Editing Transformation geometry (section)

{{Short description|Branch of mathematics concerned with movement of shapes and sets}}
 [[File:Simx2=transl OK.svg|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]].]]
[[File:Simx2=rotOK.svg|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 [[group (mathematics)|groups]] of [[geometric transformation]]s, and properties that are [[invariant (mathematics)|invariant]] under them.  It is opposed to the classical [[synthetic geometry]] approach of [[Euclidean geometry]], that focuses on proving [[theorem]]s.

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 (mathematics)|reflection]] about a certain line. This contrasts with the classical proofs by the criteria for [[triangle congruence|congruence of triangles]].<ref>[http://unesdoc.unesco.org/Ulis/cgi-bin/ulis.pl?catno=68221&set=4F331370_2_173&database=g Georges Glaeser – The crisis of geometry teaching]</ref>

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]].<ref name="russianedu"/>  These efforts culminated in the 1960s with the general reform of mathematics teaching known as the [[New Math]] movement.