Invariant
Invariant
In mathematics a quantity is said to be invariant if its value does not change following a given operation or under a set of given transformations. The property of being an invariant is called invariance. For instance, multiplication of any real number by the identity element (1) leaves it unchanged. Thus, all real numbers are invariant under the operation of multiplication by the identity element (1). In some cases, mathematical operations leave certain properties unchanged. When this occurs, those properties that are unchanged are referred to as invariants under the operation. Translation of coordinate axes (shifting of the origin from the point (0,0) to any other point in the plane) and rotation of coordinate axes are also operations. Vectors, which are quantities possessing both magnitude (size) and direction, are unchanged in magnitude and direction under a translation of axes, but only unchanged in magnitude under rotation of the axes. Thus, magnitude is an invariant property of vectors under the operation of rotation, while both magnitude and direction are invariant properties of a vector under a translation of axes.
An important objective in any branch of mathematics is to identify the invariants of a given operation, as they often lead to a deeper understanding of the mathematics involved, or to simplified analytical procedures.
Geometric invariance
In geometry, the invariant properties of points, lines, angles, and various planar and solid objects are all understood in terms of the invariant properties of these objects under such operations as translation, rotation, reflection, and magnification. For example, the area of a triangle is invariant under translation, rotation and reflection, but not under magnification. On the other hand, the interior angles of a triangle are invariant under magnification, and so are the proportionalities of the lengths of its sides.
The Pythagorean theorem states that the square of the hypotenuse of any right triangle is equal to the sum of the squares of its legs. In other words, the relationship expressing the length of the hypotenuse in terms of the lengths of the other two sides is an invariant property of right triangles, under magnification, or any other operation that results in another right triangle.
Very recently, geometric figures called fractals have gained popularity in the scientific community. Fractals are geometric figures that are invariant under magnification. That is, their fragmented shape appears the same at all magnifications. Increased interest in fractal comes from the idea that most natural objects look more like fractals than regular geometric figures. For example, clouds, trees, and mountains look more like fractal figures than they do circles (or ellipses), (upside down) triangles, and pyramids, respectively.
Algebraic invariance
Algebraic invariance refers to combinations of coefficients from certain functions that remain constant when the coordinate system in which they are expressed is translated, or rotated. An example of this kind of invariance is seen in the behavior of the conic sections (cross-sections of a right circular cone resulting from its intersection with a plane). The general equation of a conic section is ax2 + bxy + cy2 + dx + ey + f = 0. Each of the equations of a circle, or an ellipse, a parabola, or hyperbola represents a special case of this equation. One combination of coefficients, (b2 - 4ac), from this equation is called the discriminant. For a parabola, the value of the discriminant is zero, for an ellipse it is less than zero, and for a hyperbola is greater than zero. However, regardless of its value, when the axes of the coordinate system in which the figure is being graphed are rotated through an arbitrary angle, the value of the discriminant (b2 - 4ac) is unchanged. Thus, the discriminant is said to be invariant under a rotation of axes. In other words, knowing the value of the discriminant reveals the identity of a particular conic section regardless of its orientation in the coordinate system. Still another invariant of the general equation of the conic sections, under a
KEY TERMS
Conic section —A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola.
Magnification —Magnification is the operation that multiplies the dimensions of an object by a constant leaving the coordinate system unchanged, or vice versa.
rotation of axes, is the sum of the coefficients of the squared terms (a + c).
Resources
BOOKS
Jeffrey, Alan. Mathematics for Engineers and Scientists. Boca Raton, FL: Chapman & Hall/CRC, 2005.
Larson, Ron. Calculus With Analytic Geometry. Boston, MA: Houghton Mifflin College, 2002.
Noronha, Maria Helena. Euclidean and Non-Euclidean Geometries. Upper Saddle River, NJ: Prentice Hall, 2002.
Silvester, John R. Geometry: Ancient and Modern. Oxford, UK, and New York: Oxford University Press, 2001.
J. R. Maddocks
Invariant
Invariant
In mathematics a quantity is said to be invariant if its value does not change following a given operation. For instance, multiplication of any real number by the identity element (1) leaves it unchanged. Thus, all real numbers are invariant under the operation of "multiplication by the identity element (1)." In some cases, mathematical operations leave certain properties unchanged. When this occurs, those properties that are unchanged are referred to as invariants under the operation. Translation of coordinate axes (shifting of the origin from the point (0,0) to any other point in the plane ) and rotation of coordinate axes are also operations. Vectors, which are quantities possessing both magnitude (size) and direction, are unchanged in magnitude and direction under a translation of axes, but only unchanged in magnitude under rotation of the axes. Thus, magnitude is an invariant property of vectors under the operation of rotation, while both magnitude and direction are invariant properties of a vector under a translation of axes.
An important objective in any branch of mathematics is to identify the invariants of a given operation, as they often lead to a deeper understanding of the mathematics involved, or to simplified analytical procedures.
