Bibliography (as published in Experiencing Geometry)

Expanded Bibliography

A. Ancient Texts

al'Khowarizmi, Algebra , (Robert of Chester's Latin Translation), L.C. Karpinski, ed. and trans., New York: Macmillan, 1915.

This is an English translation.

Apollonius of Perga, Treatise on Conic Sections , T.L. Heath, ed., New York: Dover, 1961.

Baudhayana, Sulbasutram , G. Thibaut, trans., S. Prakash & R. M. Sharma, ed., Bombay: Ram Swarup Sharma, 1968.

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

Euclid, Elements , T.L. Heath, ed., New York: Dover, 1956.

Euclid, Optics , H. E. Burton, trans., Journal of the Optical Society of America , vol. 35, no. 5, pp. 357-372, 1945.

Euclid, Phaenomena , in Euclidis opera omnia , Heinrich Menge, ed., Lipsiae: B.G. Teubneri, 1883-1916.

The Holy Bible , NIV Zondervan Bible Publishers, 1985.

Khayyam, Omar, Algebra , D.S. Kasir, ed., New York: Columbia Teachers College, 1931. (and New York, AMS Press, 1972.)

Khayyam, Omar, Risâla fî sharh mâ ashkala min musâdarât Kitâb 'Uglîdis (Arabic: Explanation of the Difficulties in Euclid's Postulates), A.I. Sabra, ed., Alexandria, Egypt: Al Maaref, 1961.

Translated in A. R. Amir-Moez, "Discussion of Difficulties" in Euclid by Omar ibn Abrahim al-Khayyami (Omar Khayyam), Scripta Mathematica , 24 (1958-59), pp. 275-303.

Khayyam, a paper (no title).

Translated in A. R. Amir-Moez, "A Paper of Omar Khayyam," Scripta Mathematica , 26(1963), pp.323-337.

Koran (Holy Qur-An) , Abdullah Yusuf Ali, trans., New York: Harper Publishing, 1946.

Plato, The Collected Dialogues , Edith Hamilton and Huntington Carns, eds., Princeton, NJ: Bollinger, 1961.

Plotinus, The Enneads , Stephen McKenna, trans., Burdette, NY: Larson, 1992.

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

B. Art and Design

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

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

A revealing look at the artist and the ideas behind his work.

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

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

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

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

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

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

C. Differential Geometry

Dodson, C. T. J., and T. Poston, 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., A.T. Fomenko, S.P. Novikov, Modern Geometry and Applications (Part I. The Geometry of Surfaces, Transformation Groups, and Fields) , Robert G. Burns, trans., New York: Springer-Verlag, 1984.

A well-written graduate text.

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.

Penrose, Roger, "The Geometry of the Universe," Mathematics Today , Lynn Steen, ed., New York: Springer-Verlag, 1978.

Spivak, Michael, A Comprehensive Introduction to Differential Geometry , Wilmington, DE: Publish or Perish, Inc., 1970.

In four(!) volumes Spivak relates the subject back to the original sources.

Stahl, Saul, The Poincar éHalf-Plane , Boston: Jones and Bartlett Publishers, 1993.

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.

D. Dimensions and Scale

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

A fantasy about 2-dimensional being in a plane encountering the third dimension.

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

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

A sequel to Abbott's Flatland .

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

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

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

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

E. Fractals

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.

F. Geometry in Different Cultures

Albarn, K., Jenny Miall Smith, Stanford Steele, Dinah Walker, The Language of Pattern , New York: Harper & Row, 1974.

An enquiry inspired by Islamic decoration.

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

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

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

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

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

Concepts of geometry and space in the Navajo culture.

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

A presentation of the mathematics in African cultures.

G. History

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

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

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

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

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

Mostly a collection of original sources in Western mathematics.

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.

Eves, Howard, Great Moments in Mathematics (after 1650) , Dolciani Mathematical Expositions, Vol. 7, Washington, DC: M.A.A., 1981.

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

A non-Eurocentric view of the history of mathematics.

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

A complete Eurocentric history of mathematical ideas.

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

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

"The pusuit of geometry in Victorian England."

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

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

History and cultural aspects of mathematics.

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 thru dissections, golden mean, relations between geometry and music, conic sections, etc.

H. Linear Algebra and Geometry

Banchoff, T., and J. Wermer, Linear Algebra through Geometry , New York: Springer-Verlag, 1983.

For several years this was the text for Cornell's undergraduate linear algebra course.

Dodson, C. T. J., and T. Poston, 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.

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

Includes affine and projective geometry.

Taylor, Walter F., The Geometry of Computer Graphics , Pacific Grove, CA: Wadsworth and Brooks, 1992.

I. Models, Polyhedra

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

Barnette, David, Map Colouring, Polyhedra, and the Four-Colour Problem , Dociani Mathematical Expositions Vol. 8, Washington, DC: M.A.A., 1983.

Cundy, M.H., and A.P. Rollett,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.

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 George Fleck, Shaping Space: A Polyheral Approach , Design Science Collection, Boston: Birkhauser, 1988.

J. Nature

Cook, T.A., The Curves of Life: Being an Account of Spiral Formations and their Applications to Growth in Nature, to Science, and to Art , New York: Dover Publications, 1979.

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

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

The book that started the popularity of fractal geometry.

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

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

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.

K. Non-Euclidean Geometries (Mostly Hyperbolic)

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.

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

A pictoral, visual tour of non-Euclidean geometries.

Millman, Richard S., and George D. Parker, Geometry: A Metric Approach with Models , New York: Springer-Verlag, 1981.

A modern formal axiomatic approach.

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

Using transformation groups to study spherical, hyperbolic, and toroidal geometries.

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

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

L. Philosophy

Benacerraf, Paul, and Hilary Putman, Philosophy of Mathematics: Selected Readings , Cambridge: Cambridge University Press, 1964.

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

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.

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

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

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

M. Spherical and Projective Geometry

Todhunter, Isaac, Spherical Trigonometry , London: Macmillan, 1886.

All you want to know, and more, about trigonometry on the sphere. Well written with nice discussions of surveying.

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.

N. Symmetry and Groups

Budden, F.J., Fascination of Groups , Cambridge: Cambridge University Press, 1972.

This is a fascinating book that relates algebra (groups) to geometry, music, and so forth, and has a nice description of symmetry and patterns.

Bunch, Bryan H., Reality's Mirror: Exploring the Mathematics of Symmetry , New York: John Wiley, 1989.

Burn, R.P., Groups: A Pathway to Geometry , Cambridge: Cambridge University Press, 1985.

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.

O. Surveys and General Expositions

Davis, P.J., and R. Hersh, The Mathematical Experience , Boston: Birkhauser, 1981.

A very readable collection of essays by two present-day mathematicians. I think every mathematics major should own this book.

Ekeland, Ivar, Mathematics and the Unexpected , Chicago: University of Chicago Press, 1988.

Gaffney, Matthew P. and Lynn Arthur Steen, 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.

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

Hilbert, David, and S. Cohn-Vossen, 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.

Honsberger, Ross, Mathematical Gems , Dolciani Mathematical Expositions, Vol. 2, Washington, DC: M.A.A., 1973.

Honsberger, Ross, Mathematical Gems II , Dolciani Mathematical Expositions, Vol. 4, Washington, DC: M.A.A., 1976.

Honsberger, Ross, Mathematical Morsels , Dolciani Mathematical Expositions, Vol. 1, Washington, DC: M.A.A., 1978.

Honsberger, Ross,Mathematical Plums , Dolciani Mathematical Expositions, Vol. 4, Washington, DC: M.A.A., 1979.

Expository stories about mathematics.

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

A mathematical fable.

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.

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

"Mathematical explorations and excursions."

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

Expository essays.

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

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

P. Texts

Coxeter, H.S.M., Introduction to Geometry , New York: Wiley, 1969.

This is a collection of diverse topics including non-Euclidean geometry, symmetry, patterns, and much, much more. Coxeter is one of the foremost living geometers.

Eves, Howard, A Survey of Geometry, Vol. 1, Boston: Allyn & Bacon, 1963.

Eves, Howard, Modern Elementary Geometry , Boston: Jones and Bartlett Publishing, 1992.

Jacobs, Harold R., Geometry , San Fransisco: W.H. Freeman and Co., 1974.

A high-school-level text based on guided discovery.

Serra, Michael, Discovering Geometry: An Inductive Approach , Berkeley, CA: Key Curriculum Press, 1989.

Q. Miscellaneous

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

Kempe, A.B., How to Draw a Straight Line , London: Macmillan, 1877.

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