What is cup to disk asymmetry

Symmetry definition, types and exercises

A definition of symmetry, many types and examples are presented on this page. Do our exercises for free and find out how much you already know about symmetry of figures and graphs.

Definition of symmetry

In geometry, the term symmetry plays an important role when looking at one-dimensional, two-dimensional and three-dimensional objects. An object is called symmetrical if it can be mapped onto itself through movements (e.g. mirroring, rotating or shifting). Even in elementary school, mostly in the 3rd grade, symmetrical figures are considered.

Axial symmetry definition

Axial symmetry is a common form of symmetry found in math and nature. Axial symmetry is also called mirror symmetry, in which an object is congruent along one axis. There are certain properties that every axis reflection has, such as congruence, equality of distance, equality of angles.
Examples of axis symmetry
- A square has exactly four axes of symmetry. A rectangle, on the other hand, only has two axes of symmetry.
- The circle or a straight line have an infinite number of axes of symmetry
- Even three-dimensional objects such as a circle or a cylinder are axially symmetrical

Point symmetry definition

Point symmetry describes the symmetry of an object around a point. The point symmetry corresponds to a rotation of the figure by exactly 180 degrees. Thus, point symmetry is a special case of rotational symmetry.

Examples of point symmetry
- With rectangles, the rectangle, rhombus and square are point-symmetrical.
- Each circle is point-symmetrical with respect to its center.
- Two circles with the same radius are point-symmetrical to one another. The center of symmetry is the midpoint of the line connecting the two circle centers.
- There can only be several centers of symmetry if the figure is not restricted. The simplest example is the straight line. It even has an infinite number of centers of symmetry.
- A triangle is never point-symmetrical. However, two triangles can be point-symmetrical to one another.

Asymmetry and special forms of symmetry

One speaks of asymmetry if a figure has no axis of symmetry and also no point of symmetry, i.e. is not symmetrical. One speaks of translation symmetry if figures can be transformed into themselves by shifting them (periodically). These figures must be unlimited, but this is not the case in practical mathematics. In biology, one usually speaks of bilateral symmetry for the human body because the left and right halves of the body are axially symmetrical. In contrast, the radial symmetry describes an axial symmetry that starts from the center (example sea urchin).

Compute symmetry of functions

In the case of functions (curve discussion), axis symmetry and point symmetry can be demonstrated using function formulas.
1. A function is called axially symmetric to the y-axis if the following formula applies to all x-values: f (x) = f (-x)
2. A function is said to be point-symmetric to the origin if the following formula applies to all x-values: f (-x) = -f (x)

Example: Make a table and use 3 different values ​​for the formula x² + 2x -. You will find that the theorem f (x) = f (-x) applies to all values ​​here, so the function is axially symmetric. You can see this very well when you draw the function graph.