View source for Symmetry

An involutory orthogonal transformation that changes the orientation (an involutory transformation is a transformation that, when applied twice, yields the identity). For example, a reflection with respect to a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917601.png" /> in space (or with respect to a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917602.png" /> in a plane) is a symmetry under which every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917603.png" /> is mapped to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917604.png" /> such that the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917605.png" /> is perpendicular to the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917606.png" /> (the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917607.png" />) and is bisected by it. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917608.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s0917609.png" />) is called the plane (axis) of the symmetry (see ). Any orthogonal transformation, or isometry, is the composite of a finite number of reflections.

Symmetry is the following property of a geometrical figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176010.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176011.png" /> is mapped onto itself under the action of some group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176012.png" /> of transformations, called the group of symmetries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176013.png" />. Thus, symmetry reflects a certain regularity of the shape of a figure, namely, its invariance under the action of transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176014.png" />. If, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176015.png" /> is a plane figure such that rotation about some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176016.png" /> through an angle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176018.png" /> is an integer, maps it onto itself, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176019.png" /> possesses a symmetry of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176021.png" /> is called the centre of the symmetry (see Fig. b). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176022.png" /> is the cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176023.png" />. A circle has symmetry of infinite order (since it is mapped onto itself by a rotation through any angle).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760a.gif" />

Figure: s091760a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760b.gif" />

Figure: s091760b

Other simple forms of spatial symmetry are as follows.

a) A symmetry of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176025.png" /> about a straight line. Here the figure is mapped onto itself by a rotation about some straight line (the axis of symmetry) through an angle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176026.png" />. For example, if a plane figure is symmetric with respect to a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176027.png" />, then in space it has this straight line as an axis of symmetry of order 2. A cube has the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176028.png" /> as an axis of symmetry of order 3, and the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176029.png" /> as an axis of symmetry of order 4 (see Fig. c). In general, regular and semi-regular polyhedra are symmetric about several lines (see Fig. d). The position, number and orders of axes of symmetry play a fundamental role in crystallography.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760c.gif" />

Figure: s091760c

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760d.gif" />

Figure: s091760d

b) Translational symmetry. Here a figure is mapped onto itself by translation along a certain straight line (the axis of the translation) over some interval. A figure with a unique axis of translation and one plane of symmetry may have infinitely-many planes of symmetry (since any translation is a composite of two reflections) perpendicular to the axis of translation (see Fig. e). Figures having several axes of translation play an important role in the study of crystalline lattices.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760e.gif" />

Figure: s091760e

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760f.gif" />

Figure: s091760f

Symmetries composed of reflections and rotations (which exhaust the simple symmetries of bounded figures) and also of translations are of interest and are a topic of study in various fields of the natural sciences, art, etc. For example, a twist or helical symmetry, composed of rotation through some angle about an axis and translation along that axis, is observed in the arrangement of leaves in plants (see Fig. f). Symmetry is widely propagated as one of the ways of constructing embroideries and ornaments (planar figures possessing one or several translational symmetries in combination with reflections, see Fig. g, Fig. h).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760g.gif" />

Figure: s091760g

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s091760h.gif" />

Figure: s091760h

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Shubnikov,   V.A. Koptsik,   "Symmetry: in science and art" , Plenum  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Weyl,   "Symmetry" , Princeton Univ. Press  (1952)  (Translated from German)</TD></TR></table>



====Comments====
Quite generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176030.png" /> be a group acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176031.png" /> (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176032.png" /> can be the group of Euclidean motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176034.png" /> can be some set of geometrical figures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176035.png" /> closed under Euclidean motions). For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176036.png" /> the isotropy subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176037.png" /> is the symmetry group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091760/s09176038.png" />.

The statement that translational symmetry with a unique axis of translation implies symmetry with respect to planes is not correct in general.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Schattschneider,   "Visions of symmetry" , Freeman  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Grünbaum,   G.C. Shephard,   "Tilings and patterns" , Freeman  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular polytopes" , Macmillan  (1963)</TD></TR></table>