-Verlag, 1987. 

                    "This book is devoted to the theory of geometries which are locally Euclidean, in the sense they are identical to the geometry of the Euclidean plane or Euclidean 3-space... The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries."

 

Petit, Jean -Pierre. Euclid Rules OK? The Adventures of Archibald Higgins. London: John Murray, 1982. 

                    A pictorial, visual tour of non-Euclidean geometries.

 

Prenowitz, Walter and Jordan, Meyer. Basic Concepts of Geometry. New York: Blaisdell Publishing, 1965. 

                   

 

Rosenfeld, B.A. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. New York: Springer-Verlag, 1989. 

                    A extensive history of non-Euclidean geometry based on original sources.

 

Ryan, Patrick J. Euclidean and Non-Euclidean Geometry: An Analytic Approach. Cambridge: Cambridge University Press, 1986. 

                   

 

Schwerdtfeger, Hans. Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry. New York: Dover Publications, Inc., 1979. 

                    This book uses complex numbers to analyze inversions in cricles and then their relationship to hyperbolic geometry.

 

Singer, David A. Geometry: Plane and Fancy. New York: Springer, 1998. 

                    "This book is about ... the idea of curvature and how it affects the assumptions about and principles of geometry."

 

Smogorzhevsky, A.S. Lobachevskian Geometry. Moscow: Mir Publishers, 1982. 

                    From a collection of short books (phamphlets) for high school students written by Soviet mathematicians and translated into English.

 

Sommerville, D.M.Y. Bibliography of Non-Euclidean Geometry. New York: Chelsea Publishing Company, 1970. 

                    This book contains 410 pages of bibliographic references up to 1968.

 

Stahl, Saul. The Poincaré Half-Plane. Boston: Jones and Bartlett Publishers, 1993. 

                    This text is an analytic introduction to some of the ideas of intrinsic differential geometry starting from the Calculus.

 

Sved, Marta. Journey into Geometries. Washinton, DC: Mathematical Association of America, 1991. 

                    "This book, though not a text, is first and foremost about geometry. It is neither comprehensive, not can it claim to go very deep into the chosen topics, yet hopefully, it may initiate a spark to light the way into further progress. The central topic in this book is non-Euclidean geometry. The approach to it is made via the Poincare model, ..."

 

Thurston, William. Three-Dimensional Geometry and Topology, Vol. 1. Princeton, NJ: Princeton University Press, 1997. 

                    This is a detailed excursion through the geometry and topology of two- and three-manifolds. "The style of exposition in this book is intended to encourage the reader to pause, to look around and to explore.

 

Trudeau, Richard J. The Non-Euclidean Revolution. Boston: Birkhäuser, 1987. 

                    "Trudeau's book provides the reader with a non-technical description of the progress of thought from Plato and Euclid to Kant, Lobachevsky, and Hilbert."

 

Zage, Wayne: "The Geometry of Binocular Visual Space", Mathematics Magazine, 53 (1980), 289-294.

                    "... we relate the results of experiments in binocular vision to geometric models to arrive at the conclusion that the geometry of binocular visual space is [...] hyperbolic."

 

 

IN. Inversions

 

Davis, Chandler, Grunbaum, Branko, and Sherk, F.A. (editors): The Geometric Vein,  New York: Springer-Verlag, 1981.

                    Book has many addresses, essays, lectures on geometry and is dedicated to H. S. M. Coxeter. There is a chapter on Inversive Geometry written by J. B. Wilker with nice examples of the use of inversions.

 

Dupuis, N.F. Elementary Synthetic Geometry of the Point, Line and Circle in the Plane. London: Macmillan and Co., 1889. 

                    Section IV of this book is on inversion and inverse figures which gives more examples on use of inversion in solution geometric problems. Book available in Cornell Library.

 

Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillan and Co., 1893.  online:  http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Lach015&seq=7

                    Chapter XIV is on theory of inversions giving good examples of use of inversion.

 

Sommerville, D.M.Y. Bibliography of Non-Euclidean Geometry. New York: Chelsea Publishing Company, 1970. 

                    This book contains 410 pages of bibliographic references up to 1968.

 

Townsend, Richard. Chapters on the Modern Geometry of the Point, Line, and Circle; being the Substance of Lectures Delivered in the University of Dublin to the Candidates for Honors of the first Year in Arts. Dublin: Hodges, Smith, and Co., 1863.  online: (volume 1) http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00720002&seq=5 (volume 2) http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=01060002&seq=5     

Title of the book already explains what is in it, but special interest may be found in Chapter IX "Theory of Inverse Points with Respect to the Circle". 

 

 

LA. Linear Algebra and Geometry

 

Banchoff, Thomas and Wermer, John. Linear Algebra Through Geometry. New York: Springer-Verlag New York, Inc., 1983. 

                    "In this book we lead the student to an understanding of elementary linear algebra by emphasizing the geometric significance of the subject. Our experience in teaching beginning undergraduates over the years has convinced us that students learn the new ideas of linear algebra best when these ideas are grounded in the familiar geometry of two and three dimensions."

 

Dodson, C. T. J. and Poston, T. Tensor Geometry. London: Pitman, 1979. 

                    A very readable but technical text using linear (affine) algebra to study the local intrinsic geometry of spaces leading up to and including the geometry of the theory of relativity.

 

Fekete, Anton E. Real Linear Algebra. New York: Marcel Dekker, 1985. 

                   

 

Hannah, John: ""A Geometric Approach to Determinants,"", The American Mathematical Monthly, 103 (1996), 401-409.

                   

 

Hay, G. E. Vector and Tensor Analysis. New York: Dover, 1953. 

                    "First published in 1953, this is a simple clear introduction to classical vector and tensor analysis for students of engineering and mathematical physics."

 

Hestenes, David and Sobczyk, Garret. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Boston: D. Reidel Publishing Company, 1984. 

                    "Matrix algebra has been called "the arithmetic of higher mathematics" [...]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself."

 

Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton: Princeton University Press, 1999. 

                    "This book is intended for all those mathematicians, engineers, and physicists who have to know, or who want to know, more about the modern theory of quaternions. Primarily, as the title page suggests, it is an exposition of the quaternion and its primary application as a rotation operator."  Included are applications of spherical geometry.

 

Murtha, James A. and Willard, Earl R. Linear Algebra and Geometry. New York: Holt, Reinhart and Winston, Inc., 1966. 

                    Includes affine and projective geometry.

 

Postnikov, M. Lectures in Geometry: Semester II - Linear Algebra and Differential Geometry. Moscow: Mir, 1982. 

                   

 

Postnikov, M. Lectures in Geometry: Semester V - Lie Groups and Lie Algebras. Moscow: Mir Publishers, 1986. 

                    "The theory of Lie groups relies on Cartan's theorem on the equivalence of the category of simply connected Lie groups to that of Lie algebras. This book presents the proof of the Cartan theorem and the main results... The theory of Lie algebras has been developed to an extent necessary for the Cartan theorem to be proved."

 

Solow, Daniel. How to read and do proofs, an introduction to mathematical thought process. New York: John wiley and Sons, Inc., 1982. 

                   

 

Taylor, Walter F. The Geometry of Computer Graphics. Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, 1992. 

