Thus inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students.Įxperiments with concrete symmetry groups make way for abstract group theory. However, the reflection in a circle transformation seems inappropriate for lower grades. Such results show that transformation geometry includes non-commutative processes.Īn entertaining application of reflection in a line occurs in a proof of the one-seventh area triangle found in any triangle.Īnother transformation introduced to young students is the dilation. One can observe that one composition yields a counter-clockwise quarter-turn (90°) while the reverse composition yields a clockwise quarter-turn. For instance, consider reflection in a vertical line and a line inclined at 45° to the horizontal. Thus through transformations students learn about Euclidean plane isometry. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel. The first real transformation is reflection in a line or reflection against an axis. These efforts culminated in the 1960s with the general reform of mathematics teaching known as the New Math movement.Īn exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. Andrei Kolmogorov included this approach (together with set theory) as part of a proposal for geometry teaching reform in Russia. In the 20th century efforts were made to exploit it for mathematical education. For nearly a century this approach remained confined to mathematics research circles. 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. This contrasts with the classical proofs by the criteria for congruence of triangles. It is opposed to the classical synthetic geometry approach of Euclidean geometry, that focuses on proving theorems.įor 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. 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. 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. Branch of mathematics concerned with movement of shapes and sets 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.
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