﻿ Symmetry

# Symmetry

The geometric concept of symmetry ( ancient Greek συμμετρία symmetria " symmetry, regularity ," from σύν syn "together" and μέτρον metron " measure " ) refers to the property that a geometric object can be represented by movements on itself, so appears unchanged. A transformation that maps an object to itself, ie symmetry picture or symmetry operation.

Sometimes two (or more) different geometrical objects are called symmetrical to each other when viewed together, form a symmetrical figure.

Depending on the number of dimensions considered are the following different symmetries:

• 2.3.1 Examples
• 2.3.2 point symmetry of function graphs
• 3.1 The Nature
• 3.2 equivalents to the two-dimensional symmetry elements
• 3.3 rotational symmetry
• 3.4 spherical symmetry

## Symmetries in the one-dimensional

One-dimensional, in other words on a line, there is symmetry with respect to a single point, as well as symmetry with respect to the translation (displacement).

## Symmetries in two dimensions

In two dimensions must be made between point and axis of symmetry. Besides occur translational symmetries here.

### Rotational symmetry

Two-dimensional objects are rotationally symmetrical when rotated to any angle maps to a point on the object itself. This symmetry is also referred to as rotary, circle or point - symmetry.

### Axis of symmetry

The axis of symmetry, axial symmetry and mirror symmetry is a form of symmetry, which occurs in objects that are reflected along an axis of symmetry. For each axis, mirroring applies:

#### Examples

• Triangles may have one or three axes of symmetry: The isosceles triangle is axisymmetrical with respect to the mid-perpendicular to the base. Equilateral triangles have three axes of symmetry.
• Four corner may have one, two or even four axes of symmetry: At least one axis of symmetry are isosceles trapezoids ( through the midpoints of the parallel sides ) and dragons squares ( along a diagonal).
• At least two symmetry axes are in the rectangle ( the perpendicular bisectors of opposite sides ) and at diamond (both diagonals ).
• Finally, the square is rectangular and diamond at the same time and thus has four symmetry axes.

#### Axis of symmetry of function graphs

A popular especially in the school mathematics task is to detect the axis of symmetry for the graph of a function. This proof is particularly simple in the case of symmetry with respect to the y -axis of the ( Cartesian ) coordinate system. A function is axisymmetrical with respect to the y-axis, if the following applies:

It is valid for all x, there is the axis of symmetry, that is, f is an even function.

This condition boils down to is that the function values ​​for the equal and opposite arguments and must match.

More generally, the graph of a function f is axisymmetric with respect to the line with the equation when the following condition for any value of x is correct:

By substituting with yields the equivalent condition:

### Point symmetry

The point symmetry, and the central symmetry, is a property of geometric objects. A geometric object (eg a square ) is called ( in itself ) point symmetry, if there is a point reflection that maps this object to itself. The point at which this reflection is performed is referred to as center of symmetry.

#### Examples

• When a square is point symmetry ( in itself ) if and only present if it is a parallelogram. The center of symmetry, in this case the point of intersection of its diagonals. Particular cases of the parallelogram and rectangle, diamond and square are point symmetric.
• Each circle is ( in itself ) point symmetry with respect to its center.
• Two circles with the same radius are point-symmetrical to each other. The center of symmetry is the center of the connecting line between the two circle centers. The point symmetry mutually symmetrical lines are always the same length.

#### Point symmetry of function graphs

A common, especially in the school mathematics task is to prove that the graph of a given function is point symmetric. This detection can be performed with the following formula:

This equation is satisfied for all x, is point symmetrical to the point (a, b). In the special case of point symmetry about the origin ( 0,0) this equation simplifies to:

It is valid for all x, then there is point symmetry with respect to the front of the origin.

### Translational symmetry

Figures, which ( is the not the identity ) are converted into itself by a shift or translation, have a translational symmetry. They are also referred to as periodic.

• Figures, which are translational symmetry must necessarily be unlimited. In applications of mathematics, this is virtually non-existent, therefore, is referred to there also limited subsets of periodic amounts ( crystal lattice and the like. ) To be periodic.
• The graphs of periodic real functions such as the sine function have a translational symmetry in one direction.

## Symmetries in three dimensions

### In nature

The structure of the most higher organisms is more or less approximately mirror symmetrical ( in lower life forms there is often the axis of symmetry, so they form an approximate rotation of the body). Man also has a vertical plane of symmetry, the anatomical sagittal plane. However, this symmetry is not complete, the structure of the internal organs is not mirror-symmetrical. Even the seemingly mutually symmetrical body parts such as eyes, ears, arms, legs, breasts, etc. have each other more or less great location, shape and size differences.

### Correspondences with two-dimensional symmetry elements

The axis of symmetry in the two-dimensional area corresponding to the symmetry in three dimensions, the point symmetry, the axis of symmetry ( rotational symmetry of 180 °). There is also the point / central symmetry in space and how in the plane translational symmetries.

### Rotational symmetry

Three-dimensional objects are rotationally symmetrical when rotated to any angle about an axis ( the symmetry axis) maps the object onto itself.

Rotational symmetry about an axis, is also referred to as a cylindrical symmetry. Three-dimensional geometric objects with this property are also called rotary body.

### Spherical symmetry

Rotational symmetry about any axis through the same point is a special case of the rotational symmetry and will be referred to as a spherical symmetry and radial symmetry.

Stars are, for example approximately spherical symmetry, as their properties ( such as density) is not the same everywhere but depends only on the distance to the center.

Also, their gravitational fields and, for example, the electric field of a charged sphere is spherically symmetric.

## Combinations

From the possibility of combining symmetry operations, the symmetrical basic operations can be derived:

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