A Course in Modern Geometries pp 74-126 | 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 orginally 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

Linear Transformation Matrix Representation Prove Theorem Euclidean Geometry Euclidean Plane## Preview

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## Suggestions for Further Reading

- Coxford, A.F., and Usiskin, Z.P. (1971).
*Geometry: A Transformation Approach*. River Forest, IL: Laidlow Brothers. Uses transformations in its presentation of the standard topics of elementary Euclidean geometry.Google 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 high-school students to the transformations following a traditional geometry course.Google Scholar - Gans, D. (1969).
*Transformations and Geometries*. New York: Appleton-CenturyCrofts. A detailed presentation of the transformations introduced in this chapter followed by a presentation of the more general projective and topological transformations.MATHGoogle 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 Unwin. Numerous diagrams are included in this easy-to-understand presentation of isometries, similarities, and affinities.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 high-school-level introduction of isometries and similarities including their matrix representations.MATHGoogle Scholar

## Readings on Tiling the Plane and Paper Folding

- Faulkner, J.E. (1975). Paper folding as a technique in visualizing a certain class of transformations.
*Mathematics Teacher*68: 376–377.Google 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: M. Gardner,
*Mathematical Carnival*, pp. 89–102. New York: Alfred A. Knopf.Google Scholar - Grünbaum, B., and Shephard, G.C. (1987).
*Tilings and Patterns*. New York: W.H. Freeman.MATHGoogle Scholar - Haak, S. (1976). Transformation geometry and the artwork of M.C. Escher. Mathematics Teacher 69: 647–652.Google Scholar
- Johnson, D.A. (1973).
*Paper Folding for the Mathematics Class*. Reston, VA: N.C.T.M. MacGillavry, C.H. (1976).*Symmetry Aspects of M.C. Escher’s Periodic Drawings*, 2d ed. Utrecht: Bohn, Scheltema & Holkema.Google 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: N.C.T.M. Ranucci, E.R. (1974). Master of tessellations: M.C. Escher, 1898–1972. Mathematics Teacher 67: 299–306.Google Scholar
- Robertson, J. (1986). Geometric constructions using hinged mirrors. Mathematics Teacher 79: 380–386.Google Scholar
- Teeters, J.C. (1974). How to draw tessellations of the Escher type. Mathematics Teacher 67: 307–310.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 B.F.A. Educational Media.Google Scholar*Dihedral Kaleidoscopes*(1971; 13 min). Uses pairs 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*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 cube 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