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单词 Symmetry
释义

symmetry


Let V be a Euclidean vector space, FV, and E:VV be a Euclidean transformation that is not the identity map.

The following terms are used to indicate that E(F)=F if E is a rotation:

  • F has rotational symmetryMathworldPlanetmath;

  • F has point symmetry;

  • F has symmetry about a point;

  • F is symmetric about a point.

If V=2, then the last two terms may be used to indicate the specific case in which E is conjugatePlanetmathPlanetmath to (-100-1), i.e. (http://planetmath.org/Ie) the angle of rotation is 180.

The following are classic examples of rotational symmetry in 2:

  • Regular polygonsMathworldPlanetmath: A regularPlanetmathPlanetmath n-gon is symmetric about its center (http://planetmath.org/Center9) with valid angles of rotation θ=(360kn) for any positive integer k<n.

  • Circles: A circle is symmetric about its center (http://planetmath.org/Center8) with uncountably many valid angles of rotation.

As another example, let F=k=14Pk, where each Pk is defined thus:

P1={(x,y):0x41+3 and (2-3)xyx},
P2={(x,y):41+3x2 and xy(2+3)x-4},
P3={(x,y):2x431+3 and (-2+3)x+8-43y(-2-3)x+4+43},
P4={(x,y):431+3x4 and (-2+3)x+8-43y-x+4}.

Then F has point symmetry with respect to the point (2,23). The valid angles of rotation for F are 120 and 240. The boundary of F and the point (2,23) are shown in the following picture.

As a final example, the figure

{(x,y):-3x-1 and (x+1)2+y24}([-1,1]×[-2,2]){(x,y):1x3 and (x-1)2+y24} is symmetric about the origin. The boundary of this figure and the point (0,0) are shown in the following picture.

If E(F)=F and E is a reflection, then F has reflectional symmetry. In the special case that V=2, the following terms are used:

  • F has line symmetry;

  • F has symmetry about a line;

  • F is symmetric about a line.

The following are classic examples of line symmetry in 2:

  • Regular polygons: There are n lines of symmetryMathworldPlanetmath of a regular n-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 trapezoidMathworldPlanetmath defined by

T={(x,y):0x6 and 0ymin{x,2,-x+6}}

is symmetric about x=3.

In the picture above, the boundary of T is drawn in black, and the line x=3 is drawn in cyan.

Titlesymmetry
Canonical nameSymmetry
Date of creation2013-03-22 17:12:29
Last modified on2013-03-22 17:12:29
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id18
AuthorWkbj79 (1863)
Entry typeDefinition
Classificationmsc 51A10
Classificationmsc 15A04
Classificationmsc 51A15
Related topicDihedralGroup
Related topicDeterminingRotationsAndReflectionsInMathbbR2
Definessymmetry about
DefinessymmetricMathworldPlanetmath
Definessymmetric about
Definesrotational symmetry
Definespoint symmetry
Definessymmetry about a point
Definessymmetric about a point
Definesreflectional symmetry
Definesline symmetry
Definessymmetry about a line
Definessymmetric about a line
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更新时间:2025/5/5 0:26:17