Feature Column

Mathematics and Art

7. References

The literature connecting art and mathematics is especially scattered and varied. What is listed here is meant only as a small sample of what is available.

Abas, S. and A. Salman, Symmetries of Islamic Geometrical Patterns, World Scientific, Singapore, 1995.

Anderson, K., Brook Taylor's Role in the History of Linear Perspective, Springer-Verlag, New York, 1992.

Auckly, D. and J. Cleveland, Totally real origami and impossible paper folding, Amer. Math. Monthly (1995) 215-226.

Bangay, S., From virtual to physical reality with paper folding, Computational Geometry, 15 (2000) 161-174.

Bartashi, W., Linear Perspective, Van Nostrand, New York, 1981.

Berne, M. and B. Hayes, The complexity of flat origami, In Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, 1996, 175-183.

Bixler, N., A group theoretic analysis of symmetry in two-dimensional patterns from Islamic art, Ph. D. Thesis, New York University, 1980.

Boehm, W. and H. Prautzsch, Geometric concepts for Geometric Design, A. K. Peters, Wellesley, 1994.

Booker, P., A History of Engineering Drawing, Chatto & Windus, London, 1963.

Bool, F. and B. Ernst, J. Kist, J. Locher, F. Wierda, M.C. Escher, His Life and Complete Graphic Work, Harry Abrams, New York, 1982.

Botermans, J. and J. Slocum, Puzzles Old and New; University of Washington Press, Seattle, 1986.

Bourgoin, J., Arabic Geometrical Pattern & Design, Dover, New York, 1973.

Coffin, S., The Puzzling World of Polyhedral Dissections, Oxford U. Press, New York, 1990.

Coxeter, H. et al., (eds.), M.C. Escher: Art and Science, North-Holland, Amsterdam, 1986.

Crannell, A. and M. Frantz, A course in mathematics and art, J. of Geoscience Education, 48 (2000) 313-316.

Cromwell, P., Polyhedra, Cambridge U. Press, London, 1997.

Crowe, D., The geometry of African art, I. Bakuba art, Journal of Geometry 1 (1971) 169-182.

Crowe, D., The geometry of African art, II. A catalog of Benin patterns, Historia Mathematica 2 (1975) 253-271.

Crowe, D., The geometry of African art, III: The smoking pipes of Begho, in The Geometric Vein, (Coxeter Festschrift), C. Davis et al., (eds.), Springer-Verlag, New York, 1981.

Crowe, D., The mosaic patterns of H. J Woods, in Symmetry: Unifying Human Understanding, I. Hargittai (ed.), New York:, Pergamon, 1986, p. 407-411.

Crowe, D., Tongan symmetries, in Science of Pacific Island Peoples, Part IV, Education, Language, Patterns and Policy, J. Morrison, P. Garaghty, and L. Crowl, (eds.), Suva: Institute of Pacific Studies, 1994.

Crowe, D. and D. Nagy, Cakaudrove-style masi kesa of Fiji, Ars Textrina 18 (1992) 119-155.

Crowe, D. and R. Torrence, Admiralty Islands spear decorations: A minicatalog of pmm patterns, Symmetry: Culture and Science 4 (1993) 385-396.

Crowe, D. and D. Washburn, Groups and geometry in the ceramic art of San Ildefonso, Algebras, Groups and Geometries 3 (1985) 263-277.

Davies, C., A Treatise on Shades, Shadows, and Linear Perspective, A. S. Barnes and Burr, New York, 1857.

Demaine, E., Folding and Unfolding Linkages, Paper, and Polyhedra, in Discrete and Computational Geometry, J. Akiyama, M. Kano, and M. Urabe, (eds.), Volume 2098, Lecture Notes in Computer Science, Springer-Verlag, New York, 2001, p. 113-124.

Demaine, E. and M. Demaine, Recent Results in Computational Origami, Proc. 3rd. International Meeting of Origami Science, Math and Education (held in Moterey, CA., March 2001).

Demaine, E. and M. Demaine, A. Lubiw, Folding and cutting paper, In J. Akiyama, M. Kano, M. Urabe, (eds.), Volume 1763, Lecture Notes in Computer Science, Springer-Verlag, New York, 2000, p. 104-117.

Demaine, E. and M. Demaine, J. Mitchell, Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami, Comput. Geom. Theory Appl., 16 (2000) 3-21.

Descargues, P., Perspective: History, Evolution, Techniques, Van Nostrand, New York, 1982.

Dress, A. and D. Huson, Heaven and hell tilings, Structural Topology 17 (1990).

Edgerton, S., The Renaissance Rediscovery of Linear Perspective, Basic Books, New York 1975.

El-Said, I. and A. Parman, Geometric Concepts in Islamic Art, World of Islam Festival, London, 1976.

Emmer, M., (ed.), The Visual Mind, MIT Press, Cambridge, 1993.

Emmer, M., M.C. Escher: Geometries and Impossible Worlds; M.C. Escher: Symmetry and Space, 16mm films, International Telefilm Enterprises, Toronto, 1984.

Ernst, B. (Hans de Rijk), The Magic Mirror of M. C. Escher, Random House, New York, 1976.

Ernst, B. (Hans de Rijk), Optical Illusions, Benedict Taschen Verlag, Koln, 1992.

Farmer, D., Groups and Symmetry, American Mathematical Society, Providence, 1996.

Federov, E., Symmetry of regular systems of figures, Pro. of the Imperial Saint Petersburg Society, Series 2, 28 (1891) 1-146 (in Russian).

Federov, E., Symmetry in the plane, Proceedings of the Imperial Saint Petersburg Society, Series 2, 28 (1891) 345-389 (in Russian).

Field, J., Giovanni Battista Benedetti on the mathematics of linear perspective, J. of the Warburg and Courtauld Institute, 48 (1985) 71-99.

Field, J., Linear perspective and the projective geometry of Girard Desargues, Nuncius 2 (1987) 3-40.

Field, J., Perspective and the mathematicians: Alberti to Desargues, In Mathematics from Manuscript to Print, C. Hay (ed.), Oxford U. Press, New York, 1988, p. 236-263.

Field, J., Mathematics and the craft of painting: Piero della Francesca and perspective, in Renaissance and Revolution: Humanists, Craftsmen and Natural Philosophers in Early Modern Europe, J. Field and F James (eds.), Cambridge U. Press, London, 1993, p. 73-95.

Field, J., A mathematician's art. In Piero della Francesca and his Legacy, in M. Lavin, (ed.), Studies in the History of Art, Number 48, Center for the Advanced Study of the Visual Arts, National Gallery of Art, Washington, 1995, p. 177-197.

Field, J., The Invention of Infinity: Mathematics and Art in the Renaissance, Oxford U. Press, New York, 1997.

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

Frantz, M. The telescoping series in perspective, Mathematics Magazine, 71 (1998) 313-314.

Frederickson, G., Dissections Plane & Fancy, Cambridge U. Press, New York, 1997.

Frederickson, G., Geometric dissections that swing and twist, in Discrete and Computational Geometry, J. Akiyama, M. Kano, and M. Urabe, (eds.), Volume 2098, Lecture Notes in Computer Science, Springer-Verlag, New York, 2001, p. 137-148.

Frederickson, G., Hinged Dissections: Swinging & Twisting, Cambridge U. Press, New York, 2002.

Gasson, P., Geometry of Spatial Forms: Analysis, Synthesis, Concept Formulation and Space Vision for CAD, Ellis Horwood, New York, 1983.

Gerdes, P., Geometry From Africa, Mathematical Association of America, Washington, 1999.

Glassner, A., Andrew Glassner's Notebook: Recreational Computer Graphics, Morgan Kaufmann, San Francisco, 1999.

Gray, J., Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, Oxford U. Press, London, 1979.

Grünbaum, B. and G. Shepherd (1986), Is there an all-purpose tile? Amer. Math. Monthly 93, 545-551.

Grünbaum, B.. and Z. Grünbaum, G. Shephard, Symmetry in Moorish and Other Ornaments, Comp. and Math. with Appl., 12 (1986) 641-653.