                    "This book is a direct presentation of elementary analytic and projective geometry, as modeled by vectors and matrices and as applied to computer graphics."

 

Weinreich, Gabriel. Geometrical Vectors. Chicago: The University of Chicago Press, 1998. 

                    "Years of teaching Mathematical Methods of Physics at the University of Michigan to seniors and first-year graduate students convinced me that existing textbooks don't do an adequate job in the area of vector analysis: all too often, their treatment is a repetition of what students had already seen in earlier courses, with little or no insight into the essentially geometrical structure of the subject."

 

 

LS. Learning/Students

 

Solow, Daniel. How to read and do proofs, an introduction to mathematical thought process. New York: John wiley and Sons, Inc., 1982. 

                   

 

Tall, David (editors): Advanced Mathematical Thinking, .11: Boston: Kluwer Academic Publishers, 1991.

                   

 

 

ME. Mechanisms

 

Agricola, Georg. De re metallica. (Basil: E. König, 1657.) Translated from the first Latin ed. of 1556, with biographical introd., annotations, and appendices upon the development of mining methods, metallurgical processes, geology, mineralogy & mining law from the earliest times to the 16th century, by Herbert Clark Hoover and Lou Henry Hoover, New York, Dover Publications, 1950. Some excerpts can be found on various online sites.


I. I. Artobolevskii, I.I.. Mechanisms for the Generation of Plane Curves. New York: The Macmillan Company, 1964.

 

Connelly, R, Demiane, E. D., and Rote, G.. "Straightening Polygonal Arcs and Convexifying Polygonal Cycles." Discrete & Computational Geometry 2003; 30:205–239.

 

DeCamp, L. Sprague. The Ancient Engineers. New York: Ballantine Books, 1974.

 

Dyson, George B. Darwin among the Machines: The Evolution of Global Intelligence. Reading, MA: Perseus Books, 1997.

 

Ferguson, Eugene S.: "Kinematics of Mechanisms from the Time of Watt", United States National Museum Bulletin, 228Washington, D.C.: Smithsonian Institute  (1962), 185-230.

                   

 

Ferguson, Eugene S. Engineering and the Mind's Eye. Cambridge, MA: The MIT Press, 2001. 

                   

 

Finch, James Kip (editors): The Story of Engineering,  Garden City, NJ: Anchor Books, 1960.

                   

 

Florman, Samuel C. The Existential Pleasures of Engineering. New York: St, Martin's Griffin, 1994. 

                   

 

Galle, A. Mathematische Instrumente. Leipzig: B. G. Teubner, 1912. 

                   

 

Gille, Bertrand. Engineers of the Renaissance. Cambridge, MA: The MIT Press, 1966. 

                   

 

Hinkle, Rolland T. Kinematics of Machines. Englewood Cliffs, NJ: Prentice-Hall, 1960. 

                   

 

Hodges, Henry. Technology in the Ancient World. New York: Barnes & Noble Books, 1992. 

                   

 

Horsburgh, E. M. (editors): Modern Instruments and Methods of Calculation: A Handbook of the Napier Tercentenary Exhibitions,  London: G. Bell and Sons, 1914.

                   

 

Huckert, Jesse. Analytical Kinematics of Plane Motion Mechanisms. New York: The Macmillan Company, 1958. 

                   

 

Hunt, K.H. Kinematic Geometry of Mechanisms. Oxford: Clarendon Press, 1978. 

                   

 

Institution of Mechanical Engineers. Engineering Heritage: Highlights from the History of Mechanical Engineering: Volume I. London: Heinemann, 1963. 

                   

 

Institution of Mechanical Engineers. Engineering Heritage: Highlights from the History of Mechanical Engineering, Volume Two. London: Heinemann Educational Books, 1966. 

                   

 

KMODDL. Kinematic Models for Design Design Library. Cornell University Libraries, 2004.  online: http://KMODDL.library.cornell.edu

                   

 

Kempe, A. B.: "On a General Method of Describing Plane Curves of the n-th Degree by Linkages",  Proc. Lon. Math. Soc, VII (1876), 213-215.

                   

 

Kempe, A.B. How to Draw a Straight Line. London: Macmillan, 1877.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Kemp009&seq=5

                    This small book contains a discussion and description of numerous curve drawing devices including ones that will draw straight lines.

 

Kirby, R.S., Withington, S., Darling, A.B., and Kilgour, F.G. Engineering in History. New York: McGraw-Hill Book Company, 1956. 

                   

 

McCarthy, J. Michael. Geometric Design of Linkages. New York: Springer, 2000. 

                   

 

Moon, Francis C.: "Franz Reuleaux: Contributions to 19th Century Kinematics and the Theory of Machines", Applied Mechanics Reviews, 56 (2003), 1-25.

                   

 

Ramelli, Agostino. The Various and Ingenious Machines of Agostino Ramelli: A Classic Sixteenth-Century Illustrated Treatise on Technology. New York: Dover Publications, 1976. 

                   

 

Reuleaux, Franz. The Kinematics of Machinery. London: Macmillan and Co., 1876.  online:  http://historical.library.cornell.edu/cgi-bin/kmoddl/docviewer?did=029&seq=7

                   

 

Reynolds, Terry S. (editors): The Engineer in America: A Historical Anthology From "Technology and Culture",  Chicago: The University of Chicago Press, 1991.

                   

 

Shigley, Joseph Edward. Kinematics Analysis of Mechanisms. New York: McGraw-Hill Book Company, 1969. 

                   

 

Uicker, John J., Jr., Pennock, Gordon R., and Shigley, Joseph E. Theory of Machines and Mechanisms. New York: Oxford University Press, 2003. 

Williams, Trevor. A History of Inventions: From Stone Axes to Silicon Chips. New York: Checkmark Books, 2000.

                   

 

Yates, Robert. Tools: A Mathematical Sketch and Model Book. Louisiana State University, 1941. 

                   

 

 

MI. Minimal Surfaces

 

Boys, C. V. Soap-Bubbles: Their Colours and the Forces Which Mold Them. New York: Dover, 1959. 

                   

 

Hoffman, David: ""The computer aided discovery of new embedded minimal surfaces,"", Mathematcal Intelligencer, (1987), 8-21.

                   

 

Morgan, Frank. Geometric Measure Theory: A Beginner's Guide. Boston: Academic Press, 1988. 

                   

 

Morgan, Frank. "Compound soap bubbles, shortest networks, and minimal surfaces,"AMS Video. Providence, RI: AMS, 1992. 

                   

 

Morgan, Frank. The Math Chat Book. MAA, 2000. 

                    Charming little book includes short version of Double Bubble Conjecture story. Also see  http://www.maa.org/news/mathchat.html

 

Osserman, Robert. A Survey of Minimal Surfaces, 2nd edition. New York: Dover, 1986. 

                   

 

Osserman, Robert. "Minimal Surfaces in R3," Global Differential Geometry. Washington,DC: M.A.A., 1989. 

                   

 

 

MP. Models and Polyhedra

 

Barnette, David. Map Colouring, Polyhedra, and the Four-Colour Problem, Dolciani Mathematical Expositions. Washington, DC: M.A.A., 1983. 

