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

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## 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).
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*The Nature and Power of Mathematics*. Princeton, NJ: Princeton University Press.zbMATHGoogle Scholar - Devlin, K. (1994).
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*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.)zbMATHGoogle Scholar - Gardner, M. (1975). On tessellating the plane with convex polygon tiles.
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*Mathematical Carnival*, pp. 89–102. New York: Alfred A. Knopf.Google Scholar - Grünbaum, B., and Shepard, G. C. (1987).
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## 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