In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, based on the Erlangen programme. Felix Klein

Felix Klein

Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

, 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

Synthetic geometry

Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...

 remaining dominant. In the end, the reform of geometry teaching came simultaneously with the New Math

New math

New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S...

 movement.

An exploration of transformation geometry often begins with a study of reflection symmetry

Reflection symmetry

Reflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...

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

Reflection (mathematics)

In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...

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

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

 when the lines intersect, or a translation when they are parallel. Thus through transformations students learn about Euclidean plane isometry

Euclidean plane isometry

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length...

. 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

One-seventh area triangle

In plane geometry, a triangle ABC contains a triangle of one-seventh area of ABC formed as follows: the sides of this triangle lie on lines p, q, r whereA more general result is known as Routh's theorem.-References:...

 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

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

s make way for abstract group theory

Group theory

In mathematics and 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 as groups endowed with additional operations and...

. Other concrete activities use computations with complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, hypercomplex number

Hypercomplex number

In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...

s, or matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

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

Synthetic geometry

Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...

. When students then encounter analytic geometry

Analytic geometry

Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, the ideas of coordinate rotations and reflections

Coordinate rotations and reflections

In geometry, 2D coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1...

 follow easily. All these concepts prepare for linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

 where the reflection concept is expanded.