                   

 

Barr, Stephen. Experiments in Topology. New York: Crowell, 1964. 

                    Experimental topology that goes beyond the Möbius Band.

Experimental topology that goes beyond the Möbius Band.

 

Cundy, M.H. and Rollett, A.P. Mathematical Models. Oxford: Clarendon, 1961. 

                    Directions on how to make and understand various geometric models.

 

Lyusternik, L.A. Convex Figures and Polyhedra. Boston: Heath, 1966. 

                    An detailed but elementary study of convex figures.

 

Row, T. Sundra. Geometric Exercises in Paper Folding. New York: Dover, 1966. 

                    How to produce various geometric constructions merely by folding a sheet of paper.

 

Senechal, Marjorie, and Fleck, George (editors): Shaping Space: A Polyhedral Approach, Design Science Collection,  Boston: Birkhauser, 1988.

                    This book is an accessible "exploration of the world of polyhedra, beginning with [an introduction] and concluding with an examination of the significance of polyhedral models in contemporary science and a survey of some recent advances and unsolved problems in mathematics."

 

Weinreich, Gabriel. Geometrical Vectors. Chicago: The University of Chicago Press, 1998. 

                    "Years of teaching Mathematical Methods of Physics at the University of Michigan to seniors and first-year graduate students convinced me that existing textbooks don't do an adequate job in the area of vector analysis: all too often, their treatment is a repetition of what students had already seen in earlier courses, with little or no insight into the essentially geometrical structure of the subject."

 

Zawadowski, Waclaw. The Cube Made Interesting. New York: Pergamon Press, 1964. 

                    "This book arose from popular scientific talks to teachers and school children." The discussion is illustrated with 3-d pictures using special glasses.

 

 

MS. Mathematics and Social Issues

 

Cohen, Patricia Cline. A Calculating People: The Spread of Numeracy in Early America. Chicago: The University of Chicago Press, 1982. 

                   

 

Keitel, Christine, Damerow, Peter, Bishop, Alan, and Gerdes, Paulus (editors): Mathematics, Education, and Society: A Fifth Day Special Programme at the 6th International Congress on Mathematical Education, Budapest, 27 July - 3 August, 1988, Science and Technology Education Document Series.No. 35: Budapest: UNESCO, 1989.

                   

 

Restivo, Sal, Paul, Jean, Bendegem, Van, and Fischer, Roland (editors): MathWorlds,  New York: State University of New York Press, 1993.

                   

 

Swetz, Frank J. Capitalism & Arithmetic: The New Math of the 15th Century. La Salle, IL: Open Court, 1987. 

                   

 

Vatter, Terry. Civic Mathematics, Fundamentals in the Context of Social Issues. Englewood, Colorado: Teacher Ideas Press, 1996. 

                   

 

 

NA. Geometry in Nature

 

Cook, T.A. The Curves of Life. New York: Dover Publications, 1979. 

                    Subtitle: Being an Account of Spiral Formations and their Applications to Growth in Nature, to Science, and to Art.

 

Doczi, György. The Power of Limits. Boulder, CO: Shambhala, 1981. 

                   

 

Ghyka, Matila. The Geometry of Art and Life. New York: Dover Publications, 1977. 

                   

 

Hildebrandt, Stefan and Tromba, Anthony. Mathematics and Optimal Form. New York: Scientific American Books, Inc., 1985. 

                    "Combining striking photographs with a compelling text, authors ... give us a thoughtful account of the symmetry and regularity of nature's forms and patterns."

 

Kemp, Martin. Visualizations: The Nature Book of Art and Science. Berkeley, CA: The University of California Press, 2000. 

                   

 

Kohl, Judith and Kohl, Herbert. The View from the Oak: The Private Worlds of Other Creatures. New York: Sierra Club Books/Charles Scribner's Sons, 1977. 

                    This delightful books describes the various experiential worlds of different creatures and is a good illustration of intrinsic ways of thinking. Included are differing dimensions and scales of these worlds.

 

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: W.H. Freeman and Company, 1983. 

                    The book that started the popularity of fractal geometry.

 

McMahon, Thomas and Bonner, James. On Size and Life. New York: Scientific American Library, 1983. 

                    A geometric discussion of the shapes and sizes of living things.

 

Neihardt, John G. Black Elk Speaks: Being the Life Story of a Holy Man of the Oglala Sioux. Lincoln, NE: University of Nebraska Press, 1961. 

                    Contains descriptions of geometric ideas in Oglala Sioux culture

 

Osserman, Robert. A Survey of Minimal Surfaces, 2nd edition. New York: Dover, 1986. 

                   

 

Ravielli, Anthony. Adventure in Geometry. New York: Viking Press, 1966. 

                    A beautifully illustrated children's book. "Whether we realize it or not, much of the beauty we admire in the world around us is a result of nature's geometric skill. Every living thing -- a tree, a flower, or an insect -- is a lesson in geometry at its exquisite best."

 

Row, T. Sundra. Geometric Exercises in Paper Folding. New York: Dover, 1966. 

                    How to produce various geometric constructions merely by folding a sheet of paper.

 

Schneider, Michael S. A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperPerennial, 1994. 

                   

 

Thom, Rene. Structural Stabililty and Morphogenesis. Redwood City, CA: Addison-Wesley, 1989. 

                    A geometric and analytic treatment of "Catastrophe Theory."

 

Thompson, D'Arcy. On Growth and Form. Cambridge: Cambridge University Press, 1961. 

                    A classic on the geometry of the natural world.

 

Tomilin, Anatoly. How People Discovered the Shape of the Earth. Moscow: Raduga Publishers, 1984. 

                    A childrens book with nice colored illustrations.

 

Zebrowski, Ernest, Jr. A History of the Circle: Mathematical Reasoning and the Physical Universe. New Brunswick, NJ: Rutgers University Press, 1999. 

                   

 

 

PA. Projective and Affine Geometry

 

Brannan, David A., Esplen, Matthew F., and Gray, Jeremy J. Geometry. Cambridge: Cambridge University Press, 1999. 

                    Geometry textbook for the Open University.

 

Coxeter, H.S.M. The Real Projective Plane. New York: Cambridge University Press, 1955. 

                    "This introduction to projective geometry can be understood by anyone familiar with high-school geometry and algebra. The restriction to real geometry of two dimensions makes it possible for every theorem to be illustrated by a diagram."

 

Farin, Gerald E. NURBS: From Projective Geometry to Practical Use, Second Edition. Natick, MA: A. K. Peters, 1999. 

                    NURBS is an industry standard for curve and surface design based on projective geometry.

 

Krylov, N., Lobandiyevsky, P., and Men, S. Descriptive Geometry. Moscow: MIR Publishers, 1974. 

                    "This book is a text for students of civil engineering colleges and embraces a full course of descriptive geometry. It gives detailed information on orthogonal projections, axonometry, linear perspective and projections with elevations."

 

Prasolov, V.V. and Tikhomirov, V.M. Geometry. Providence, RI: American Mathematical Society, 2001. 

                   

 

Prenowitrz, Walter. A Contemporary Approach to Classical Geometry. Published as a supplement to the American Mathematical Monthly, Vol 68, No 1, January 1961, 1961. 

