Transformation geometry

In

An entertaining application of reflection in a line occurs in a proof of the one-seventh area triangle found in any triangle.

Another transformation introduced to young students is the

Experiments with concrete symmetry groups make way for abstract group theory. Other concrete activities use computations with complex numbers, hypercomplex numbers, or matrices to express transformation geometry. Such transformation geometry lessons present an alternate view that contrasts with classical

Educators have shown some interest and described projects and experiences with transformation geometry for children from kindergarten to high school. In the case of very young age children, in order to avoid introducing new terminology and to make links with students' everyday experience with concrete objects, it was sometimes recommended to use words they are familiar with, like "flips" for line reflections, "slides" for translations, and "turns" for rotations, although these are not precise mathematical language. In some proposals, students start by performing with concrete objects before they perform the abstract transformations via their definitions of a mapping of each point of the figure.^[3][4][5][6]

In an attempt to restructure the courses of geometry in Russia, Kolmogorov suggested presenting it under the point of view of transformations, so the geometry courses were structured based on set theory. This led to the appearance of the term "congruent" in schools, for figures that were before called "equal": since a figure was seen as a set of points, it could only be equal to itself, and two triangles that could be overlapped by isometries were said to be congruent.^[2]

One author expressed the importance of group theory to transformation geometry as follows:

I have gone to some trouble to develop from first principles all the group theory that I need, with the intention that my book can serve as a first introduction to transformation groups, and the notions of abstract group theory if you have never seen these.^[7]

See also

• Chirality (mathematics)
• Geometric transformation
• Euler's rotation theorem
• Motion (geometry)
• Transformation matrix

References

Further reading

The Wikibook Geometry has a page on the topic of: Unified Angles

• Heinrich Guggenheimer (1967) Plane Geometry and Its Groups, Holden-Day.
• ISBN 978-0-8218-3900-3
  .
• American Mathematical Monthly
  118:565–8.
• ISBN 0-521-31694-4
  .
• P.S. Modenov and A.S. Parkhomenko (1965) Geometric Transformations, translated by Michael B.P. Slater, Academic Press.
• George E. Martin (1982) Transformation Geometry: An Introduction to Symmetry,
  Springer Verlag
  .
• Isaak Yaglom (1962) Geometric Transformations, Random House (translated from the Russian).
• Max Jeger (1966) Transformation Geometry (translated from the German).
• Transformations teaching notes from Gatsby Charitable Foundation
• Kristin A. Camenga (NCTM's 2011 Annual Meeting & Exposition) - Transforming Geometric Proof with Reflections, Rotations and Translations.^{[permanent dead link]}
• Nathalie Sinclair (2008) The History of the Geometry Curriculum in the United States, pps. 63–66.
• Zalman P. Usiskin and Arthur F. Coxford. A Transformation Approach to Tenth Grade Geometry, The Mathematics Teacher, Vol. 65, No. 1 (January 1972), pp. 21-30.
• Zalman P. Usiskin. The Effects of Teaching Euclidean Geometry via Transformations on Student Achievement and Attitudes in Tenth-Grade Geometry, Journal for Research in Mathematics Education, Vol. 3, No. 4 (Nov., 1972), pp. 249-259.
• A. N. Kolmogorov. Геометрические преобразования в школьном курсе геометрии, Математика в школе, 1965, Nº 2, pp. 24–29. (Geometric transformations in a school geometry course) (in Russian)
• Alton Thorpe Olson (1970). High School Plane Geometry Through Transformations: An Exploratory Study, Vol. I. University of Wisconsin--Madison.
• Alton Thorpe Olson (1970). High School Plane Geometry Through Transformations: An Exploratory Study, Vol II. University of Wisconsin--Madison.