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Improper rotation
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<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an <a href="/facts/Isometry/K5wX8ah6">isometry</a> in <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> that is a combination of a <a href="/facts/Rotation_(geometry)/6gtkkcy0">rotation</a> about an axis and a <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflection</a> in a plane perpendicular to that axis. Reflection and <a href="/facts/Point_reflection/UzswAm8N">inversion</a> are each a special case of improper rotation. Any improper rotation is an <a href="/facts/Affine_transformation/BD7yDr10">affine transformation</a> and, in cases that keep the coordinate origin fixed, a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a>. It is used as a <a href="/facts/Symmetry_operation/umWYbPVH">symmetry operation</a> in the context of <a href="/facts/Symmetry_(geometry)/TljpwtNQ">geometric symmetry</a>, <a href="/facts/Molecular_symmetry/BrDxAkTK">molecular symmetry</a> and <a href="/facts/Crystallographic_point_group/43NqYrf0">crystallography</a>, where an object that is unchanged by a combination of rotation and reflection is said to have <i>improper rotation symmetry</i>. </p> Example polyhedra with rotoreflection symmetry<table><tbody><tr><th>Group</th><th>S4</th><th><i>S</i>6</th><th><i>S</i>8</th><th><i>S</i>10</th><th><i>S</i>12</th></tr><tr><th>Subgroups</th><td><i>C</i>2</td><td><i>C</i>3, <i>S</i>2 = <i>C</i>i</td><td><i>C</i>4, <i>C</i>2</td><td><i>C</i>5, <i>S</i>2 = <i>C</i>i</td><td><i>C</i>6, <i>S</i>4, <i>C</i>3, <i>C</i>2</td></tr><tr><th>Example</th><td>beveled digonal antiprism</td><td><a href="/facts/Triangular_antiprism/G38q92xW">triangular antiprism</a></td><td><a href="/facts/Square_antiprism/4514Q02s">square antiprism</a></td><td><a href="/facts/Pentagonal_antiprism/gR8aMBKN">pentagonal antiprism</a></td><td><a href="/facts/Hexagonal_antiprism/bt1W4cs6">hexagonal antiprism</a></td></tr><tr><td colspan="6"><a href="/facts/Antiprism/FgP5jIVd">Antiprisms</a> with directed edges have rotoreflection symmetry.<i>p</i>-antiprisms for odd <i>p</i> contain <a href="/facts/Inversion_symmetry/UzswAm8N">inversion symmetry</a>, <i>C</i>i.</td></tr></tbody></table>