                   

 

Watson, Ernest W. Creative Perspective for Artists and Illustrators. Mineola, NY: Dover Publications, 1992. 

                   

 

Whicher, Olive. Projective Geometry: Creative Polarities in Space and Time. London: Rudolf Steiner Press, 1971. 

                    Projective geometry is the geometry of perception and prospective drawings.

 

 

PH. Philosophy of Mathematics

 

Abraham, Ralph H. Chaos. Gaia. Eros: A Chaos Pioneer Uncovers the Three Great Streams of History. New York: HarperSanFrancisco, A Division of HarperCollins Publishers, 1994. 

                   

 

Changeux, Jean -Pierre and Connes, Alain. Conversations on Mind, Matter, and Mathematics. Princeton: Princeton University Press, 1995. 

                    "Why order should exist in the world at all--and why it should be comprehensible by human beings--is the question that lies at the heart of these remarkable dialogues" between a neurobiologist and a mathematician.

 

Derrida, Jacques, and Allison, David B. (editors): Edmund Husserl's Origin of Geometry: An Introduction,  USA: Nicolas Hays,Ltd., 1978.

                    "In this commentary-interpretation of the famous appendix to Husserl's Crisis, Derrida relates writing to such key concepts as differing, consciousness, presence, and historicity."

 

Dewdney, A.K. A Mathematical Mystery Tour: Discovering the Truth and Beauty of the Cosmos. New York City: John Wiley & Sons, 1999. 

                    "The mathematical odyssey herein explores two key questions about mathematics and its relationship to reality: Why is mathematics so amazingly successful in describing the structure of physical reality? Is mathematics created, or is it discovered?" Chapter 4 of this book is about mapping the spheres, and whole book is written in a language accessible to general audience, not only mathematicians.

 

Fitzgerald, Janet A. Alfred North Whitehead's Early Philosophy of Space and Time. Washington, D.C.: University Press of America, 1979. 

                   

 

Franz, Marie-Louise von. Number and Time: Reflections Leading toward a Unification of Depth Psychology and Physics. Evanston, IL: Northwestern University Press, 1974. 

                   

 

Gray, Jeremy. Ideas of Space: Euclidean, Non-Euclidean and Relativistic. Oxford: Oxford University Press, 1989. 

                    A mostly historical account of Euclidean, non-Euclidean and relativistic geometry. "I shall discuss Greek and modern geometry, in particular what came to be known as the problem of parallels, that 'blot on geometry' as Saville called it in 1621."

 

Grunbaum, Adolf. Geometry and Chronometry in Philosophical Perspective. Minneapolis: University of Minnesota Press, 1968. 

                    This is a philosophical treatise on the empiracle(physical) status of spatial geometry.

 

Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books, 1979. 

                    A general audience book which probes the meaning of mind.

 

Iamblichus. The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers. Grand Rapids: Phanes Press, 1988. 

                   

 

Kaplan, Robert. The Nothing That Is: A Natural History of Zero. New York: Oxford University Press, 2000. 

                    "Look at zero and yoou see nothing, but look through it and you see the world."

 

Kline, Morris. Mathematics: The Loss of Certainty. New York: Oxford University Press, 1980. 

                    "This book treats the fundamental changes that man has been forced to make in his understanding of the nature and role of mathematics."

 

Koyré, Alexandre. From the Closed World to the Infinite Universe. Baltimore: Johns Hopkins University Press, 1976. 

                    "During the sixteenth and seventeenth centuries a radical change occurred in the patterns and framework of European thought. The root and fruit of this revolution are modern science and modern philosophy. Dr. Koyré interprets this revolution in terms of the change that occurred in man's conception of his universe and of his own place in it and demonstrates the primacy of this change in the development of the modern world."

 

Kyburg, Jr., Henry E. Theory and Measurement. Cambridge, UK: Cambridge University Press, 1984. 

                    "[The author] proposes here an original, carefully worked out theory the foundations of measurement, to show how quantities can be defined, why certain mathematical structures are appropriate to them and what meaning attaches to the results generated. Crucial to his approach is the notion of error -- it can not be eliminated entirely and from its introduction and control, he argues, arises the very possibility of measurement."

 

Lachterman, David Rapport. The Ethics of Geometry: A Genealogy of Modernity. New York: Routledge, 1989. 

                   

 

Lakatos, I. Proofs and Refutations. Cambridge: Cambridge University Press, 1976. 

                    A deep but accessible book that uses an imaginary classroom dialogue in which the actual historical words of mathematicians are used to explore the evolving nature of mathematical ideas and to support the author's quasi empirical view of mathematics.

 

Lakatos, Imre. Mathematics, Science and Epistomology: Philosophical Papers, vol.2. Cambridge: Cambridge University Press, 1978. 

                   

 

Lakatos, Imre. The Methodology of Scientific Research Programmes: Philosophical Papers, vol.1. Cambridge: Cambridge University Press, 1978. 

                   

 

Posy, Carl J. (editors): Kant's Philosophy of Mathematics: Modern Essays,  Boston: Kluwer, 1992.

                    "The present volume includes the classic papers from the 1960s and 1970s which sparked this renaissance of interest, together with updated postscripts by their authors. It includes the most important recent work on Kant's philosophy of mathematics"

 

Reichenbach, Hans. The Philosophy of Space and Time. New York: Dover Publications, Inc., 1958. 

                   

 

Rothstein, Edward. Emblems of Mind: The Inner Life of Music and Mathematics. New York: Times Books, 1995. 

                   

 

Rucker, Rudy. Infinity and the Mind: The Science and Philosophy of the Infinite. Boston: Birkhauser, 1982. 

                    "This book discusses every kind of infinity: potential and actual, mathematical and physical, theological and mundane. Talking about infinity leads to many fascinating paradoxes. By closely examining these paradoxes we learn a great deal about the human mind, its powers, and its limitations."

 

Russell, Bertrand. An Essay on the Foundations of Geometry. New York: Dover Publications, Inc., 1956. 

                    "The problem Russell analyzes and solves, at least to his satisfaction in 1897, is: What geometrical knowledge must be the logical starting point for a sceince of space and must also be logically necessary to the experience of any form of externality?"

 

Sacks, Oliver. The Man Who Mistook His Wife for a Hat: and Other Clinical Tales. New York: Harper & Row, 1987. 

                   

 

Shapiro, Stewart. Thinking About Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press, 2000. 

                   

 

Sklar, Lawrence. Space, Time, And Spacetime. Berkley: University of California Press, 1977. 

                    "The major aim of this book is to cast as much doubt as possible on the view that science and philosophy are independent pursuits that can be carried out in total ignorance of each other."

 

Stein, Charles (editors): Being = Space X Action, IO.41: Berkeley, CA: North Atlantic Books, 1988.

                    "Searches for Freedom of Mind through Mathematics, Art, and Mysticism."

 

Tragesser, Robert S. Husserl and Realism In Logic and Mathematics. Cambridge: Cambridge University Press, 1984. 

                   

 

Tymoczko, Thomas. New Directions in the Philosophy of Mathematics. Boston: Birkhauser, 1986. 

