mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

by focusing on

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...

s, 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 the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass ...

approach of

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

, that focuses on proving

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

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 Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

about a certain line. This contrasts with the classical proofs by the criteria for

congruence of triangles Congruence of triangles may refer to: * Congruence (geometry)#Congruence of triangles * Solution of triangles Solution of triangles ( la, solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles ...

. The first systematic effort to use transformations as the foundation of geometry was made by

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

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 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 mathematics, is mostly concern ...

) as part of a proposal for geometry teaching reform in

Russia Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-ei ...

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

Pedagogy

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. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D the ...

as found in daily life. The first real transformation is ''

in a line'' or ''reflection against an axis''. The composition of two reflections results in a

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

when the lines intersect, or a

translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

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 Dilation (or dilatation) may refer to: Physiology or medicine * Cervical dilation, the widening of the cervix in childbirth, miscarriage etc. * Coronary dilation, or coronary reflex * Dilation and curettage, the opening of the cervix and surgi ...

. However, the reflection in a circle transformation seems inappropriate for lower grades. Thus

inversive geometry Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotion Emotions are mental states brought on by neurophysiological changes, variou ...

, 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 groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...

. 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 fo ...

s, hypercomplex numbers, or

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

to express transformation geometry. Such transformation geometry lessons present an alternate view that contrasts with classical

. When students then encounter

analytic geometry In classical 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 enginee ...

, 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 matrice ...

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 Miles Anthony Reid FRS (born 30 January 1948) is a mathematician who works in algebraic geometry. Education Reid studied the Cambridge Mathematical Tripos at Trinity College, Cambridge and obtained his Ph.D. in 1973 under the supervision of Pe ...

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

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...

, ,

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

Kristin A. Camenga (NCTM's 2011 Annual Meeting & Exposition) - Transforming Geometric Proof with Reflections, Rotations and Translations.
* 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

Pedagogy

See also

References

Further reading