Let be a Euclidean vector space, , and be a Euclidean transformation that is not the identity map.
The following terms are used to indicate that if is a rotation:
has rotational symmetry;
has point symmetry;
has symmetry about a point;
is symmetric about a point.
If , then the last two terms may be used to indicate the specific case in which is conjugate to , i.e. (http://planetmath.org/Ie) the angle of rotation is .
The following are classic examples of rotational symmetry in :
Regular polygons: A regular -gon is symmetric about its center (http://planetmath.org/Center9) with valid angles of rotation for any positive integer .
Circles: A circle is symmetric about its center (http://planetmath.org/Center8) with uncountably many valid angles of rotation.
As another example, let , where each is defined thus:
Then has point symmetry with respect to the point . The valid angles of rotation for are and . The boundary of and the point are shown in the following picture.
As a final example, the figure
is symmetric about the origin. The boundary of this figure and the point are shown in the following picture.
If and is a reflection, then has reflectional symmetry. In the special case that , the following terms are used:
has line symmetry;
has symmetry about a line;
is symmetric about a line.
The following are classic examples of line symmetry in :
Regular polygons: There are lines of symmetry of a regular -gon. Each of these pass through its center and at least one of its vertices.
Circles: A circle is symmetric about any line passing through its center.
As another example, the isosceles trapezoid defined by
is symmetric about .
In the picture above, the boundary of is drawn in black, and the line is drawn in cyan.
|Date of creation||2013-03-22 17:12:29|
|Last modified on||2013-03-22 17:12:29|
|Last modified by||Wkbj79 (1863)|
|Defines||symmetry about a point|
|Defines||symmetric about a point|
|Defines||symmetry about a line|
|Defines||symmetric about a line|