                    An updated (to 1986) collection of readings.

 

Uspensky, V.A. Gödel's Incompleteness Theorem. Moscow: Mir Publishers, 1987. 

                   

 

Webb, Judson Chambers. Mechanism, Mentalism, and Metamathematics: An Essay on Finitism. Dordrecht: D. Reidel Publishing Company, 1980. 

                   

 

Wertheim, Margaret. The Pearly Gates of Cyberspace: A History of Space from Dante to the Internet. New York: W.W. Norton & Company, 1999. 

                    "Can cyberspace be a new realm for the soul? In this povocative book Margaret Wertheim traces the evolution of our concept of space from the Middle Ages through the rise of modern science and on to cyberspace. Linking the science of space to wider cultrual history, Wertheim challenges the current spiritualizing of cyberspace and suggests that it cannot sustain religious dreams."

 

Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics. Cambridge, MA: The MIT Press, 1983. 

                   

 

Zebrowski, Ernest, Jr. A History of the Circle: Mathematical Reasoning and the Physical Universe. New Brunswick, NJ: Rutgers University Press, 1999. 

                   

 

 

RN. Real Numbersal, Ebbinghaus et. Numbers. New York: Springer-Verlag, 1991. 

                    A lively story about the concept of number.

 

Goldblatt, Robert. Lectures on the Hyperreals. New York: Springer, 1998. 

                   

 

Laugwitz, Detlef: ""Infinitely Small Quantities in Cauchy's Textbooks,"", Historia Mathematica, 14 (1987), 258-274.

                   

 

Moore, Ramon. Methods and Applications of Interval Analysis. Philadelphia: Society of Industrial and Applied Mathematics, 1979. 

                    Interval analysis "an approach to computing that treats an interval as a new kind of number."

 

Russell, Bertrand. An Essay on the Foundations of Geometry. New York: Dover Publications, Inc., 1956. 

                    "The problem Russell analyzes and solves, at least to his satisfaction in 1897, is: What geometrical knowledge must be the logical starting point for a sceince of space and must also be logically necessary to the experience of any form of externality?"

 

Simpson: ""The Infidel Is Innocent,"", The Mathematical Intelligencer, 12 (1990), 42-51.

                    An accessible expositon of the nonstandard reals.

 

Turner, Peter R.: ""Will the 'Real' Real Arithmetic Please Stand Up?"", Notices of the A.M.S., 38 (1991), 298-304.

                    An article about various finite representations of real numbers used in computing.

 

 

SA. Sacred and Spiritual Geometry

 

al, Ebbinghaus et. Numbers. New York: Springer-Verlag, 1991. 

                    A lively story about the concept of number.

 

Howell, Alice O. The Web in the Sea: Jung, Sophia, and the Geometry of the Soul. Wheaton: Quest Books, 1993. 

                   

 

Iamblichus. The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers. Grand Rapids: Phanes Press, 1988. 

                   

 

Koyré, Alexandre. From the Closed World to the Infinite Universe. Baltimore: Johns Hopkins University Press, 1976. 

                    "During the sixteenth and seventeenth centuries a radical change occurred in the patterns and framework of European thought. The root and fruit of this revolution are modern science and modern philosophy. Dr. Koyré interprets this revolution in terms of the change that occurred in man's conception of his universe and of his own place in it and demonstrates the primacy of this change in the development of the modern world."

 

Lachterman, David Rapport. The Ethics of Geometry: A Genealogy of Modernity. New York: Routledge, 1989. 

                   

 

Lawlor, Robert. Sacred Geometry: Philosophy and practice. New York: Crossroad, 1982. 

                    "This is an introduction to the geometry which, as the ancients taught and modern science confirms, underlies the structure of the universe."

 

Lundy, Miranda. Sacred Geometry. New York: Walker & Company, 1998. 

                   

 

Mann, A.T. The Round Art: The Astrology of Time and Space. New York: Galley Press, 1979. 

                   

 

Mohen, Jean -Pierre. Standing Stones: Stonehenge, Carnac and the World of Megaliths. London: Thames & Hudson, 1999. 

                    "... this book considers the special significance -- religious and cultural, architectural and scientific -- of these enigmatic Neolithic stone structures ..."

 

Pennick, Nigel. Sacred Geometry: Symbolism and Purpose in Religious Structures. San Francisco: Harper & Row, 1982. 

                   

 

Perkins, James S. A geometry of Space-Consciousness. Adyar, Madras 600020, India: Theosophical Publishing House, 1978. 

                    "There are two geometries: the geometry of form in physical space, and the geometry of motion in man's consciousness."

 

Plummer, L. Gordon. The Mathematics of the Cosmic Mind. Wheaton, IL: The Theosophical Publishing House, 1970. 

                    "To study the Pythagorean Solids as embodying mathematically the keys to the mysteries about Man and the Universe."

 

Plummer, L. Gordon. By The Holy Tetraktys!: Symbol and Reality in Man and Universe. Point Loma, CA: Point Loma Publications, 1982. 

                   

 

Vandenbroeck, André. Philosophical Geometry. Rochester, VT: Inner Traditions International, Ltd., 1987. 

                    "Philosophical Geometry covers the activity of establishing a necessary conduct for mind through a set of signs denoting a necessary conduct of facts."

 

Wertheim, Margaret. The Pearly Gates of Cyberspace: A History of Space from Dante to the Internet. New York: W.W. Norton & Company, 1999. 

                    "Can cyberspace be a new realm for the soul? In this povocative book Margaret Wertheim traces the evolution of our concept of space from the Middle Ages through the rise of modern science and on to cyberspace. Linking the science of space to wider cultrual history, Wertheim challenges the current spiritualizing of cyberspace and suggests that it cannot sustain religious dreams."

 

 

SG. Symmetry and Groups

 

Vision Geometry, Contemporary Mathematics.119: Washington DC: American Mmathematical Society, 1989.

                    "Computer vision is concerned with obtaining descriptive information about a scene by computer analysis of images of the scene."

 

Field, Mike and Golubitsky, Martin. Symmetry in Chaos: A Search for Pattern in Mathematics, Art, Aand Nature. New York: Oxford University Press, 1992. 

                   

 

Gardner, Martin. Penrose Tiles to Trapdoor Ciphers. W.H. Freeman & Co, 1989. 

                   

 

Giacovazzo, C. (editors): Fundamentals of Crystallography,  New York: Oxford University Press, 1985.

                   

 

Grünbaum, Branko and Shepard, G.C. Tilings and Patterns. New York: W.H. Freeman, 1987. 

                    A 700 page book detailing what is know about plane tilings and patterns.

 

Hargittai, István and Hargittai, Magdolna. Symmetry: A Unifying Concept. Bolinas, CA: Shelter Publications, 1994. 

                    "The single, most important purpose of this book is to help you notice the world around you, to train your eye and mind to see new patterns and make new connections."

 

Horne, Clare E. Geometric symmetry in patterns and tilings. Boca Raton, FL: CRC Press, 2000. 

                   

 

Jones, Owen. The Grammar of Ornament. New York: Dorling Kindersley, 2001. 

