Annotated Bibliography
Related to Geometry

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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'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.

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.

Annotated Bibliography Related to Geometry

Last Updated: August 2004

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

Annotated Bibliography
Related to Geometry