Geometric invariance
In geometry , the invariant properties of points, lines, angles, and various planar and solid objects are all understood in terms of the invariant properties of these objects under such operations as translation, rotation, reflection, and magnification. For example, the area of a triangle is invariant under translation, rotation and reflection, but not under magnification. On the other hand, the interior angles of a triangle are invariant under magnification, and so are the proportionalities of the lengths of its sides.
The Pythagorean theorem states that the square of the hypotenuse of any right triangle is equal to the sum of the squares of its legs. In other words, the relationship expressing the length of the hypotenuse in terms of the lengths of the other two sides is an invariant property of right triangles, under magnification, or any other operation that results in another right triangle.
Very recently, geometric figures called fractals have gained popularity in the scientific community. Fractals are geometric figures that are invariant under magnification. That is, their fragmented shape appears the same at all magnifications.
Algebraic invariance
Algebraic invariance refers to combinations of coefficients from certain functions that remain constant when the coordinate system in which they are expressed is translated, or rotated. An example of this kind of invariance is seen in the behavior of the conic sections (cross sections of a right circular cone resulting from its intersection with a plane). The general equation of a conic section is ax2 + bxy + cy2 + dx + ey + f = 0. Each of the equations of a circle , or an ellipse , a parabola , or hyperbola represents a special case of this equation. One combination of coefficients, (b2-4ac), from this equation is called the discriminant. For a parabola, the value of the discriminant is zero , for an ellipse it is less than zero, and for a hyperbola is greater than zero. However, regardless of its value, when the axes of the coordinate system in which the figure is being graphed are rotated through an arbitrary angle , the value of the discriminant (b2-4ac) is unchanged. Thus, the discriminant is said to be invariant under a rotation of axes. In other words, knowing the value of the discriminant reveals the identity of a particular conic section regardless of its orientation in the coordinate system. Still another invariant of the general equation of the conic sections, under a rotation of axes, is the sum of the coefficients of the squared terms (a+c).
Resources
books
Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.
J.R. Maddocks
KEY TERMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .- Conic section
—A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola.
- Magnification
—Magnification is the operation that multiplies the dimensions of an object by a constant leaving the coordinate system unchanged, or vice versa.
Invariant
INVARIANT
In the first chapter of his book Transformations: Change from Learning to Growth (1965), Wilfred Bion defined the idea of the invariant and elucidated the link between transformations and the invariant. He used the metaphor of a painter, a painting, and a field of poppies to explain that the patient and the analyst have to "repaint" new realizations, starting from the primal scene, onto the canvas of the transference. In this process, a constant must be recognized for meaning to emerge. What remains unchanged between the real field of poppies and the field of poppies on the painter's canvas, that is, between the primal scene and a transferential realization, constitutes the invariant.
To develop further his ideas on the invariant, Bion turned to the notion of mathematical invariance. By way of illustration, Bion explained that the mathematical use of symbols such as ellipses, circles, dots, and lines, no matter how coherent such a use may be, must not cause us to forget that this does not involve "things in themselves," and that a change in vertex can change the data. Thus, the eye sees two parallel lines as converging at a point that, for the mathematician, does not exist. While in algebraic projective geometry, invariance makes it possible for there to be an object common to a circular object and the elliptical representation of it, this does not apply in the case of lengths, angles, and congruence, even though these data are also a part of Euclidean geometry. Why should not the same be true of psychoanalysis? What are the invariants specific to it? How are they related to one another? According to Bion, psychoanalysis involves transformations. Through interpretations, which are themselves transformations, the analyst gains access to the analysand's original experience and its realization.
In the work of interpretation, theories are like the painter's tools, and they give access to meaning only if they have invariants. These theories vary in different methods of interpretation, so that a Kleinian transformation is different from a classically Freudian one. The meaning transmitted by the theory is also different, even if the material transformed is the same in the two cases. To illustrate his thinking, Bion offered a clinical example: In a first stage, the analysis seemed to reveal a patient suffering from hypochondriacal pain, but then in the second stage, external events (a family crisis, hospitalization) disrupted the mechanism and put the treatment at risk. What had changed? What had remained unchanged, invariant? In Bion's view, the analyst must think in terms of the fate of the external and internal objects. What appeared during the second stage as external emotional objects (worried relatives, the hospital) corresponded to what remained contained, in the form of internal objects, in pains in the knee, leg, or abdomen.
What the analyst considered internal transformed into something external, but the emotional element, though disguised, had remained unchanged. Thus, the invariant provides a link that goes beyond an apparent disconnect. It ensures the continuity necessary for psychic growth and gives the interpretation its effectiveness.
Jean-Claude Guillaume
See also: Transformations.
Bibliography
Bion, Wilfred R. (1965). Transformations: Change from learning to growth. London: Heinemann.
invariant
Note that invariants cannot accurately be described as TRUE AT ALL TIMES since individual operations may destroy and subsequently restore the invariant condition. However the invariant is always TRUE between such operations, and therefore provides a static characterization by which the element can be analyzed and understood.