                    "A unique collection of more than 2,350 classic patterns"

 

Lyndon, Roger C. Groups and Geometry. New York: Cambridge University Press, 1985. 

                    "This book is intended as an introduction, demanding a minimum of background, to some of the central ideas in the theory of groups and in geometry. It grew out of a course, for advanced undergraduates and beginning graduate students"

 

Macgillavry, Caorline H. Symmetry Aspects of M.C. Escher's Periodic Drawings. Utrecht: Published for the International Union of Crystallography by A. Oosthoek's Uitgeversmaatschappij NV, 1965. 

                   

 

Martin, George E. Transformation Geometry: An Introduction to Symmetry. New York: Springer-Verlag, 1982. 

                    "Our study of the automorphisms of the plane and of space is based on only the most elementary high-school geometry. In particular, group theory is not a prerequisite here. On the contrary, this modern approach to Euclidean geometry gives the concrete examples that are necessary to appreciate an introduction to group theory."

 

Montesinos, José María. Classical Tessellations and Three-Manifolds. New York: Springer-Verlag, 1985. 

                    "This book explores a relationship between classical tessellations and three-manifolds."

 

Radin, Charles. Miles of Tiles. Providence, RI: American Mathematical Society, 1999. 

                    "In this book, we try to display the value (and joy!) of starting from a mathematically amorphous problem and combining ideas from diverse sources to produce new and significant mathematics -- mathematics unforeseen from the motivating problem ..." The common thread throughout this book is aperiodic tilings.

 

Robertson, Stewart A. Polytopes and Symmetry. New York: Cambridge University Press, 1985. 

                    "These notes are intended to give a fairly systematic exposition of an approach to the symmetry classification of convex polytopes that casts some fresh light on classical ideas and generates a number of new theorems."

 

Senechal, M. Quasicrystals and Geometry, , 1995. Cambridge, UK: Cambridge University Press, 1995. 

                   

 

Weyl, Hermann. Symmetry. Princeton, NJ: Princeton University Press, 1952. 

                    A readable discussion of all mathematical aspects of symmetry especially its relation to art and nature - nice pictures. Weyl is a leading mathematician of this century.

 

Yaglom, I.M. Geometric Transformations III. New York: Random House, Inc., 1973. 

                   

 

Yale, Paul B. Geometry and Symmetry. New York: Dover, 1988. 

                    "This book is an introduction to the geometry of Euclidean, affine and projective spaces with special emphasis on the important groups of symmetries of these spaces."

 

 

SP. Spherical Geometry

 

Bonnet, Pierre Ossian. Astronomie sphérique; notes sur le cours professé pendant l'année 1887. The Cornell Library Historical Mathematics Monographs, 1887.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=03500002&seq=9  

 

Coxeter, H.S.M. The Real Projective Plane. New York: Cambridge University Press, 1955. 

                    "This introduction to projective geometry can be understood by anyone familiar with high-school geometry and algebra. The restriction to real geometry of two dimensions makes it possible for every theorem to be illustrated by a diagram."

 

Davies, Charles. Elements of Descriptive Geometry, with their application to spherical trigonometry, spherical projections, and warped surfaces. The Cornell Library Historical Mathematics Monographs, 1859.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=03190002&seq=3

                   

 

Davies, Charles. Elements of Geometry and Trigonometry from the works of A. M. Legendre: adapted to the course of mathematical instruction in the United States. The Cornell Library Historical Mathematics Monographs, 1890.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=04850002&seq=5&frames=0&view=50

                   

 

Farin, Gerald E. NURBS: From Projective Geometry to Practical Use, Second Edition. Natick, MA: A. K. Peters, 1999. 

                    NURBS is an industry standard for curve and surface design based on projective geometry.

 

Hartshorne, Robin: "Non-Euclidean III.36", American Mathematical Monthly, 110 (2003), 495-502.

                    Power of a point on sphere and hyperbolic plane

 

Hearn, George Whitehead. Researches on curves of the second order. London: G. Bell, 1846. 

                    Subtitle:  also on cones and spherical conics treated analytically, in which the tangencies of Apollonius are investigated, and general geometrical constructions deduced from analysis; also several of the geometrical conclusions of M. Chasles are analytically resolved, together with many properties entirely original.  Electronic Access: http://resolver.library.cornell.edu/math/1849296

 

Henderson, David W. Experiencing Geometry on Plane and Sphere. Upper Saddle River, NJ: Prentice Hall, 1996. 

                    "This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the plane and on the sphere."

 

Lénárt, István. Non-Euclidean Adventures on the Lénárt Sphere: Activities comparing planar and spherical geometry. Berkley: Key Curriculum Press, 1995. 

                   

 

Singer, David A. Geometry: Plane and Fancy. New York: Springer, 1998. 

                    "This book is about ... the idea of curvature and how it affects the assumptions about and principles of geometry."

 

Tan, A.: "A Bird's Eye View of Spherical Triangles", Mathematical Spectrum, 32:2 (1999-2000), 25-28.

                   

 

Taylor, Charles. An introduction to the ancient and modern geometry of conics. Cambridge [Eng.]: Deighton, Bell and co, 1881. 

                    Subtitle: being a geometrical treatise on the conic sections with a collection of problems and historical notes and prolegomena. Electronic Access: http://resolver.library.cornell.edu/math/1849306

 

Todhunter, Isaac. Spherical Trigonometry: For the Use of Colleges and Schools. London: Macmillan, 1886.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00640001&seq=5

                     All you want to know, and more, about trigonometry on the sphere. Well written with nice discussions of surveying.

 

Yale, Paul B. Geometry and Symmetry. New York: Dover, 1988. 

                    "This book is an introduction to the geometry of Euclidean, affine and projective spaces with special emphasis on the important groups of symmetries of these spaces."

 

 

TG. Teaching Geometry

 

Case, Bette Anne (editors): You're the Professor, What Next?, Ideas and Resources For Preparing College Teachers,  Washington DC.: The Mathematical Association of America, 1994.

                   

 

Malkevitch, Joseph and Proceedings of a COMAP conference "of a small group of geometers to study what could be done to revitalize geometry in our colleges, and what effects this might have on the teaching of geometry in general.". Geometry's Future, second edition. USA: COMAP, 1991. 

                   

 

Mammana, Carmelo, and Villiani, Vinicio (editors): Perspectives on the Teaching of Geometry for the 21st Century: An ICMI Study, New ICMI Study Series Dordrecht: Kluwer, 1998.

                   

 

Todhunter, Isaac. Spherical Trigonometry: For the Use of Colleges and Schools. London: Macmillan, 1886.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00640001&seq=5

                     All you want to know, and more, about trigonometry on the sphere. Well written with nice discussions of surveying.

 

Zimmermann, Walter and Cunningham, Steve. Visualization in Teaching and Learning Mathematics. Washinton, DC: Mathematical Association of America, 1991. 

                    "A project sponsored by the Committee on Computers in Mathematics Education of M.A.A."

 

 

TM. Teaching MathematicsKrantz, Steven G. How to Teach Mathematics: A Personal Perspective. Providence, RI: American Mathematical Society, 1993. 

