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<title>Point reflection</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Point reflection</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Central inversion" redirects here; not to be confused with <a href="Circle_inversion" class="mw-redirect" title="Circle inversion">Circle inversion</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Reflection_point" class="mw-redirect" title="Reflection point">Reflection point</a>.</div>

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:222px;max-width:222px"><div class="trow"><div class="tsingle" style="width:109px;max-width:109px"><div class="thumbimage" style="height:116px;overflow:hidden"><span typeof="mw:File"></span></div></div><div class="tsingle" style="width:109px;max-width:109px"><div class="thumbimage" style="height:116px;overflow:hidden"><span typeof="mw:File"></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Dual tetrahedra that are centrally symmetric to each other</div></div></div></div>

<p>In <a href="Geometry" title="Geometry">geometry</a>, a <b>point reflection</b> (also called a <b>point inversion</b> or <b>central inversion</b>) is a <a href="Geometric_transformation" title="Geometric transformation">geometric transformation</a> of <a href="Affine_space" title="Affine space">affine space</a> in which every <a href="Point_(geometry)" title="Point (geometry)">point</a> is reflected across a designated <b>inversion center</b>, which remains <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed</a>. In <a href="Euclidean_space" title="Euclidean space">Euclidean</a> or <a href="Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean spaces</a>, a point reflection is an <a href="Isometry" title="Isometry">isometry</a> (preserves <a href="Euclidean_distance" title="Euclidean distance">distance</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In the <a href="Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, a point reflection is the same as a <a href="Half_turn" class="mw-redirect" title="Half turn">half-turn</a> <a href="Rotation" title="Rotation">rotation</a> (180° or <span class="texhtml mvar" style="font-style:italic;">π</span> <a href="Radian" title="Radian">radians</a>), while in three-dimensional Euclidean space a point reflection is an <a href="Improper_rotation" title="Improper rotation">improper rotation</a> which preserves distances but <a href="Orientation-reversing" class="mw-redirect" title="Orientation-reversing">reverses orientation</a>. A point reflection is an <a href="Involution_(mathematics)" title="Involution (mathematics)">involution</a>: applying it twice is the <a href="Identity_transformation" class="mw-redirect" title="Identity transformation">identity transformation</a>.

</p><p>An object that is invariant under a point reflection is said to possess <b>point symmetry</b> (also called <b>inversion symmetry</b> or <b>central symmetry</b>). A <a href="Point_group" title="Point group">point group</a> including a point reflection among its symmetries is called <i><a href="Centrosymmetry" title="Centrosymmetry">centrosymmetric</a></i>. Inversion symmetry is found in many <a href="Crystal_structure" title="Crystal structure">crystal structures</a> and <a href="Molecule" title="Molecule">molecules</a>, and has a major effect upon their physical properties.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div>

<p>The term <i>reflection</i> is loose, and considered by some an abuse of language, with <i>inversion</i> preferred; however, <i>point reflection</i> is widely used. Such maps are <a href="Involution_(mathematics)" title="Involution (mathematics)">involutions</a>, meaning that they have order 2 – they are their own inverse: applying them twice yields the <a href="Identity_function" title="Identity function">identity map</a> – which is also true of other maps called <i>reflections</i>. More narrowly, a <i><a href="Reflection_(linear_algebra)" class="mw-redirect" title="Reflection (linear algebra)">reflection</a></i> refers to a reflection in a <a href="Hyperplane" title="Hyperplane">hyperplane</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

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</math></span><img src="./fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span> dimensional <a href="Affine_subspace" class="mw-redirect" title="Affine subspace">affine subspace</a> – a point on the <a href="Line_(geometry)" title="Line (geometry)">line</a>, a line in the <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, a plane in 3-space), with the hyperplane being fixed, but more broadly <i>reflection</i> is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension <i>k</i>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq k\leq n-1}">

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<annotation encoding="application/x-tex">{\displaystyle 1\leq k\leq n-1}</annotation>

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</math></span><img src="./8f9adcd3d085311183b8f0dff7ec086c26e82b14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.968ex; height:2.343ex;" alt="{\displaystyle 1\leq k\leq n-1}" loading="lazy"></span>) is called the <i>mirror</i>. In dimension 1 these coincide, as a point is a hyperplane in the line.

</p><p>In terms of linear algebra, assuming the origin is fixed, involutions are exactly the <a href="Diagonalizable" class="mw-redirect" title="Diagonalizable">diagonalizable</a> maps with all <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

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</p><p>The term <i>inversion</i> should not be confused with <a href="Inversive_geometry" title="Inversive geometry">inversive geometry</a>, where <i>inversion</i> is defined with respect to a circle.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<caption>2D examples

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<td><span typeof="mw:File"></span><br>Hexagonal <a href="Parallelogon" title="Parallelogon">parallelogon</a>

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<td><span typeof="mw:File"></span><br><a href="Octagon" title="Octagon">Octagon</a>

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<p>In two dimensions, a point reflection is the same as a <a href="Rotation" title="Rotation">rotation</a> of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation <a href="Composition_of_functions" class="mw-redirect" title="Composition of functions">composed</a> with reflection across the plane of rotation, perpendicular to the axis of rotation. In dimension <i>n</i>, point reflections are <a href="Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a>-preserving if <i>n</i> is even, and orientation-reversing if <i>n</i> is odd.

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<div class="mw-heading mw-heading2"><h2 id="Formula">Formula</h2></div>

<p>Given a vector <b>a</b> in the Euclidean space <b>R</b><sup><i>n</i></sup>, the formula for the reflection of <b>a</b> across the point <b>p</b> is

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Ref} _{\mathbf {p} }(\mathbf {a} )=2\mathbf {p} -\mathbf {a} .}</annotation>

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</math></span><img src="./0d26d96b4003306d34f2d9b19ec2b2e0e36f13e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.379ex; height:3.009ex;" alt="{\displaystyle \mathrm {Ref} _{\mathbf {p} }(\mathbf {a} )=2\mathbf {p} -\mathbf {a} .}" loading="lazy"></span></dd></dl>

<p>In the case where <b>p</b> is the origin, point reflection is simply the negation of the vector <b>a</b>.

</p><p>In <a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the <b>inversion</b> of a <a href="Point_(geometry)" title="Point (geometry)">point</a> <i>X</i> with respect to a point <i>P</i> is a point <i>X</i>* such that <i>P</i> is the midpoint of the <a href="Line_segment" title="Line segment">line segment</a> with endpoints <i>X</i> and <i>X</i>*. In other words, the <a href="Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> from <i>X</i> to <i>P</i> is the same as the vector from <i>P</i> to <i>X</i>*.

</p><p>The formula for the inversion in <i>P</i> is

</p>

<p>where <b>p</b>, <b>x</b> and <b>x</b>* are the position vectors of <i>P</i>, <i>X</i> and <i>X</i>* respectively.

</p><p>This <a href="Function_(mathematics)" title="Function (mathematics)">mapping</a> is an <a href="Isometry" title="Isometry">isometric</a> <a href="Involution_(mathematics)" title="Involution (mathematics)">involutive</a> <a href="Affine_transformation" title="Affine transformation">affine transformation</a> which has exactly one <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a>, which is <i>P</i>.

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<div class="mw-heading mw-heading2"><h2 id="Point_reflection_as_a_special_case_of_uniform_scaling_or_homothety">Point reflection as a special case of uniform scaling or homothety</h2></div>

<p>When the inversion point <i>P</i> coincides with the origin, point reflection is equivalent to a special case of <a href="Uniform_scaling" class="mw-redirect" title="Uniform scaling">uniform scaling</a>: uniform scaling with scale factor equal to −1. This is an example of <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>.

</p><p>When <i>P</i> does not coincide with the origin, point reflection is equivalent to a special case of <a href="Homothetic_transformation" class="mw-redirect" title="Homothetic transformation">homothetic transformation</a>: homothety with <a href="Homothetic_center" title="Homothetic center">homothetic center</a> coinciding with P, and scale factor −1. (This is an example of non-linear <a href="Affine_transformation" title="Affine transformation">affine transformation</a>.)

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<div class="mw-heading mw-heading2"><h2 id="Point_reflection_group">Point reflection group </h2></div>

<p>The <a href="Composition_of_functions" class="mw-redirect" title="Composition of functions">composition</a> of two point reflections is a <a href="Translation_(geometry)" title="Translation (geometry)">translation</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Specifically, point reflection at <b>p</b> followed by point reflection at <b>q</b> is translation by the vector 2(<b>q</b> − <b>p</b>).

</p><p>The set consisting of all point reflections and translations is <a href="Lie_subgroup" class="mw-redirect" title="Lie subgroup">Lie subgroup</a> of the <a href="Euclidean_group" title="Euclidean group">Euclidean group</a>. It is a <a href="Semidirect_product" title="Semidirect product">semidirect product</a> of <b>R</b><sup><i>n</i></sup> with a <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of order 2, the latter acting on <b>R</b><sup><i>n</i></sup> by negation. It is precisely the subgroup of the Euclidean group that fixes the <a href="Line_at_infinity" title="Line at infinity">line at infinity</a> pointwise.

</p><p>In the case <i>n</i> = 1, the point reflection group is the full <a href="Euclidean_group" title="Euclidean group">isometry group</a> of the line.

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<div class="mw-heading mw-heading2"><h2 id="Point_reflections_in_mathematics">Point reflections in mathematics</h2></div>

<ul><li>Point reflection across the center of a sphere yields the <a href="Antipodal_map" class="mw-redirect" title="Antipodal map">antipodal map</a>.</li>

<li>A <a href="Riemannian_symmetric_space" class="mw-redirect" title="Riemannian symmetric space">symmetric space</a> is a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> with an isometric reflection across each point. Symmetric spaces play an important role in the study of <a href="Lie_group" title="Lie group">Lie groups</a> and <a href="Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Point_reflection_in_analytic_geometry">Point reflection in analytic geometry</h2></div>

<p>Given the point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle P(x,y)}</annotation>

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</math></span><img src="./d5b3d8f37f5458c22b61eaf26e5af0523acb63e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.074ex; height:2.843ex;" alt="{\displaystyle P(x,y)}" loading="lazy"></span> and its reflection <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'(x',y')}">

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<annotation encoding="application/x-tex">{\displaystyle P'(x',y')}</annotation>

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</math></span><img src="./6ef542f849d3fb2e73ff45a8706615db44d47797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.209ex; height:3.009ex;" alt="{\displaystyle P'(x',y')}" loading="lazy"></span> with respect to the point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(x_{c},y_{c})}">

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<annotation encoding="application/x-tex">{\displaystyle C(x_{c},y_{c})}</annotation>

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</math></span><img src="./caf112dcd26f035c06c3339b5ed9085f6b58bf48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.967ex; height:2.843ex;" alt="{\displaystyle C(x_{c},y_{c})}" loading="lazy"></span>, the latter is the <a href="Midpoint" title="Midpoint">midpoint</a> of the segment <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {PP'}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {PP'}}}</annotation>

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</math></span><img src="./416e9b55522ac457d5d224fa5eceecef6749cd1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.367ex; height:3.176ex;" alt="{\displaystyle {\overline {PP'}}}" loading="lazy"></span>;

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x_{c}={\frac {x+x'}{2}}\\y_{c}={\frac {y+y'}{2}}\end{cases}}}</annotation>

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</math></span><img src="./af36e22e07f7acbcff41b12f31d5326d0e8b59a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.234ex; margin-bottom: -0.271ex; width:12.524ex; height:8.176ex;" alt="{\displaystyle {\begin{cases}x_{c}={\frac {x+x'}{2}}\\y_{c}={\frac {y+y'}{2}}\end{cases}}}" loading="lazy"></span></dd></dl>

<p>Hence, the equations to find the coordinates of the reflected point are

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x'=2x_{c}-x\\y'=2y_{c}-y\end{cases}}}</annotation>

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</math></span><img src="./8c75ddf97127e0be4bf143491e0411ded1b3d4fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.214ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}x'=2x_{c}-x\\y'=2y_{c}-y\end{cases}}}" loading="lazy"></span></dd></dl>

<p>Particular is the case in which the point C has coordinates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}">

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<annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x'=-x\\y'=-y\end{cases}}}</annotation>

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</math></span><img src="./6509f9a8b2e2c7bde6d32fa716acd4f1b0c41fd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.745ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}x'=-x\\y'=-y\end{cases}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>In even-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, say 2<i>N</i>-dimensional space, the inversion in a point <i>P</i> is equivalent to <i>N</i> rotations over angles <span class="texhtml mvar" style="font-style:italic;">π</span> in each plane of an arbitrary set of <i>N</i> mutually orthogonal planes intersecting at <i>P</i>. These rotations are mutually commutative. Therefore, inversion in a point in even-dimensional space is an orientation-preserving isometry or <a href="Euclidean_group" title="Euclidean group">direct isometry</a>.

</p><p>In odd-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, say (2<i>N</i> + 1)-dimensional space, it is equivalent to <i>N</i> rotations over <span class="texhtml mvar" style="font-style:italic;">π</span> in each plane of an arbitrary set of <i>N</i> mutually orthogonal planes intersecting at <i>P</i>, combined with the reflection in the 2<i>N</i>-dimensional subspace spanned by these rotation planes. Therefore, it <i>reverses</i> rather than preserves <a href="Orientation_(mathematics)" class="mw-redirect" title="Orientation (mathematics)">orientation</a>, it is an <a href="Euclidean_group" title="Euclidean group">indirect isometry</a>.

</p><p>Geometrically in 3D it amounts to <a href="Rotation" title="Rotation">rotation</a> about an axis through <i>P</i> by an angle of 180°, combined with reflection in the plane through <i>P</i> which is perpendicular to the axis; the result does not depend on the <a href="Orientation_(rigid_body)" class="mw-redirect" title="Orientation (rigid body)">orientation</a> (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {1}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {1}}}</annotation>

</semantics>

</math></span><img src="./f50d1053ea4c96ed679d1deca6105c442b54bf11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.277ex; height:2.843ex;" alt="{\displaystyle {\overline {1}}}" loading="lazy"></span>, <i>C</i><sub><i>i</i></sub>, <i>S</i><sub>2</sub>, and 1×. The group type is one of the three <a href="Symmetry_group" title="Symmetry group">symmetry group</a> types in 3D without any pure <a href="Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a>, see <a href="Cyclic_symmetries" class="mw-redirect" title="Cyclic symmetries">cyclic symmetries</a> with <i>n</i> = 1.

</p><p>The following <a href="Point_groups_in_three_dimensions" title="Point groups in three dimensions">point groups in three dimensions</a> contain inversion:

</p>

<p>Closely related to inverse in a point is <a href="Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> in respect to a <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a>, which can be thought of as an "inversion in a plane".

</p>

<div class="mw-heading mw-heading2"><h2 id="Inversion_centers_in_crystals_and_molecules">Inversion centers in crystals and molecules</h2></div>

<p>Inversion symmetry plays a major role in the properties of materials, as also do other symmetry operations.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Some molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In many cases they can be considered as polyhedra, categorized by their coordination number and bond angles. For example, four-coordinate polyhedra are classified as <a href="Tetrahedral_molecular_geometry" title="Tetrahedral molecular geometry">tetrahedra</a>, while five-coordinate environments can be <a href="Square_pyramidal_molecular_geometry" title="Square pyramidal molecular geometry">square pyramidal</a> or <a href="Trigonal_bipyramidal_molecular_geometry" title="Trigonal bipyramidal molecular geometry">trigonal bipyramidal</a> depending on the bonding angles. Six-coordinate octahedra are an example of centrosymmetric polyhedra, as the central atom acts as an inversion center through which the six bonded atoms retain symmetry. Tetrahedra, on the other hand, are non-centrosymmetric as an inversion through the central atom would result in a reversal of the polyhedron. Polyhedra with an odd (versus even) coordination number are not centrosymmtric. Polyhedra containing inversion centers are known as centrosymmetric, while those without are non-centrosymmetric. The presence or absence of an inversion center has a strong influence on the optical properties;<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> for instance molecules without inversion symmetry have a <a href="Dipole_moments_of_molecules" class="mw-redirect" title="Dipole moments of molecules">dipole moment</a> and can directly interact with photons, while those with inversion have no dipole moment and only interact via <a href="Raman_scattering" title="Raman scattering">Raman scattering</a>.<sup id="cite_ref-raman1928_5-0" class="reference"><a href="#cite_note-raman1928-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The later is named after <a href="C._V._Raman" title="C. V. Raman">C. V. Raman</a> who was awarded the 1930 <a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> for his discovery.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>In addition, in <a href="Crystallography" title="Crystallography">crystallography</a>, the presence of inversion centers for periodic structures distinguishes between <a href="Centrosymmetric" class="mw-redirect" title="Centrosymmetric">centrosymmetric</a> and non-centrosymmetric compounds. All crystalline compounds come from a repetition of an atomic building block known as a unit cell, and these unit cells define which polyhedra form and in what order. In many materials such as oxides these polyhedra can link together via corner-, edge- or face sharing, depending on which atoms share common bonds and also the valence. In other cases such as for <a href="Metals" class="mw-redirect" title="Metals">metals</a> and <a href="Alloys" class="mw-redirect" title="Alloys">alloys</a> the structures are better considered as arrangements of close-packed atoms. Crystals which do not have inversion symmetry also display the <a href="Piezoelectric_effect" class="mw-redirect" title="Piezoelectric effect">piezoelectric effect</a>. The presence or absence of inversion symmetry also has numerous consequences for the properties of solids,<sup id="cite_ref-:0_2-2" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> as does the mathematical relationships between the different crystal symmetries.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>Real polyhedra in crystals often lack the uniformity anticipated in their bonding geometry. Common irregularities found in crystallography include distortions and disorder. Distortion involves the warping of polyhedra due to nonuniform bonding lengths, often due to differing electrostatic interactions between heteroatoms or electronic effects such as <a href="Jahn%E2%80%93Teller_effect" title="Jahn–Teller effect">Jahn–Teller distortions</a>. For instance, a titanium center will likely bond evenly to six oxygens in an octahedra, but distortion would occur if one of the oxygens were replaced with a more <a href="Electronegative" class="mw-redirect" title="Electronegative">electronegative</a> fluorine. Distortions will not change the inherent geometry of the polyhedra—a distorted octahedron is still classified as an octahedron, but strong enough distortions can have an effect on the centrosymmetry of a compound. Disorder involves a split occupancy over two or more sites, in which an atom will occupy one crystallographic position in a certain percentage of polyhedra and the other in the remaining positions. Disorder can influence the centrosymmetry of certain polyhedra as well, depending on whether or not the occupancy is split over an already-present inversion center.

</p><p>Centrosymmetry applies to the crystal structure as a whole, not just individual polyhedra. Crystals are classified into thirty-two <a href="Crystallographic_point_groups" class="mw-redirect" title="Crystallographic point groups">crystallographic point groups</a> which describe how the different polyhedra arrange themselves in space in the bulk structure. Of these thirty-two point groups, eleven are centrosymmetric. The presence of noncentrosymmetric polyhedra does not guarantee that the point group will be the same—two non-centrosymmetric shapes can be oriented in space in a manner which contains an inversion center between the two. Two tetrahedra facing each other can have an inversion center in the middle, because the orientation allows for each atom to have a reflected pair. The inverse is also true, as multiple centrosymmetric polyhedra can be arranged to form a noncentrosymmetric point group.

</p>

<div class="mw-heading mw-heading2"><h2 id="Inversion_with_respect_to_the_origin">Inversion with respect to the origin</h2></div>

<p>Inversion with respect to the origin corresponds to <a href="Additive_inverse" title="Additive inverse">additive inversion</a> of the position vector, and also to <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> by −1. The operation commutes with every other <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>, but not with <a href="Translation_(geometry)" title="Translation (geometry)">translation</a>: it is in the <a href="Center_(group_theory)" title="Center (group theory)">center</a> of the <a href="General_linear_group" title="General linear group">general linear group</a>. "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion; in physics 3-dimensional reflection through the origin is also called a <a href="Parity_(physics)" title="Parity (physics)">parity transformation</a>.

</p><p>In mathematics, <b>reflection through the origin</b> refers to the point reflection of <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <b>R</b><sup><i>n</i></sup> across the <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a> of the <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>. Reflection through the origin is an <a href="Orthogonal_transformation" title="Orthogonal transformation">orthogonal transformation</a> corresponding to <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math></span><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"></span>, and can also be written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -I}</annotation>

</semantics>

</math></span><img src="./1af1f812fd44b9f513f3c163c2bcc5f3f738f98d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.98ex; height:2.343ex;" alt="{\displaystyle -I}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math></span><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> is the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>. In three dimensions, this sends <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}</annotation>

</semantics>

</math></span><img src="./58621041fdf3a9a1f65c92e16495e596fe76dc38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.939ex; height:2.843ex;" alt="{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}" loading="lazy"></span>, and so forth.

</p>

<div class="mw-heading mw-heading3"><h3 id="Representations">Representations</h3></div>

<p>As a <a href="Scalar_matrix" class="mw-redirect" title="Scalar matrix">scalar matrix</a>, it is represented in every basis by a matrix with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math></span><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"></span> on the diagonal, and, together with the identity, is the <a href="Center_(group_theory)" title="Center (group theory)">center</a> of the <a href="Orthogonal_group" title="Orthogonal group">orthogonal group</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n)}</annotation>

</semantics>

</math></span><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy"></span>.

</p><p>It is a product of <i>n</i> orthogonal reflections (reflection through the axes of any <a href="Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a>); note that orthogonal reflections commute.

</p><p>In 2 dimensions, it is in fact rotation by 180 degrees, and in dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

</semantics>

</math></span><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy"></span>, it is rotation by 180 degrees in <i>n</i> orthogonal planes;<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> note again that rotations in orthogonal planes commute.

</p>

<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>

<p>It has determinant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1)^{n}}</annotation>

</semantics>

</math></span><img src="./c490525b94310eb9d66c0282f8d28f652af9f40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.998ex; height:2.843ex;" alt="{\displaystyle (-1)^{n}}" loading="lazy"></span> (from the representation by a matrix or as a product of reflections). Thus it is orientation-preserving in even dimension, thus an element of the <a href="Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> SO(2<i>n</i>), and it is orientation-reversing in odd dimension, thus not an element of SO(2<i>n</i> + 1) and instead providing a <a href="Split_short_exact_sequence" class="mw-redirect" title="Split short exact sequence">splitting</a> of the map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(2n+1)\to \pm 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(2n+1)\to \pm 1}</annotation>

</semantics>

</math></span><img src="./f575dc00d48b77a10948ba9ffc7d4b985ddcfa9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.727ex; height:2.843ex;" alt="{\displaystyle O(2n+1)\to \pm 1}" loading="lazy"></span>, showing that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(2n+1)=SO(2n+1)\times \{\pm I\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(2n+1)=SO(2n+1)\times \{\pm I\}}</annotation>

</semantics>

</math></span><img src="./abcfde7bef4e9a321580949ef7d719072735590e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.028ex; height:2.843ex;" alt="{\displaystyle O(2n+1)=SO(2n+1)\times \{\pm I\}}" loading="lazy"></span> as an <a href="Internal_direct_product" class="mw-redirect" title="Internal direct product">internal direct product</a>.

</p>

<ul><li>Together with the identity, it forms the <a href="Center_(group_theory)" title="Center (group theory)">center</a> of the <a href="Orthogonal_group" title="Orthogonal group">orthogonal group</a>.</li>

<li>It preserves every quadratic form, meaning <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(-v)=Q(v)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(-v)=Q(v)}</annotation>

</semantics>

</math></span><img src="./aeca5d23c18bd20859f9805fb79b43c52ff0cb66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.457ex; height:2.843ex;" alt="{\displaystyle Q(-v)=Q(v)}" loading="lazy"></span>, and thus is an element of every <a href="Indefinite_orthogonal_group" title="Indefinite orthogonal group">indefinite orthogonal group</a> as well.</li>

<li>It equals the identity if and only if the characteristic is 2.</li>

<li>It is the <a href="Longest_element_of_a_Coxeter_group" title="Longest element of a Coxeter group">longest element</a> of the <a href="Coxeter_group" title="Coxeter group">Coxeter group</a> of <a href="Signed_permutations" class="mw-redirect" title="Signed permutations">signed permutations</a>.</li></ul>

<p>Analogously, it is a longest element of the orthogonal group, with respect to the generating set of reflections: elements of the orthogonal group all have <a href="Length_function" title="Length function">length</a> at most <i>n</i> with respect to the generating set of reflections,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> and reflection through the origin has length <i>n,</i> though it is not unique in this: other maximal combinations of rotations (and possibly reflections) also have maximal length.

</p>

<div class="mw-heading mw-heading3"><h3 id="Geometry">Geometry</h3></div>

<p>In SO(2<i>r</i>), reflection through the origin is the farthest point from the identity element with respect to the usual metric. In O(2<i>r</i> + 1), reflection through the origin is not in SO(2<i>r</i>+1) (it is in the non-identity component), and there is no natural sense in which it is a "farther point" than any other point in the non-identity component, but it does provide a <a href="Base_point" class="mw-redirect" title="Base point">base point</a> in the other component.

</p>

<div class="mw-heading mw-heading3"><h3 id="Clifford_algebras_and_spin_groups">Clifford algebras and spin groups</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Spin_group" title="Spin group">Spin group</a></div>

<p>It should <i>not</i> be confused with the element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\in \mathrm {Spin} (n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1\in \mathrm {Spin} (n)}</annotation>

</semantics>

</math></span><img src="./1a621f1f6e10f6f86b7def7a0fb04cb3a88c93c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.54ex; height:2.843ex;" alt="{\displaystyle -1\in \mathrm {Spin} (n)}" loading="lazy"></span> in the <a href="Spin_group" title="Spin group">spin group</a>. This is particularly confusing for even spin groups, as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -I\in SO(2n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -I\in SO(2n)}</annotation>

</semantics>

</math></span><img src="./113d36aa18e076ae785eced9021d657f2eef3533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.46ex; height:2.843ex;" alt="{\displaystyle -I\in SO(2n)}" loading="lazy"></span>, and thus in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (n)}</annotation>

</semantics>

</math></span><img src="./049140eca932e4de307c8bf196ed124234ae0286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.728ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (n)}" loading="lazy"></span> there is both <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math></span><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"></span> and 2 lifts of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -I}</annotation>

</semantics>

</p><p>Reflection through the identity extends to an automorphism of a <a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a>, called the <i>main involution</i> or <i>grade involution.</i>

</p><p>Reflection through the identity lifts to a <a href="Pseudoscalar_(Clifford_algebra)" class="mw-redirect" title="Pseudoscalar (Clifford algebra)">pseudoscalar</a>.

