symmetry
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:
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has rotational symmetry;
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has point symmetry;
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has symmetry about a point;
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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 :
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Regular polygons: A regular -gon is symmetric about its center (http://planetmath.org/Center9) with valid angles of rotation for any positive integer .
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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:
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has line symmetry;
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has symmetry about a line;
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is symmetric about a line.
The following are classic examples of line symmetry in :
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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.
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Circles: A circle is symmetric about any line passing through its center.
In the picture above, the boundary of is drawn in black, and the line is drawn in cyan.
Title | symmetry |
Canonical name | Symmetry |
Date of creation | 2013-03-22 17:12:29 |
Last modified on | 2013-03-22 17:12:29 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 18 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 51A10 |
Classification | msc 15A04 |
Classification | msc 51A15 |
Related topic | DihedralGroup |
Related topic | DeterminingRotationsAndReflectionsInMathbbR2 |
Defines | symmetry about |
Defines | symmetric |
Defines | symmetric about |
Defines | rotational symmetry |
Defines | point symmetry |
Defines | symmetry about a point |
Defines | symmetric about a point |
Defines | reflectional symmetry |
Defines | line symmetry |
Defines | symmetry about a line |
Defines | symmetric about a line |