                   

 

Smith, Seaton E., Jr., and Backman, Carl A. (editors): Games and Puzzles for Elementary and Middle School Mathematics: Readings from the Arithmetic Teacher,  Reston, VA: National Council of Teachers of Mathematics, 1975.

                   

 

Stanley and Usiskin. Mathematics for High School Teachers, An Advanced Perspective. Pre-Print,

                   

 

Stueben, Michael and Sandford, Diane. Twenty Years Before the Blackboard: The Lessons and Humor of a Mathematics Teachers. Washington, DC: The Mathematical Association of America, 1998. 

                   

 

Sciences, The ConferenceBoardoftheMathematical. The Mathematical Education of Teachers. Providence, RI: American Mathematical Society, 2001. 

                   

 

Zimmermann, Walter and Cunningham, Steve. Visualization in Teaching and Learning Mathematics. Washinton, DC: Mathematical Association of America, 1991. 

                    "A project sponsored by the Committee on Computers in Mathematics Education of M.A.A."

 

 

TP. Topology

 

Barr, Stephen. Experiments in Topology. New York: Crowell, 1964. 

                    Experimental topology that goes beyond the Möbius Band.

Experimental topology that goes beyond the Möbius Band.

 

Farmer, David W. and Stanford, Theodore B. Knots and Surfaces, A Guide to Disovering Mathematics. Washington DC: American Mathematical Society, 1996. 

                   

 

Francis, G.K. A Topological Picturebook. New York: Springer Verlag, 1987. 

                    Francis presents elaborate and illustrative drawings of surfaces and provides guidelines for those who wish to produce such drawings.

 

Francis, G.K., and Weeks, Jeffrey R.: "Conway's ZIP Proof", American Mathematical Monthly, 106 (1999), 393-399.

                    A new proof of the classification of (triangulated) surfaces (2-manifolds).

 

Hurewics, W. and Wallman, H. Dimension Theory. Princeton: Princeton University Press, 1941. 

                    Contains a proof of the Invariance of Domain in the context of the theory of the dimension of topological spaces.

 

Newman, M.H.A. Elements of the Topology of Plane Sets of Points. Cambridge: Cambridge University Press, 1964. 

                    Contains a proof of the Invariance of Domain that is the most geometric.

 

Prasolov, V.V. Intuituve Topology. Washington DC.: American Mathematical Society, 1995. 

                   

 

Singer, I.M. and Thorpe, John A. Lecture Notes on Elementary Topology and Geometry. Glenview: Scott, Foresman and Company, 1967. 

                    "What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topology, and group theory."

 

Sossinsky, Alexei. Knots: Mathematics with a Twist. Cambridge, MA: Harvard University Press, 2002. 

                   

 

Spanier. Algebraic Topology. New York: McGraw Hill Book Company, 1966. 

                    Contains a proof of the Invariance of Domain based on algebraic topology.

 

 

TX. Geometry Texts

 

Brannan, David A., Esplen, Matthew F., and Gray, Jeremy J. Geometry. Cambridge: Cambridge University Press, 1999. 

                    Geometry textbook for the Open University.

 

Coxeter, H.S.M. Non-Euclidean Geometry. Toronto: University of Toronto Press, 1965. 

                   

 

Davies, Charles. Elements of Geometry and Trigonometry from the works of A. M. Legendre: adapted to the course of mathematical instruction in the United States. The Cornell Library Historical Mathematics Monographs, 1890.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=04850002&seq=5&frames=0&view=50

                   

 

Eves, Howard. A Survey of Geometry. Boston: Allyn & Bacon, 1963. 

                    A textbook that contains an extensive coverage of the dissection theory of polygons.

 

Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History. New York: Freeman, 1980. 

                    This is a very readable textbook that includes some philosophical discussions.

 

Hansen, Vagn Lundsgaard. Shadows of the circle: Conic Sections, Optimal Figures and Non-Euclidean Geometry. Singapore: World Scientific, 1998. 

                    "It is my hope that these topics will be an inspiration in connection with teaching of geometry at various levels including upper secondary school and college education."

 

Hartshorne, Robin. Companion to Euclid: A course of geometry, based on Euclid's Elements and its modern descendants. Providence, RI: American Mathematical Society, 1997. 

                    "The course begins ... with a critical of Euclid's Elements... Then ... we study Hilbert's axiom system to bring the subject up to a modern standard of rigor... Then, depending on the taste of the instructor, one can follow a more geometric path by going directly to non-Euclidean [hyperbolic] geometry ..., or a more algebraic one, exploring the relation between geometric constructions and field extensions ..."

 

Hartshorne, Robin. Geometry: Euclid and Beyond. New York: Springer, 2000. 

                   

 

Henderson, David W. Experiencing Geometry on Plane and Sphere. Upper Saddle River, NJ: Prentice Hall, 1996. 

                    "This book will lead the reader on an exploration of the notion of straightness and the closely related notion of parallel on the plane and on the sphere."

 

Henderson, David W. and Taimina, with Daina. Experiencing Geometry: in Euclidean, Spherical, and Hyperbolic Spaces. Upper Saddle River, NJ: Prentice Hall, 2000. 

                    revised and expanded second edition of Experiencing Geometry

 

Henle, Michael. Modern Geometries: The Analytic Approach. Upper Saddle River: Prentice-Hall, 1997. 

                   

 

Jacobs, Harold R. Geometry. San Fransisco: W.H. Freeman and Co., 1974. 

                    A high-school-level text based on guided discovery.

 

Jennings, George. Modern Geometry with Applications. Springer-Verlag, 1994. 

                   

 

Katz, Victor (editors): Using History to Teach Mathematics: An International Perspective, MAA Notes.#51: Washington, D.C.: Mathematical Association of America, 2000.

                   

 

Kay, David C. College Geometry. New York: Holt, Rinehart and Winston, 1969. 

                    A unified treatment of axiomatic spherical and hyperbolic geometries

 

Kay, David C. College Geometry: A Discovery Approach. New York: HarperCollins College Publishers, 1994. 

                    "This book was written for an introductory, college level course in geometry for mathematics majors or students in mathematics education seeking teachers certification in secondary school mathematics. The latter purpose is fully recognized throughout the text, with the development traditional lines, and numerous problems and examples coming from current secondary school textbooks."

 

King, James. Geometry Through the Circle with The Geometer's Sketchpad. Key Curriculum Press, 1994. 

                   

 

Martin, George E. Geometric Constructions. New York: Springer, 1998. 

                    A geometry textbook based on ruler and compass constructions.

 

Meserve, Bruce E. Fundamental Concepts of Geometry. New York: Dover, 1983. 

                    "The primary purpose of this book is to help the reader (i) to discover how Euclidean plane geometry is related to, and often a special case of, many other geometries, (ii) to obtain a practical understanding of "proof," (iii) to obtain the concept of geometry as a logical system based upon postulates and undefined elements, and (iv) to appreciate the historical evolution of our geometrical concepts and the relation of Euclidean geometry to the space in which we live."

 

Meyer, Walter. Geometry and Its Applications. San Diego: Academic Press, 1999. 

                    "... a solid introduction to axiomatic Euclidean geometry, some non-Euclidean geometry, and a substantial amount of transformation geometry. ... We pay significant attention to applications, we provide optional dynamic geometry courseware for use with The Geometer's Sketchpad ..."

