Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. More one can be obtained from the other by uniformly scaling with additional translation and reflection; this means that either object can be rescaled and reflected, so as to coincide with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level. For example, all circles are similar to each other, all squares are similar to each other, all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, isosceles triangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle the triangles are similar.
Corresponding sides of similar polygons are in proportion, corresponding angles of similar polygons have the same measure. This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are similar, but some school textbooks exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar. In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional, it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of, necessary and sufficient for two triangles to be similar: The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:If ∠BAC is equal in measure to ∠B′A′C′, ∠ABC is equal in measure to ∠A′B′C′ this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar. All the corresponding sides have lengths in the same ratio:AB/A′B′ = BC/B′C′ = AC/A′C′; this is equivalent to saying. Two sides have lengths in the same ratio, the angles included between these sides have the same measure. For instance:AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′; this is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; when two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′. There are several elementary results concerning similar triangles in Euclidean geometry: Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other.
Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if one other side have lengths in the same ratio. Given a triangle △ABC and a line segment DE one can, with ruler and compass, find a point F such that △ABC ∼ △DEF; the statement that the point F satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by G. D. Birkhoff the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms. Similar triangles provide the basis for many synthetic proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles provide the foundations for right triangle trigonometry.
The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles. Equality of all angles in sequence is not sufficient to guarantee similarity. A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given n, all regular n-gons are similar. Several types of curves have the property; these include: Circles Parabolas Hyperbolas of a specific eccentricity Ellipses of a specific eccentricity Catenaries Graphs of the logarithm function for different bases Graphs of the exponential function for different bases Logarithmic spirals are self-similar A similarity of a Euclidean space is a bijection f from the space onto itself that multiplies all distances
Non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries; when the metric requirement is relaxed there are affine planes associated with the planar algebras which give rise to kinematic geometries that have been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, not on ℓ, there is one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ.
Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: In Euclidean geometry, the lines remain at a constant distance from each other and are known as parallels. In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular. In elliptic geometry, the lines intersect. Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, geometries that deviated from this were not accepted as legitimate until the 19th century; the debate that led to the discovery of the non-Euclidean geometries began as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions and sought to prove all the other results in the work; the most notorious of the postulates is referred to as "Euclid's Fifth Postulate," or the "parallel postulate", which in Euclid's original formulation is: If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it appears to be more complicated than Euclid's other postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another. For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī, Giovanni Girolamo Saccheri; the theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries." These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the development of non-Euclidean geometry.
These early attempts at challenging the fifth postulate had a considerable influence on its development among European geometers, including Witelo, Levi ben Gerson, John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry, provided flawed proofs of the parallel postulate, containing assumptions that were equivalent to the parallel postulate; these early attempts did, provide some early properties of the hyperbolic and elliptic geometries. Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher": "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge." Khayyam considered the three cases right and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid which he didn't realize was equivalent to his own postulate.
Another example is al-Tusi's son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusi's thoughts, which presented another hypothesis equivalent to the parallel postulate. "He revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements." His work was published in Rome in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis. Giordano Vitale, in his book Euclide restituo, used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summ
Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal, it can be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square; the term oblong is used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD; the word rectangle comes from the Latin rectangulus, a combination of rectus and angulus. A crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals, it is a special case of an antiparallelogram, its angles are not right angles. Other geometries, such as spherical and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons. A convex quadrilateral is a rectangle if and only if it is any one of the following: a parallelogram with at least one right angle a parallelogram with diagonals of equal length a parallelogram ABCD where triangles ABD and DCA are congruent an equiangular quadrilateral a quadrilateral with four right angles a quadrilateral where the two diagonals are equal in length and bisect each other a convex quadrilateral with successive sides a, b, c, d whose area is 1 4.
A convex quadrilateral with successive sides a, b, c, d whose area is 1 2. A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a special case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a convex quadrilateral. A convex quadrilateral is Simple: The boundary does not cross itself. Star-shaped: The whole interior is visible from a single point, without crossing any edge. De Villiers defines a rectangle more as any quadrilateral with axes of symmetry through each pair of opposite sides; this definition crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, another, the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides.
