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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Regular polygonal symmetry</div> <table class="wikitable" style="float:right; margin-left:1em" width="300"> <caption>Selected <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">point groups in three dimensions</a> </caption> <tbody><tr style="text-align:center;"> <td><span typeof="mw:File"><a href="/wiki/File:Sphere_symmetry_group_cs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Sphere_symmetry_group_cs.png/100px-Sphere_symmetry_group_cs.png" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Sphere_symmetry_group_cs.png/150px-Sphere_symmetry_group_cs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Sphere_symmetry_group_cs.png/200px-Sphere_symmetry_group_cs.png 2x" data-file-width="636" data-file-height="622" /></a></span><br /><a href="/wiki/List_of_spherical_symmetry_groups#Involutional_symmetry" title="List of spherical symmetry groups">Involutional symmetry</a><br />C<sub>s</sub>, (*)<br />[ ] = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Sphere_symmetry_group_c3v.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Sphere_symmetry_group_c3v.png/100px-Sphere_symmetry_group_c3v.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Sphere_symmetry_group_c3v.png/150px-Sphere_symmetry_group_c3v.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Sphere_symmetry_group_c3v.png/200px-Sphere_symmetry_group_c3v.png 2x" data-file-width="621" data-file-height="620" /></a></span><br /><a href="/wiki/Cyclic_symmetry_in_three_dimensions" title="Cyclic symmetry in three dimensions">Cyclic symmetry</a><br />C<sub>nv</sub>, (*nn)<br />[n] = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/46/CDel_n.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Sphere_symmetry_group_d3h.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Sphere_symmetry_group_d3h.png/100px-Sphere_symmetry_group_d3h.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Sphere_symmetry_group_d3h.png/150px-Sphere_symmetry_group_d3h.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Sphere_symmetry_group_d3h.png/200px-Sphere_symmetry_group_d3h.png 2x" data-file-width="621" data-file-height="620" /></a></span><br /><a class="mw-selflink selflink">Dihedral symmetry</a><br />D<sub>nh</sub>, (*n22)<br />[n,2] = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/46/CDel_n.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_2.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span></span> </td></tr> <tr> <th colspan="4"><a href="/wiki/Polyhedral_group" title="Polyhedral group">Polyhedral group</a>, [n,3], (*n32) </th></tr> <tr style="text-align:center;"> <td><span typeof="mw:File"><a href="/wiki/File:Sphere_symmetry_group_td.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Sphere_symmetry_group_td.png/100px-Sphere_symmetry_group_td.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Sphere_symmetry_group_td.png/150px-Sphere_symmetry_group_td.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Sphere_symmetry_group_td.png/200px-Sphere_symmetry_group_td.png 2x" data-file-width="649" data-file-height="625" /></a></span><br /><a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry">Tetrahedral symmetry</a><br />T<sub>d</sub>, (*332)<br />[3,3] = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Sphere_symmetry_group_oh.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Sphere_symmetry_group_oh.png/100px-Sphere_symmetry_group_oh.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Sphere_symmetry_group_oh.png/150px-Sphere_symmetry_group_oh.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Sphere_symmetry_group_oh.png/200px-Sphere_symmetry_group_oh.png 2x" data-file-width="649" data-file-height="625" /></a></span><br /><a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">Octahedral symmetry</a><br />O<sub>h</sub>, (*432)<br />[4,3] = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/80/CDel_node_c1.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Sphere_symmetry_group_ih.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/100px-Sphere_symmetry_group_ih.png" decoding="async" width="100" height="92" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/150px-Sphere_symmetry_group_ih.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Sphere_symmetry_group_ih.png/200px-Sphere_symmetry_group_ih.png 2x" data-file-width="671" data-file-height="617" /></a></span><br /><a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">Icosahedral symmetry</a><br />I<sub>h</sub>, (*532)<br />[5,3] = <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/CDel_node_c2.png" decoding="async" width="7" height="23" class="mw-file-element" data-file-width="7" data-file-height="23" /></span></span></span> </td></tr></tbody></table> <p>In <a href="/wiki/Geometry" title="Geometry">geometry</a>, <b>dihedral symmetry in three dimensions</b> is one of three infinite sequences of <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">point groups in three dimensions</a> which have a <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> that as an abstract group is a <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> Dih<sub><i>n</i></sub> (for <i>n</i> ≥ 2). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Types">Types</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_symmetry_in_three_dimensions&action=edit&section=1" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: <a href="/wiki/Sch%C3%B6nflies_notation" class="mw-redirect" title="Schönflies notation">Schönflies notation</a>, <a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a>, and <a href="/wiki/Orbifold_notation" title="Orbifold notation">orbifold notation</a>. </p> <dl><dt>Chiral</dt> <dd></dd></dl> <ul><li><i>D<sub>n</sub></i>, [<i>n</i>,2]<sup>+</sup>, (22<i>n</i>) of order 2<i>n</i> – <b>dihedral symmetry</b> or <b>para-n-gonal group</b> (abstract group: <a href="/wiki/Dihedral_group" title="Dihedral group"><i>Dih<sub>n</sub></i></a>).</li></ul> <dl><dt>Achiral</dt> <dd></dd></dl> <ul><li><i>D<sub>nh</sub></i>, [<i>n</i>,2], (*22<i>n</i>) of order 4<i>n</i> – <b>prismatic symmetry</b> or <b>full ortho-n-gonal group</b> (abstract group: <i>Dih<sub>n</sub></i> × <i>Z</i><sub>2</sub>).</li> <li><i>D<sub>nd</sub></i> (or <i>D<sub>nv</sub></i>), [2<i>n</i>,2<sup>+</sup>], (2*<i>n</i>) of order 4<i>n</i> – <b>antiprismatic symmetry</b> or <b>full gyro-n-gonal group</b> (abstract group: <i>Dih</i><sub>2<i>n</i></sub>).</li></ul> <p>For a given <i>n</i>, all three have <i>n</i>-fold <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> about one axis (<a href="/wiki/Rotation" title="Rotation">rotation</a> by an angle of 360°/<i>n</i> does not change the object), and 2-fold rotational symmetry about a perpendicular axis, hence about <i>n</i> of those. For <i>n</i> = ∞, they correspond to three <a href="/wiki/Frieze_group" title="Frieze group">Frieze groups</a>. <a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schönflies notation</a> is used, with <a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a> in brackets, and <a href="/wiki/Orbifold_notation" title="Orbifold notation">orbifold notation</a> in parentheses. The term horizontal (h) is used with respect to a vertical axis of rotation. </p><p>In 2D, the symmetry group <i>D<sub>n</sub></i> includes reflections in lines. When the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection through a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°. In 3D, the two operations are distinguished: the group <i>D<sub>n</sub></i> contains rotations only, not reflections. The other group is <a href="/wiki/Pyramidal_symmetry" class="mw-redirect" title="Pyramidal symmetry">pyramidal symmetry</a> <i>C<sub>nv</sub></i> of the same order, 2<i>n</i>. </p><p>With <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a> in a plane perpendicular to the <i>n</i>-fold rotation axis, we have <i>D<sub>nh</sub></i>, [n], (*22<i>n</i>). </p><p><i>D<sub>nd</sub></i> (or <i>D<sub>nv</sub></i>), [2<i>n</i>,2<sup>+</sup>], (2*<i>n</i>) has vertical mirror planes between the horizontal rotation axes, not through them. As a result, the vertical axis is a 2<i>n</i>-fold <a href="/wiki/Rotoreflection" class="mw-redirect" title="Rotoreflection">rotoreflection</a> axis. </p><p><i>D<sub>nh</sub></i> is the symmetry group for a regular <i>n</i>-sided <a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prism</a> and also for a regular n-sided <a href="/wiki/Bipyramid" title="Bipyramid">bipyramid</a>. <i>D<sub>nd</sub></i> is the symmetry group for a regular <i>n</i>-sided <a href="/wiki/Antiprism" title="Antiprism">antiprism</a>, and also for a regular n-sided <a href="/wiki/Trapezohedron" title="Trapezohedron">trapezohedron</a>. <i>D<sub>n</sub></i> is the symmetry group of a partially rotated prism. </p><p><i>n</i> = 1 is not included because the three symmetries are equal to other ones: </p> <ul><li><i>D</i><sub>1</sub> and <i>C</i><sub>2</sub>: group of order 2 with a single 180° rotation.</li> <li><i>D</i><sub>1<i>h</i></sub> and <i>C</i><sub>2<i>v</i></sub>: group of order 4 with a reflection in a plane and a 180° rotation about a line in that plane.</li> <li><i>D</i><sub>1<i>d</i></sub> and <i>C</i><sub>2<i>h</i></sub>: group of order 4 with a reflection in a plane and a 180° rotation about a line perpendicular to that plane.</li></ul> <p>For <i>n</i> = 2 there is not one main axis and two additional axes, but there are three equivalent ones. </p> <ul><li><i>D</i><sub>2</sub>, [2,2]<sup>+</sup>, (222) of order 4 is one of the three symmetry group types with the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a> as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a <a href="/wiki/Cuboid" title="Cuboid">cuboid</a> with an S written on two opposite faces, in the same orientation.</li> <li><i>D</i><sub>2<i>h</i></sub>, [2,2], (*222) of order 8 is the symmetry group of a cuboid.</li> <li><i>D</i><sub>2<i>d</i></sub>, [4,2<sup>+</sup>], (2*2) of order 8 is the symmetry group of e.g.: <ul><li>A square cuboid with a diagonal drawn on one square face, and a perpendicular diagonal on the other one.</li> <li>A regular <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> scaled in the direction of a line connecting the midpoints of two opposite edges (<i>D</i><sub>2<i>d</i></sub> is a subgroup of <a href="/wiki/Tetrahedral_symmetry" title="Tetrahedral symmetry"><i>T<sub>d</sub></i></a>; by scaling, we reduce the symmetry).</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Subgroups">Subgroups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_symmetry_in_three_dimensions&action=edit&section=2" title="Edit section: Subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right"> <tbody><tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Order_2_dihedral_symmetry_subgroup_tree.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Order_2_dihedral_symmetry_subgroup_tree.png/200px-Order_2_dihedral_symmetry_subgroup_tree.png" decoding="async" width="200" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Order_2_dihedral_symmetry_subgroup_tree.png/300px-Order_2_dihedral_symmetry_subgroup_tree.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Order_2_dihedral_symmetry_subgroup_tree.png/400px-Order_2_dihedral_symmetry_subgroup_tree.png 2x" data-file-width="441" data-file-height="486" /></a></span><br /><i>D<sub>2h</sub></i>, [2,2], (*222) </td> <td><span typeof="mw:File"><a href="/wiki/File:Order_4_dihedral_symmetry_subgroup_tree.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Order_4_dihedral_symmetry_subgroup_tree.png/200px-Order_4_dihedral_symmetry_subgroup_tree.png" decoding="async" width="200" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Order_4_dihedral_symmetry_subgroup_tree.png/300px-Order_4_dihedral_symmetry_subgroup_tree.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Order_4_dihedral_symmetry_subgroup_tree.png/400px-Order_4_dihedral_symmetry_subgroup_tree.png 2x" data-file-width="523" data-file-height="606" /></a></span><br /><i>D<sub>4h</sub></i>, [4,2], (*224) </td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Cyclic_symmetry_in_three_dimensions" title="Cyclic symmetry in three dimensions">Cyclic symmetry in three dimensions</a></div> <p>For <i>D<sub>nh</sub></i>, [n,2], (*22n), order 4n </p> <ul><li><i>C<sub>nh</sub></i>, [n<sup>+</sup>,2], (n*), order 2n</li> <li><i>C<sub>nv</sub></i>, [n,1], (*nn), order 2n</li> <li><i>D<sub>n</sub></i>, [n,2]<sup>+</sup>, (22n), order 2n</li></ul> <p>For <i>D<sub>nd</sub></i>, [2n,2<sup>+</sup>], (2*n), order 4n </p> <ul><li><i>S</i><sub>2<i>n</i></sub>, [2n<sup>+</sup>,2<sup>+</sup>], (n×), order 2n</li> <li><i>C<sub>nv</sub></i>, [n<sup>+</sup>,2], (n*), order 2n</li> <li><i>D<sub>n</sub></i>, [n,2]<sup>+</sup>, (22n), order 2n</li></ul> <p><i>D<sub>nd</sub></i> is also subgroup of <i>D</i><sub>2<i>nh</i></sub>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_symmetry_in_three_dimensions&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" width="600"> <tbody><tr> <th>D<sub>2h</sub>, [2,2], (*222)<br />Order 8 </th> <th>D<sub>2d</sub>, [4,2<sup>+</sup>], (2*2)<br />Order 8 </th> <th>D<sub>3h</sub>, [3,2], (*223)<br />Order 12 </th></tr> <tr align="center"> <td><span typeof="mw:File"><a href="/wiki/File:Basketball.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Basketball.png/200px-Basketball.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Basketball.png/300px-Basketball.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7a/Basketball.png 2x" data-file-width="340" data-file-height="340" /></a></span><br /><a href="/wiki/Basketball" title="Basketball">basketball</a> seam paths </td> <td><span typeof="mw:File"><a href="/wiki/File:Baseball_(crop).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Baseball_%28crop%29.png/200px-Baseball_%28crop%29.png" decoding="async" width="200" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Baseball_%28crop%29.png/300px-Baseball_%28crop%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Baseball_%28crop%29.png/400px-Baseball_%28crop%29.png 2x" data-file-width="1522" data-file-height="1501" /></a></span><br /><a href="/wiki/Baseball" title="Baseball">baseball</a> seam paths <br />(ignoring directionality of seam) </td> <td><span typeof="mw:File"><a href="/wiki/File:BeachBall.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/BeachBall.jpg/200px-BeachBall.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/BeachBall.jpg/300px-BeachBall.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/BeachBall.jpg/400px-BeachBall.jpg 2x" data-file-width="1584" data-file-height="1584" /></a></span><br /><a href="/wiki/Beach_ball" title="Beach ball">Beach ball</a><br />(ignoring colors) </td></tr></tbody></table> <p><b><i>D<sub>nh</sub></i>, [<i>n</i>], (*22<i>n</i>):</b> </p> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Geometricprisms.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Geometricprisms.gif/250px-Geometricprisms.gif" decoding="async" width="250" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Geometricprisms.gif/375px-Geometricprisms.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Geometricprisms.gif/500px-Geometricprisms.gif 2x" data-file-width="693" data-file-height="263" /></a></span><br /><a href="/wiki/Prism_(geometry)" title="Prism (geometry)">prisms</a> </td></tr></tbody></table> <p><b><i>D</i><sub>5<i>h</i></sub>, [5], (*225):</b> </p> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Pentagrammic_prism.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Pentagrammic_prism.png/100px-Pentagrammic_prism.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Pentagrammic_prism.png/150px-Pentagrammic_prism.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Pentagrammic_prism.png/200px-Pentagrammic_prism.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Pentagrammic_prism" title="Pentagrammic prism">Pentagrammic prism</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Pentagrammic_antiprism.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Pentagrammic_antiprism.png/100px-Pentagrammic_antiprism.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Pentagrammic_antiprism.png/150px-Pentagrammic_antiprism.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Pentagrammic_antiprism.png/200px-Pentagrammic_antiprism.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Pentagrammic_antiprism" title="Pentagrammic antiprism">Pentagrammic antiprism</a> </td></tr></tbody></table> <p><b><i>D</i><sub>4<i>d</i></sub>, [8,2<sup>+</sup>], (2*4):</b> </p> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Snub_square_antiprism.