Grünbaum, B. and G. Shephard, Tilings and Patterns, Freeman. New York, 1987.

Grünbaum, B., Regular polyhedra, in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guinness, (ed.), Routledge, London, 1994, p. 866-876.

Hanson, R., Molecular Origami: Precision Scale Models from Paper, University Science Books, Sausalitio, 1995.

Hargettai, I., (ed.), Symmetry1: Unifying Human Understanding, Pergamon, Oxford, 1986.

Hargettai, I., (ed.), Symmetry2, Pergamon, Oxford, 1989.

Hargittai, I., (ed.), Fivefold Symmetry, World Scientific, Singapore, 1992.

Holden, A., Shapes Space and Symmetry, Dover Press, New York, 1991.

Hull, T., On the mathematics of flat origamis, Congr. Number., 100 (1994) 215-224.

Hull, T., A note on "impossible" paper folding, Amer. Math. Monthly (1996) 240-241.

Jablon, S., Mirror generated curves, Symmetry: Culture and Science, 6 (1995) 275-278.

Jones, O., The Grammar of Ornament, Day and Son, London, 1856, reprint, Studio Editions, London, 1988.

Kaplan, C. and D. Salesin, Escherization, International Conference on Computer Graphics and Interactive Techniques, Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, Association of Machinery, 2000.

Kappraff, J., Connections, The Geometric Bridge between Art and Science, McGraw Hill, 1990.

Kemp, M., The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat, Yale U. Press, New Haven, 1990.

Kinsey, L. and T. Moore, Symmetry, Shape and Space, Key Curriculum Press, Emeryville, 2002.

Lang, R., A computational algorithm for origami design, in Proc. 12th Symposium Comput. Geom., ACM., New York, 1996, p. 98-105.

Lindberg, D., Theories of Vision from Al-Kindi to Kepler, U. of Chicago Press, Chicago, 1976.

Liu, Y., Symmetry groups in robotic assembly planning, Ph.D. Thesis, U. of Massachusetts, Amherst, 1990.

Locher, J. (ed.), The World of M. C. Escher, Harry Abrams, New York, 1972.

MacGillavery, C., Fantasy and Symmetry: The Periodic Drawings of M.C. Escher, Harry Abrams, New York, 1976.

Mainzer, K., Symmetries in mathematics, in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guinness, (ed.), Routledge, London, p. 1994-1623.

Makovicky, E., Ornamental brick work, theoretical and applied symmetrology and classification of pattern, Comp. Math with Appl. 17 (1989) 995-999.

Martin, G., Transformation Geometry, Springer-Verlag, Berlin, 1992.

Miura, K., (ed.), Origami Science and Art, Proceedings of the Second International Meeting of Origami Science and Scientific Origami, Seian University, Otsu, Shiga, Japan, 1997.

Gurkewitz, R. and B. Arnstein, 3-D Geometric Origami: Modular Polyhedra, Dover, New York, 1995.

Niggli, P., Die Flachensymmetrien homogener diskontinuen, Zeit. f. Kristallographie, 60 (1924) 283-98.

Niggli, P., Die regelmassige Punkverteilung langs einer Geraden in einer Ebene, Zeit. f. Kristallographie, 63 (1926) 255-74.

Ouchi, Jamime, Japanese Optical and Geometric Art, Dover, New York, 1977.

Penrose, L. and R. Penrose, Impossible objects: A special type of illusion, British J. of Psychology, 49 (1958) 31.

Peterson, I., Fragments of Infinity: A kaleidoscope of math and art, Wiley, New York, 2001.

Polya, G. Uber die Analogie der Kristallsymmetrie in der ebene, Z. Kristall., 60 (1924) 278-282.

Rowe, C. and J. McFarland, Engineering Descriptive Geometry, Princeton U. Press, Princeton, 1939 (2nd edition, 1953).

Salenius, T., Elementart Bevis for Pohlkes Sats, Nordisk Matematisk Tidskrift, 25-26 (1978) 150-152.

Sarhangi, R. (ed.), Bridges: Mathematical Connections in Art, Music, and Science, Conference Proceedings, yearly, 1998-2001.