These quadrilaterals crossed isosceles trapezia. A rectangle is cyclic: all corners lie on a single circle, it is equiangular: all its corner angles are equal. It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit, it has two lines of reflectional symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus; the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa. A rectangle is rectilinear: its sides meet at right angles. A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position, one for shape, one for overall size. Two rectangles, neither of which will fit inside the other, are said to be incomparable. If a rectangle has length ℓ and width w it has area A = ℓ w, it has perimeter P = 2 ℓ + 2 w = 2, each diagonal has length d = ℓ 2 + w 2, when ℓ = w, the rectangle is a square; the isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area.
The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle. A parallelogram with equal diagonals is a rectangle; the Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The British flag theorem states that with vertices denoted A, B, C, D, for any point P on the same plane of a rectangle: 2 + 2 = 2 + 2
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, C is denoted △ A B C. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. In other words, there is only one plane that contains that triangle, every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; this article is about triangles in Euclidean geometry, in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: An equilateral triangle has all sides the same length. An equilateral triangle is a regular polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of the same measure, namely the angles opposite to the two sides of the same length; some mathematicians define an isosceles triangle to have two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.
The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure. Hatch marks called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of short line segments in the form of tally marks. In a triangle, the pattern is no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, a scalene triangle has different patterns on all sides since no sides are equal. Patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, a scalene triangle has different patterns on all angles since no angles are equal.
Triangles can be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°; the side opposite to the right angle is the longest side of the triangle. The other two sides are called the catheti of the triangle. Right triangles obey the Pythagorean theorem: the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. Special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 + 42 = 52. In this situation, 3, 4, 5 are a Pythagorean triple; the other one is an isosceles triangle. Triangles that do not have an angle measuring 90° are called oblique triangles. A triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle.
If c is the length of the longest side a2 + b2 > c2, where a and b are the lengths of the other sides. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side a2 + b2 < c2, where a and b are the lengths of the other sides. A triangle with an interior angle of 180° is degenerate. A right degenerate triangle has collinear vertices. A triangle that has two angles with the same measure has two sides with the same length, therefore it is an isosceles triangle, it follows that in a triangle where all angles have the same measure, all three sides have the same length, such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures. In rigorous treatments, a triangle is therefore called a 2-simplex. Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC; the sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.
This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle given the measure of two angles. An exterior angle of a triangle is an angle, a linear pair to an interior angle; the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it. The sum of the measures of the three exterior angles of any triangle is 360 degrees. Two triangles are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle; the corresponding sides of similar triangles have lengths that are in the same proportion, this property is sufficient to establish similarity. Some basic theorems about similar triangles are: If and only if one pair of internal angles of two triangles have the sam
Polygon
In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".
It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.
The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.
This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1
Euclidean geometry
Euclidean geometry is a mathematical system attributed to 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, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.
Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.
Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv
Symmetry
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together. Mathematical symmetry may be observed with respect to the passage of time; this article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people. The opposite of symmetry is asymmetry. A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion; this means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry if there is a line going through it which divides it into two pieces which are mirror images of each other.
An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has translational symmetry. An object has helical symmetry if it can be translated and rotated in three-dimensional space along a line known as a screw axis. An object contracted. Fractals exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection rotoreflection symmetry. A dyadic relation R is only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary Mary is the same age as Paul. Symmetric binary logical connectives are and, or, nand and nor. Generalizing from geometrical symmetry in the previous section, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object; the set of operations that preserve a given property of the object form a group.
In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include and odd functions in calculus. In statistics, it appears as symmetric probability distributions, as skewness, asymmetry of distributions. Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations; this concept has become one of the most powerful tools of theoretical physics, as it has become evident that all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his read 1972 article More is Different that "it is only overstating the case to say that physics is the study of symmetry." See Noether's theorem. Important symmetries in physics include discrete symmetries of spacetime. In biology, the notion of symmetry is used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves.
Animals that move in one direction have upper and lower sides and tail ends, therefore a left and a right. The head becomes specialized with a mouth and sense organs, the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs remain asymmetric. Plants and sessile animals such as sea anemones have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, sea lilies. In biology, the notion of symmetry is used as in physics, to say to describe the properties of the objects studied, including their interactions. A remarkable property of biological evolution is the changes of symmetry corresponding to the appearance of new parts and dynamics. Symmetry is important to chemistry because it undergirds all specific interactions between molecules in nature; the control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer the