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 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.

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 symmetricMathworldPlanetmath
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