Geometric Transformations of the Euclidean Plane

  • Judith N. Cederberg
Part of the Undergraduate Texts in Mathematics book series (UTM)


The presentation of non-Euclidean geometry in Chapter 2 was synthetic, that is, figures were studied directly and without use of their algebraic representations. This reflects the manner in which both Euclidean and non-Euclidean geometries were originally developed. However, in the 17th century, French mathematicians Pierre de Fermat (1601–1665) and René Descartes (1596–1650) began using algebraic representations of figures. They realized that by assigning to each point in the plane an ordered pair of real numbers, algebraic techniques could be employed in the study of Euclidean geometry. This study of figures in terms of their algebraic representations by equations is known as analytic geometry.


Equilateral Triangle Prove Theorem Affine Transformation Euclidean Plane Invariant Point 
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Suggestions for Further Reading

  1. Caldwell, J. H. (1966). Chapter 11: The plane symmetry groups. In Topics in Recreational Mathematics. Cambridge, U.K.: Cambridge University Press.Google Scholar
  2. Coxford, A. E, and Usiskin, Z. P. (1971). Geometry: A Transformation Approach. River Forest, IL: Laidlow Brothers. (Uses transformations in presentation of the standard topics of elementary Euclidean geometry.)Google Scholar
  3. Cromwell, P. R. (1997). Polyhedra. Cambridge, U.K.: Cambridge University Press.zbMATHGoogle Scholar
  4. Crowe, D. (1986). HiMAP Module 4: Symmetry, Rigid Motions, and Patterns. Arlington, MA: COMAEGoogle Scholar
  5. Davis, D. M. (1993). The Nature and Power of Mathematics. Princeton, NJ: Princeton University Press.zbMATHGoogle Scholar
  6. Devlin, K. (1994). Mathematics: The Science of Patterns. New York: Scientific American Library.zbMATHGoogle Scholar
  7. Dodge, C. W. (1972). Euclidean Geometry and Transformations. Reading, MA: Addison-Wesley. (Chapters 2 and 3 contain an elementary presentation of isometries and similarities and include applications.)Google Scholar
  8. Eccles, F. M. (1971). An Introduction to Transformational Geometry. Menlo Park, CA. Addison-Wesley. (Intended to introduce secondary-school students to transformations following a traditional geometry course.)Google Scholar
  9. Farmer, D. W. (1996). Groups and Symmetry, Vol. 5, Mathematical World. AMS. (A beginning undergraduate guide to discovery of groups and symmetry.)Google Scholar
  10. Faulkner, J. E. (1975). Paper folding as a technique in visualizing a certain class of transformations. Mathematics Teacher 68: 376–377.Google Scholar
  11. Gans, D. (1969). Transformations and Geometries. New York: Appleton-Century-Crofts. (A detailed presentation of the transformations introduced in this chapter followed by a presentation of the more general projective and topological transformations.)zbMATHGoogle Scholar
  12. Gardner, M. (1975). On tessellating the plane with convex polygon tiles. Scientific American 233(1):112–117.CrossRefGoogle Scholar
  13. Gardner, M. (1978). The art of M. C. Escher. In Mathematical Carnival, pp. 89–102. New York: Alfred A. Knopf.Google Scholar
  14. Gardner, M. (1989). Penrose Tiles to Trapdoor Ciphers. New York: W. H. Freeman & Co.Google Scholar
  15. Grünbaum, B., and Shepard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. (The authoritative source on the subject of tilings and polyhedra.)zbMATHGoogle Scholar
  16. Haak, S. (1976). Transformation geometry and the artwork of M. C. Escher. Mathematics Teacher 69:647–652.Google Scholar
  17. Iaglom, I. M. (1962). Geometric Transformations, Vols. 1, 2, 3. New York: Random House. (Numerous problems of elementary Euclidean geometry are solved through transformations.)Google Scholar
  18. Jeger, M. (1969). Transformation Geometry. London: Allen and Un-win. (Numerous diagrams are included in this easy-to-understand presentation of isometries, similarities, and affinities.Google Scholar
  19. Johnson, D. A. (1973). Paper Folding for the Mathematics Class. Reston, VA: NCTM.Google Scholar
  20. Johnston, B. L., and Richman, F. (1997). Numbers and Symmetry: An Introduction to Algebra. New York: CRC Press. (Nice introductory chapter on symmetries and another on wallpaper patterns.)zbMATHGoogle Scholar
  21. Jones, O. (1986). The Grammar of Ornament. Ware England: Omega Books. (Wonderful collection of ornamental patterns and designs from different civilizations.)Google Scholar
  22. King, J., and Schattschneider, D. (1997). Geometry Turned On! Dynamic Software in Learning, Teaching and Research, MAA Notes 41. MAA. (A collection of articles by people at the forefront of dynamic geometry.)Google Scholar
  23. Lockwood, E. H., and Macmillan, R. H. (1978). Geometric Symmetry. Cambridge: Cambridge University Press. (Great source of information about frieze, wallpaper, and space patterns.)zbMATHGoogle Scholar
  24. MacGillavry, C. H. (1976). Symmetry Aspects of M.C. Escher’s Periodic Drawings, 2d ed. Utrecht: Bohn, Scheltema & Holkema.Google Scholar
  25. Martin, G. E. (1982b). Transformation Geometry: An Introduction to Symmetry. New York: Springer-Verlag. (Introduces isometries and applies them to ornamental groups and tessellations.)zbMATHCrossRefGoogle Scholar
  26. Maxwell, E. A. (1975). Geometry by Transformations. Cambridge: Cambridge University Press. (A secondary-school-level introduction of isometries and similarities including their matrix representations.)zbMATHGoogle Scholar
  27. O’Daffer, P. G., and Clemens, S. R. (1976). Geometry: An Investigative Approach. Menlo Park, CA: Addison-Wesley.Google Scholar
  28. Olson, A. T. (1975). Mathematics Through Paper Folding. Reston, VA: NCTM.Google Scholar
  29. Radin, C. (1995). Symmetry and Tilings. Notices of the AMS, 42(1). pp. 26–31.MathSciNetzbMATHGoogle Scholar
  30. Ranucci, E. R. (1974). Master of tessellations: M. C. Escher, 1898–1972. Mathematics Teacher 67:299–306.Google Scholar
  31. Ranucci, E. R., and Teeters, J. E. (1977). Creating Escher-Type Drawings. Palo Alto, CA: Creative Publications (Straightforward, easy to follow directions.)Google Scholar
  32. Robertson, J. (1986). Geometric constructions using hinged mirrors. Mathematics Teacher 79: 380–386.Google Scholar
  33. Rosen, J. (1975). Symmetry Discovered: Concepts and Application in Nature and Science. Cambridge: Cambridge University Press.Google Scholar
  34. Schattschneider, D. (1978). The plane symmetry groups: Their recognition and notation. The American Mathematical Monthly, 85:439–450.MathSciNetzbMATHCrossRefGoogle Scholar
  35. Schattschneider, D. (1990). M. C. Escher: Visions of Symmetry. New York: W H. Freeman and Company. (Contains all of Escher’s notebook patterns with extensive commentary by the Escher expert.)Google Scholar
  36. Senechal, M., and Fleck, G. (eds.) (1988). Shaping Space: A Polyhedral Approach. Cambridge MA: Birkhäuser Boston.zbMATHGoogle Scholar
  37. Singer, D. (1997). Geometry: Plane and Fancy. New York: Springer-Verlag (Contains information about tessellations in non-Euclidean geometry.)Google Scholar
  38. Steen, L.A. (Ed.) (1990). On the Shoulders of Giants: New Approaches to Numeracy. Washington DC: National Academy Press.Google Scholar
  39. Stewart, I., and Golubitsky, M. (1993). Fearful Symmetry: Is God A Geometer? London: Penguin Books. (Analyzes the role of “symmetry breaking” in a wide range of natural patterns.)Google Scholar
  40. Teeters, J. C. (1974). How to draw tessellations of the Escher type. Mathematics Teacher 67: 307–310.Google Scholar
  41. Washburn, D., and Crowe, D. (1988). Symmetries of Culture; Theory and Practice of Plane Patterns Analysis. Seattle: University of Washington Press. (Careful and nontechnical presentation of pattern analysis with examples from numerous cultures.)Google Scholar
  42. Watson, A. (1990). The mathematics of symmetry. New Scientist 17, October 1990: 45–50. (Survey article describing the group concept, its history and its application in mathematics, chemistry, and physics.)Google Scholar
  43. Weyl, H. (1989). Symmetry. Princeton: Princeton University Press. (Original copyright in 1952. This classic explores symmetry as a geometrical concept and as an underlying principle in art and nature.)Google Scholar

