right|thumb|A reflection against an axis followed by a reflection against a second axis parallel to the first one results in a total motion that is a [[Translation (geometry)|translation.]]

right|thumb|A reflection against an axis followed by a reflection against a second axis not parallel to the first one results in a total motion that is a [[rotation around the point of intersection of the axes.]]

In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geometry, which focuses on proving theorems.

For example, within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line. This contrasts with the classical proofs by the criteria for congruence of triangles.

The first systematic effort to use transformations as the foundation of geometry was made by Felix Klein in the 19th century, under the name Erlangen programme. For nearly a century, this approach remained confined to mathematics research circles. In the 20th century, efforts were made to exploit it for mathematical education. Andrei Kolmogorov included this approach (together with set theory) as part of a proposal for geometry teaching reform in Russia.

In an attempt to restructure the courses of geometry in Russia, Kolmogorov suggested presenting it from 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.

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.

See also

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

References

Further reading

  • Heinrich Guggenheimer (1967) Plane Geometry and Its Groups, Holden-Day.
  • Roger Evans Howe & William Barker (2007) Continuous Symmetry: From Euclid to Klein, American Mathematical Society, .
  • Robin Hartshorne (2011) Review of Continuous Symmetry, American Mathematical Monthly 118:565–8.
  • Roger Lyndon (1985) Groups and Geometry, #101 London Mathematical Society Lecture Note Series, Cambridge University Press .
  • 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
  • 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)