 

Millman, Richard S. and Parker, George D. Geometry: A Metric Approach with Models. New York: Springer-Verlag, 1981. 

                    "This book is intended as a first rigorous course in geometry. As the title indicates, we have adopted Birkoff's metric approach (i.e., through use of real numbers) rather than Hilbert's synthetic approach to the subject."

 

Moise, Edwin E. Elementary Geometry from an Advanced Standpoint. Reading, MA: Addison-Wesley Publishing, 1963. 

                   

 

Noronha, M. Helena. Euclidean and Non-Euclidean Geometries. Upper Saddle River, NJ: Prentice Hall, 2002. 

                   

 

Pedoe, Dan. Geometry: A Comprehensive Course. New York: Dover Publications, Inc., 1970.  

                    "The main purpose of the course was to increase geometrical, and therefore mathematical understanding, and to help students to enjoy geometry. This is also the purpose of my book."

 

Pogorelov, A. Geometry. Moscow: MIR publishers, 1987. 

                    "The book is... aimed at professional training of the school or university teacher-to-be. The first part, analytic geometry, is easy to assimilate, and actually reduced to acquiring skills in applying algebraic methods to elementary geometry. The second part, differential geometry, contains the basics of the theory of curves and surfaces. The third part, foundations of geometry, is original. The fourth part is devoted to certain topics of elementary geometry."

 

Posamentier, Alfred S. Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students. Emeryville, CA: Key College Publishing, 2002. 

                   

 

Prasolov, V.V. Intuituve Topology. Washington DC.: American Mathematical Society, 1995. 

                   

 

Prasolov, V.V. and Tikhomirov, V.M. Geometry. Providence, RI: American Mathematical Society, 2001. 

                   

 

Prenowitz, Walter and Jordan, Meyer. Basic Concepts of Geometry. New York: Blaisdell Publishing, 1965. 

                   

 

Serra, Michael. Discovering Geometry: An Inductive Approach. Berkeley, CA: Key Curriculum Press, 1989. 

                   

 

Shurman, Jerry. Geometry of the Quintic. New York: John Wiley & Sons, Inc., 1997. 

                    An advanced undergraduate text which uses the icosahedron to solve quintic equations. Along the way, he explores the Riemann sphere, group representations, and invariant functions.

 

Sibley, Thomas Q. The Geometric Viewpoint: A Survey of Geometries. Reading, MA: Addison Wesley, 1998. 

                    "Geometry combines visual delights and powerful abstractions, concrete intuitions and general theories, historical perspective and contemporary applications, and surprising insights and satisfying certainty. In this textbook, I try to weave together these facets of geometry. I also want to convey the multiple connections that different topics in geometry have with each other and that geometry has with other areas of mathematics."

 

Singer, David A. Geometry: Plane and Fancy. New York: Springer, 1998. 

                    "This book is about ... the idea of curvature and how it affects the assumptions about and principles of geometry."

 

Smart, James R. Modern Geometries, Fifth Edition. Pacific Grove: Brooks/Cole Publishing Company, 1998. 

                    "This fifth edition of Modern Geometries is designed for one or more courses in modern geometry at the junior-senior level in universities. The central text of the theme is the presentation of many different geometries, rather than any single geometry. The use of both groups of transformations and sets of axioms to classify geometries continues to be of central importance."

 

Stahl, Saul. Geometry From Euclud to Knots. Upper Saddle River, New Jersey: Pearson Education, Inc., 2003. 

                   

 

Stanley and Usiskin. Mathematics for High School Teachers, An Advanced Perspective. Pre-Print,

                   

 

Wallace, Edward C. and West, Stephen F. Roads to Geometry. Upper Saddle River, NJ: Prentice Hall, Inc., 1998. 

                    "The goal of this book is to provide a geometric experience which clarifies, extends, and unifies concepts which are generally discussed in traditional high school geometry courses and to present additional topics which assist in gaining a better understanding of elementary geometry."

 

 

UN. The Physical Universe

 

Ferguson, Kitty. Measuring the Universe. New York: Walker & Co, 1999. 

                    A history of the attempts to measure the universe.

 

Ferris, Timothy. The Whole Shebang: A State-of-the-Universe(s) Report. New York City: Simon & Schuster, 1997. 

                    "This book aims to summarize the picture of the universe that science has adduced ..., and to forecast an exciting if unsettling new picture that may emerge in the near future."

 

Guth, Alan H. The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. Reading, MA: Perseus Books, 1997. 

                    "The inflationary universe is a theory of the 'bang' of the big bang."

 

Osserman, Robert. Poetry of the Universe: A Mathematical Exploration of the Cosmos. New York: Anchor Books, 1995. 

                    "What is the shape of the universe, and what do we mean by the curvature of space? One aim of this book is to make absolutely clear and understandable both the meanings of those questions and the answers to them. Little or no mathematical background is needed..."

 

Penrose, Roger: The Geometry of the Universe. Mathematics Today. Steen L, (eds). New York, Springer-Verlag, 1978,

                    An expository discussion of the geometry of the universe.

An expository discussion of the geometry of the universe.

 

Penrose, Roger, Shimony, Abner, Cartwright, Nancy, and Hawking, Stephen. The Large, The Small and the Human Mind. Cambridge, UK: Cambridge University Press, 1997. 

                    "This volume provides an accessible, illuminating and stimiulating introdution to Roger Penrose's vision of theoretical physics for the 21st Century."

 

Proctor, Richard A. A Treatise on The Cycloid and all forms of Cycloidal Curves and on the use of such curves in dealing with the motions of planets, comets, &c. and of matter projected from the sun. The Cornell Library Historical Mathematics Monographs, 1878.  online: http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=02260001&view=50&frames=0&seq=9

                   

 

Rees, Martin. Before the Beginning: Our Universe and Others. Reading, MA: Perseus Books, 1997. 

                    "This book presents an individual view on cosmology -- how we perceive our universe, what the current debates are about, and the scope and limits of our future knowledge."

 

Schneider, Michael S. A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperPerennial, 1994. 

                   

 

Stahl, Saul. Geometry From Euclud to Knots. Upper Saddle River, New Jersey: Pearson Education, Inc., 2003. 

                   

 

Toben, Bob. Space-Time and Beyond: Toward an Explanation of the Unexplainable. New York: E.P. Dutton, 1975. 

                   

 

Wertheim, Margaret. The Pearly Gates of Cyberspace: A History of Space from Dante to the Internet. New York: W.W. Norton & Company, 1999. 

                    "Can cyberspace be a new realm for the soul? In this provocative book Margaret Wertheim traces the evolution of our concept of space from the Middle Ages through the rise of modern science and on to cyberspace. Linking the science of space to wider cultrual history, Wertheim challenges the current spiritualizing of cyberspace and suggests that it cannot sustain religious dreams."

 

Zebrowski, Ernest, Jr. A History of the Circle: Mathematical Reasoning and the Physical Universe. New Brunswick, NJ: Rutgers University Press, 1999. 

                   

 

The University of Michigan historic books collection has following books on-line, which can be useful in geometry class: