Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Symmetry operation
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a symmetry operation is a <a href="/facts/Geometric_transformation/qRjrasJf">geometric transformation</a> of an object that leaves the object looking the same after it has been carried out. For example, a 1⁄3 turn <a href="/facts/Rotation/yChkz49x">rotation</a> of a <a href="/facts/Regular_triangle/MJomsYDS">regular triangle</a> about its <a href="/facts/Center_(geometry)/oKPIwkY3">center</a>, a <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a> of a <a href="/facts/Square/JHJeMvmL">square</a> across its <a href="/facts/Diagonal/39z9sIRb">diagonal</a>, a <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a> of the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean plane</a>, or a <a href="/facts/Point_reflection/UzswAm8N">point reflection</a> of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some <a href="/facts/Symmetry_element/dA8rPCVK">symmetry element</a> (a point, line or plane). </p><p>In the context of <a href="/facts/Molecular_symmetry/BrDxAkTK">molecular symmetry</a>, a symmetry operation is a <a href="/facts/Permutation/JSw9ukmv">permutation</a> of atoms such that the <a href="/facts/Molecule/N9ljSfFq">molecule</a> or <a href="/facts/Crystal/e2R2mk3E">crystal</a> is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasizes its usefulness. </p> <ol><li>Physical properties must be <a href="/facts/Invariant_(physics)/UfHcehYs">invariant</a> with respect to symmetry operations.</li> <li>Symmetry operations can be collected together in groups which are <a href="/facts/Isomorphism/pj8KgkxU">isomorphic</a> to <a href="/facts/Permutation_groups/vRsNNwwx">permutation groups</a>.</li></ol> <p>In the context of molecular symmetry, quantum <a href="/facts/Wavefunctions/KgM5ykjY">wavefunctions</a> need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property. </p>