Transformation geometry
In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. Felix Klein, who pioneered this point of view, was himself interested in mathematical education. It took many years, though, for his "modern" point of view to have much effect, with the synthetic geometry remaining dominant. In the end, the reform of geometry teaching came simultaneously with the New Math movement.
An exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel. Thus through transformations students learn about Euclidean plane isometry. For instance, consider reflection in a vertical line and a line inclined at 45° to the horizontal. One can observe that one composition yields a counter-clockwise quarter-turn (90°) while the reverse composition yields a clockwise quarter-turn. Such results show that transformation geometry includes non-commutative processes.
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 dilation. However, the reflection in a circle transformation seems inappropriate for lower grades. Thus inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students.
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 synthetic geometry. When students then encounter analytic geometry, the ideas of coordinate rotations and reflections follow easily. All these concepts prepare for linear algebra where the reflection concept is expanded.
[edit] References
- 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, ISBN 978-0821839003 .
- Robin Hartshorne (2011) Review of Continuous Symmetry, American Mathematical Monthly 118:565–8.
- 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.
- Transformations teaching notes from Gatsby Charitable Foundation