Suggestions for Viewing

  1. Adventures in Perception (1973, 22 min). An especially effective presentation of the work of M. C. Escher. Produced by Hans Van Gelder, Film Producktie, N. V., The Netherlands. Available from Phoenix/B.F.A. Films, 468 Park Ave. S., New York, NY 10016 (800) 221–1274.Google Scholar
  2. Dihedral Kaleidoscopes (1971 ; 13 min). Uses pair of intersecting mirrors (dihedral kaleidoscopes) to demonstrate several regular figures and their stellations and tilings of the plane. Produced by the College Geometry Project at the University of Minnesota. Available from International Film Bureau, 332 South Michigan Ave., Chicago, IL 60604.Google Scholar
  3. Isometries (1971; 26 min). Demonstrates that every plane isometry is a translation, rotation, reflection, or glide reflection and that each is the product of at most three reflections. Produced by the College Geometry Project at the University of Minnesota. Available from International Film Bureau, 332 South Michigan Ave., Chicago, IL 60604.Google Scholar
  4. Similarity (1990; 25 min). A Project Mathematics video, produced by and available from California Institute of Technology, Caltech 1–70, Pasadena, CA 91125.Google Scholar
  5. Symmetries of the Cube (1971; 13.5 min). Uses mirrors to exhibit the symmetries of a square as a prelude to the analogous generation of the cubeGoogle Scholar
  6. by reflections. Produced by the College Geometry Project at the University of Minnesota, Available from International Film Bureau, 332 South Michigan Ave., Chicago, IL 60604.Google Scholar
  7. Three-Dimensional Symmetry (1995; 17 min). Shows how transformations create symmetries in two and three dimensions. Computer animation is used to show the relationships found in symmetrical objects. Available from Key Curriculum Press, Berkeley, CA.Google Scholar
  8. The Fantastic World of M. C. Escher (1994; 50 min). Explores the man, his inspirations, and the mathematical principles found in so much of his art through first-person accounts by Escher’s friends and mathematicians, computer animated recreations of his work, and a look at his sources of inspiration. Published by Film 7 International, Rome, Italy. Available from Atlas Video.Google Scholar

Suggested Software

  1. The Geometry Center ( is a great source of downloadable geometry software. In particular, you may want the following:Google Scholar
  2. Geomview—A 3D object viewer.Google Scholar
  3. Kali—A 2D symmetry pattern editor.Google Scholar
  4. Kaleido Tile—Creates tilings of the sphere, plane, or hyperbolic space.Google Scholar
  5. KaleidoMania!—A tool for dynamically creating symmetric designs and exploring the mathematics of symmetry. Available from Key Curriculum Press ( Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Judith N. Cederberg
    • 1
  1. 1.Department of MathematicsNorthfieldUSA

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