mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical

synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates ...

approach of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, that focuses on proving

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

s. For example, within transformation geometry, the properties of an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

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 Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

) as part of a proposal for geometry teaching reform in

Russia Russia, or the Russian Federation, is a country spanning Eastern Europe and North Asia. It is the list of countries and dependencies by area, largest country in the world, and extends across Time in Russia, eleven time zones, sharing Borders ...

. These efforts culminated in the 1960s with the general reform of mathematics teaching known as the New Math movement.

Use in mathematics teaching

An exploration of transformation geometry often begins with a study of

reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...

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 Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

when the lines intersect, or a

translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

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 group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

s make way for abstract

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

. Other concrete activities use computations with

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

hypercomplex number In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. ...

s, or matrices to express transformation geometry. Such transformation geometry lessons present an alternate view that contrasts with classical

. When students then encounter

analytic geometry In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and als ...

, the ideas of coordinate rotations and reflections follow easily. All these concepts prepare for

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

where the reflection concept is expanded. 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. 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

. 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.Alexander Karp & Bruce R. Vogeli – Russian Mathematics Education: Programs and Practices, Volume 5
pgs. 100–102 One author expressed the importance of

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. Miles Reid & Balázs Szendröi (2005) ''Geometry and Topology'', pg. xvii,

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, ,

References

. * 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)'' * *{{cite book, author=Alton Thorpe Olson, title=High School Plane Geometry Through Transformations: An Exploratory Study, Vol II, url=https://books.google.com/books?id=6bvQAAAAMAAJ, year=1970, publisher=University of Wisconsin--Madison Fields of geometry Symmetry Geometry education

Use in mathematics teaching

See also

References

Further reading