Schaaf, W., Art and mathematics: A brief guide to source materials, Amer. Math. Monthly 58 (1951) 167-177.

Schattschneider, D. The plane symmetry groups. Their recognition and notation, Amer. Math. Monthly 85 (1978) 439-450.

Schattschneider, D., Tiling the plane with congruent pentagons, Mathematics Magazine 51 (1978) 29-44.

Schattschneider, D., Will it tile/ Try the Conway criterion!, Mathematics Magazine 53 (1980) 224-233.

Schattschneider, D., In black and white: how to create perfectly colored symmetric patterns, Comp. and Math. with Appl. 12B (1986) 673-695.

Schattschneider, D., The Polya-Escher connection, Mathematics Magazine 60 (1987) 293-298.

Schattschneider, D., Visions of Symmetry, W. H. Freeman, New York, 1990.

Schattschneider, D., Escher: A mathematician in spite of himself, in The Lighter Side of Mathematics, R. Guy, and R. Woodrow, (eds.), Mathematical Association of America, 1994, p. 91-100. (Reprinted from Structural Topology 15 (1988) 9-22).

Schattschneider, D. and W. Walker, M.C. Escher Kaleidocycles, Pomegranate Artbooks, Rohnert Park, 1987.

Schreiber, P., Art and Architecture, in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guiness, (ed.), Routledge, London, 1994, p. 1593-1611,

Senechal, M. and G. Fleck, (eds.), Patterns of Symmetry, U. Massachusetts Press, Amherst, 1974.

Senechal, M., Point groups and color symmetry, Z. Kristall., 142 (1975) 1-23.

Senechal, M., Color groups, Disc. Appl. Math., 1 (1979) 51-73.

Shubnikov, A. and V. Koptsik, Symmetry in Science and Art, Nauka, Moscow, 1972, Plenum Press, New York, 1974.

Stevens, P., Handbook of Regular Patterns, MIT Press, Cambridge, 1981.

Stewart, I. and M. Golubitsky, Fearful Symmetry - Is God a Geometer?, Blackwell, Oxford, 1992.

Taylor, R. and A. Micolich, D. Jones, Fractal analysis of Pollock's drip paintings, Nature, 399 (1999) 422.

Termes, D., New Perspective Systems, (privately published), Spearfish, South Dakota, 1998.

Van Delft, P. and J. Botermans, Creative Puzzles of the World, Harry Abrams, New York, 1978.

Veltman, K., Linear Perspective and the Visual Dimension of Science and Art, Deutscher Kunstverlag, Munich, 1986.

Videla, C., On points constructible from conics, Mathematical Intelligencer 19 (1997) 53-57.

Washburn, D., Style, classification and ethnicity: design categories on Bakuba raffia cloth, American Philosophical Society, Philadelphia, 1990.

Washburn, D. and D. Crowe, Symmetries of Culture, U. Washington Press, Seattle, 1988.

Wenninger, M., Polyhedron Models, Cambridge U. Press, New York, 1971.

Wenninger, M., Dual Models, Cambridge U. Press, New York, 1983.

Weyl, H., Symmetry, Princeton U. Press, Princeton, 1952.

White, J. The Birth and Rebirth of Pictorial Space, reprinted, Harvard University Press, Cambridge, 1987.

Wittkower, R. and B. Carter, The perspective of Piero della Francesca's "Flagellation," Journal of the Warburg and Courtauld Institutes, 16 (1953) 292-302.

Yen, J. and C. Sequin, Escher sphere construction kit, Proceedings of the 2001 Symposium on Interactive 3D Graphics, ACM, 2001, p. 95-98.

Zaslavsky, C., Africa Counts: Number and Pattern in African Culture, Lawrence Hill Books, Brooklyn, 1973.

Those who can access JSTOR can find some of the papers mentioned above there.


  1. Introduction
  2. Mathematical tools for artists
  3. Symmetry
  4. Mathematical artists and artist mathematicians
  5. Polyhedra, tilings, and dissections
  6. Origami
  7. References

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