A Course in Modern Geometries pp 99-211 | Cite as

# Geometric Transformations of the Euclidean Plane

## Abstract

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*.

## Keywords

Equilateral Triangle Prove Theorem Affine Transformation Euclidean Plane Invariant Point## Preview

Unable to display preview. Download preview PDF.

## Suggestions for Further Reading

- Caldwell, J. H. (1966). Chapter 11: The plane symmetry groups. In
*Topics in Recreational Mathematics*. Cambridge, U.K.: Cambridge University Press.Google Scholar - 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 - Cromwell, P. R. (1997).
*Polyhedra*. Cambridge, U.K.: Cambridge University Press.MATHGoogle Scholar - Crowe, D. (1986).
*HiMAP Module 4: Symmetry, Rigid Motions, and Patterns*. Arlington, MA: COMAEGoogle Scholar - Davis, D. M. (1993).
*The Nature and Power of Mathematics*. Princeton, NJ: Princeton University Press.MATHGoogle Scholar - Devlin, K. (1994).
*Mathematics: The Science of Patterns*. New York: Scientific American Library.MATHGoogle Scholar - 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 - 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 - Farmer, D. W. (1996).
*Groups and Symmetry*, Vol. 5, Mathematical World. AMS. (A beginning undergraduate guide to discovery of groups and symmetry.)Google Scholar - Faulkner, J. E. (1975). Paper folding as a technique in visualizing a certain class of transformations.
*Mathematics Teacher*68: 376–377.Google Scholar - 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.)MATHGoogle Scholar - Gardner, M. (1975). On tessellating the plane with convex polygon tiles.
*Scientific American*233(1):112–117.CrossRefGoogle Scholar - Gardner, M. (1978). The art of M. C. Escher. In
*Mathematical Carnival*, pp. 89–102. New York: Alfred A. Knopf.Google Scholar - 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.)MATHGoogle Scholar - Haak, S. (1976). Transformation geometry and the artwork of M. C. Escher.
*Mathematics Teacher*69:647–652.Google Scholar - 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 - 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 - Johnson, D. A. (1973).
*Paper Folding for the Mathematics Class*. Reston, VA: NCTM.Google Scholar - 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.)MATHGoogle Scholar - Jones, O. (1986).
*The Grammar of Ornament*. Ware England: Omega Books. (Wonderful collection of ornamental patterns and designs from different civilizations.)Google Scholar - 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 - Lockwood, E. H., and Macmillan, R. H. (1978).
*Geometric Symmetry*. Cambridge: Cambridge University Press. (Great source of information about frieze, wallpaper, and space patterns.)MATHGoogle Scholar - MacGillavry, C. H. (1976).
*Symmetry Aspects of M.C. Escher’s Periodic Drawings*, 2d ed. Utrecht: Bohn, Scheltema*&*Holkema.Google Scholar - Martin, G. E. (1982b).
*Transformation Geometry: An Introduction to Symmetry*. New York: Springer-Verlag. (Introduces isometries and applies them to ornamental groups and tessellations.)MATHCrossRefGoogle Scholar - Maxwell, E. A. (1975).
*Geometry by Transformations*. Cambridge: Cambridge University Press. (A secondary-school-level introduction of isometries and similarities including their matrix representations.)MATHGoogle Scholar - O’Daffer, P. G., and Clemens, S. R. (1976).
*Geometry: An Investigative Approach*. Menlo Park, CA: Addison-Wesley.Google Scholar - Olson, A. T. (1975).
*Mathematics Through Paper Folding*. Reston, VA: NCTM.Google Scholar - Radin, C. (1995). Symmetry and Tilings.
*Notices of the AMS*, 42(1). pp. 26–31.MathSciNetMATHGoogle Scholar - Ranucci, E. R. (1974). Master of tessellations: M. C. Escher, 1898–1972.
*Mathematics Teacher*67:299–306.Google Scholar - Ranucci, E. R., and Teeters, J. E. (1977).
*Creating Escher-Type Drawings*. Palo Alto, CA: Creative Publications (Straightforward, easy to follow directions.)Google Scholar - Robertson, J. (1986). Geometric constructions using hinged mirrors.
*Mathematics Teacher*79: 380–386.Google Scholar - Rosen, J. (1975).
*Symmetry Discovered: Concepts and Application in Nature and Science*. Cambridge: Cambridge University Press.Google Scholar - Schattschneider, D. (1978). The plane symmetry groups: Their recognition and notation.
*The American Mathematical Monthly*, 85:439–450.MathSciNetMATHCrossRefGoogle Scholar - 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 - Senechal, M., and Fleck, G. (eds.) (1988).
*Shaping Space: A Polyhedral Approach*. Cambridge MA: Birkhäuser Boston.MATHGoogle Scholar - Singer, D. (1997).
*Geometry: Plane and Fancy*. New York: Springer-Verlag (Contains information about tessellations in non-Euclidean geometry.)Google Scholar - Steen, L.A. (Ed.) (1990).
*On the Shoulders of Giants: New Approaches to Numeracy*. Washington DC: National Academy Press.Google Scholar - 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 - Teeters, J. C. (1974). How to draw tessellations of the Escher type.
*Mathematics Teacher*67: 307–310.Google Scholar - 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 - 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 - 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

*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*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*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*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*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- 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
*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*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

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