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Point group
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<table><tbody><tr><td>The <i><a href="/facts/Bauhinia_blakeana/vHmIztQB">Bauhinia blakeana</a></i> flower on the <a href="/facts/Hong_Kong/4nAf3mqQ">Hong Kong</a> region flag has C5 symmetry; the star on each petal has D5 symmetry.</td><td>The <a href="/facts/Yin_and_Yang/b0YQXM7U">Yin and Yang</a> symbol has C2 symmetry of geometry with inverted colors</td></tr></tbody></table> <p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a point group is a <a href="/facts/Group_(mathematics)/IQTYqINt">mathematical group</a> of <a href="/facts/Symmetry_operation/umWYbPVH">symmetry operations</a> (<a href="/facts/Isometry/K5wX8ah6">isometries</a> in a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>) that have a <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> in common. The <a href="/facts/Origin_(mathematics)/Lm26tRCB">coordinate origin</a> of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension <i>d</i> is then a subgroup of the <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> O(<i>d</i>). Point groups are used to describe the <a href="/facts/Symmetry_(geometry)/TljpwtNQ">symmetries</a> of geometric figures and physical objects such as <a href="/facts/Molecular_symmetry/BrDxAkTK">molecules</a>. </p><p>Each point group can be <a href="/facts/Group_representation/FsuuDagQ">represented</a> as sets of <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrices</a> <i>M</i> that transform point <i>x</i> into point <i>y</i> according to <i>y</i> = <i>Mx</i>. Each element of a point group is either a <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotation</a> (<a href="/facts/Determinant/eGYqJ2TF">determinant</a> of <i>M</i> = 1), or it is a <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a> or <a href="/facts/Improper_rotation/kcLdCD9r">improper rotation</a> (determinant of <i>M</i> = −1). </p><p>The geometric symmetries of <a href="/facts/Crystal/e2R2mk3E">crystals</a> are described by <a href="/facts/Space_group/2XQx0J7M">space groups</a>, which allow <a href="/facts/Translation_(geometry)/KlpWmnfX">translations</a> and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the <a href="/facts/Crystallographic_restriction_theorem/tojdCRI8">crystallographic restriction theorem</a> and <a href="/facts/Space_group/2XQx0J7M">one of Bieberbach's theorems</a>, each number of dimensions has only a finite number of point groups that are symmetric over some <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> or grid with that number of dimensions. These are the <a href="/facts/Crystallographic_point_group/43NqYrf0">crystallographic point groups</a>. </p>