</p>

<ul><li><a href="Affine_involution" title="Affine involution">Affine involution</a></li>

<li><a href="Circle_inversion" class="mw-redirect" title="Circle inversion">Circle inversion</a></li>

<li><a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li>

<li><a href="Congruence_(geometry)" title="Congruence (geometry)">Congruence (geometry)</a></li>

<li><a href="Estermann_measure" title="Estermann measure">Estermann measure</a></li>

<li><a href="Euclidean_group" title="Euclidean group">Euclidean group</a></li>

<li><a href="Kovner%E2%80%93Besicovitch_measure" title="Kovner–Besicovitch measure">Kovner–Besicovitch measure</a></li>

<li><a href="Orthogonal_group" title="Orthogonal group">Orthogonal group</a></li>

<li><a href="Parity_(physics)" title="Parity (physics)">Parity (physics)</a></li>

<li><a href="Reflection_(mathematics)" title="Reflection (mathematics)">Reflection (mathematics)</a></li>

<li><a href="Riemannian_symmetric_space" class="mw-redirect" title="Riemannian symmetric space">Riemannian symmetric space</a></li>

<li><a href="Spin_group" title="Spin group">Spin group</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">"Orthogonal planes" meaning all elements are orthogonal and the planes intersect at 0 only, not that they intersect in a line and have <a href="Dihedral_angle" title="Dihedral angle">dihedral angle</a> 90°.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">This follows by classifying orthogonal transforms as direct sums of rotations and reflections, which follows from the <a href="Spectral_theorem" title="Spectral theorem">spectral theorem</a>, for instance.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://new.math.uiuc.edu/public403/isometries/reflections.html#_geometrical_definition_of_a_reflection">"Reflections in Lines"</a>. <i>new.math.uiuc.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-04-27</span></span>.</cite></span>

</li>

<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFNye1984" class="citation book cs1">Nye, J. F. (1984). <i>Physical properties of crystals: their representation by tensors and matrices</i> (1st published in pbk. with corrections, 1984 ed.). Oxford [Oxfordshire] : New York: Clarendon Press ; Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-851165-6</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://sites.math.washington.edu/~king/coursedir/m444a02/lab/lab09.html">"Lab 9 Point Reflection"</a>. <i>sites.math.washington.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-04-27</span></span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHarris_and_Bertolucci1989" class="citation book cs1">Harris and Bertolucci (1989). <i>Symmetry and Spectroscopy</i>. Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-66144-5</bdi>.</cite></span>

</li>

<li id="cite_note-raman1928-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-raman1928_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRaman1928" class="citation journal cs1">Raman, C. V. (1928). "A new radiation". <i>Indian Journal of Physics</i>. <b>2</b>: <span class="nowrap">387–</span>398. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10821%2F377">10821/377</a>. <q>Inaugural Address delivered to the South Indian Science Association on Friday, the 16th March, 1928</q></cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFSingh2002" class="citation journal cs1">Singh, R. (2002). "C. V. Raman and the Discovery of the Raman Effect". <i>Physics in Perspective</i>. <b>4</b> (4): <span class="nowrap">399–</span>420. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002PhP.....4..399S">2002PhP.....4..399S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs000160200002">10.1007/s000160200002</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121785335">121785335</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFMüllerWondratschekBärnighausen2017" class="citation book cs1">Müller, Ulrich; Wondratschek, Hans; Bärnighausen, Hartmut (2017). <i>Symmetry relationships between crystal structures: applications of crystallographic group theory in crystal chemistry</i>. International Union of Crystallography texts on crystallography (first published in paperback ed.). Oxford: Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-880720-9</bdi>.</cite></span>

</li>

</ol></div></div>

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