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Snub_square_antiprism.png/100px-Snub_square_antiprism.png" decoding="async" width="100" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Snub_square_antiprism.png/150px-Snub_square_antiprism.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Snub_square_antiprism.png/200px-Snub_square_antiprism.png 2x" data-file-width="440" data-file-height="349" /></a></span><br /><a href="/wiki/Snub_square_antiprism" title="Snub square antiprism">Snub square antiprism</a> </td></tr></tbody></table> <p><b><i>D</i><sub>5<i>d</i></sub>, [10,2<sup>+</sup>], (2*5):</b> </p> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Antiprism5.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Antiprism5.jpg/100px-Antiprism5.jpg" decoding="async" width="100" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Antiprism5.jpg/150px-Antiprism5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Antiprism5.jpg/200px-Antiprism5.jpg 2x" data-file-width="835" data-file-height="666" /></a></span><br /><a href="/wiki/Pentagonal_antiprism" title="Pentagonal antiprism">Pentagonal antiprism</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Pentagrammic_crossed_antiprism.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Pentagrammic_crossed_antiprism.png/100px-Pentagrammic_crossed_antiprism.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Pentagrammic_crossed_antiprism.png/150px-Pentagrammic_crossed_antiprism.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Pentagrammic_crossed_antiprism.png/200px-Pentagrammic_crossed_antiprism.png 2x" data-file-width="1000" data-file-height="1000" /></a></span><br /><a href="/wiki/Pentagrammic_crossed-antiprism" title="Pentagrammic crossed-antiprism">Pentagrammic crossed-antiprism</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Trapezohedron5.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Trapezohedron5.jpg/60px-Trapezohedron5.jpg" decoding="async" width="60" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Trapezohedron5.jpg/90px-Trapezohedron5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Trapezohedron5.jpg/120px-Trapezohedron5.jpg 2x" data-file-width="452" data-file-height="773" /></a></span><br /><a href="/wiki/Pentagonal_trapezohedron" title="Pentagonal trapezohedron">pentagonal trapezohedron</a> </td></tr></tbody></table> <p><b><i>D</i><sub>17<i>d</i></sub>, [34,2<sup>+</sup>], (2*17):</b> </p> <table class="wikitable"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/File:Antiprism17.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Antiprism17.jpg/100px-Antiprism17.jpg" decoding="async" width="100" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Antiprism17.jpg/150px-Antiprism17.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Antiprism17.jpg/200px-Antiprism17.jpg 2x" data-file-width="877" data-file-height="527" /></a></span><br /><a href="/wiki/Antiprism" title="Antiprism">Heptadecagonal antiprism</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_symmetry_in_three_dimensions&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_spherical_symmetry_groups" title="List of spherical symmetry groups">List of spherical symmetry groups</a></li> <li><a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">Point groups in three dimensions</a></li> <li><a href="/wiki/Cyclic_symmetry_in_three_dimensions" title="Cyclic symmetry in three dimensions">Cyclic symmetry in three dimensions</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_symmetry_in_three_dimensions&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCoxeter,_H._S._M._and_Moser,_W._O._J.1980" class="citation book cs1"><a href="/wiki/Coxeter" class="mw-redirect" title="Coxeter">Coxeter</a>, H. S. M. and Moser, W. O. J. (1980). <i>Generators and Relations for Discrete Groups</i>. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-09212-9" title="Special:BookSources/0-387-09212-9"><bdi>0-387-09212-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generators+and+Relations+for+Discrete+Groups&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1980&rft.isbn=0-387-09212-9&rft.au=Coxeter%2C+H.+S.+M.+and+Moser%2C+W.+O.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+symmetry+in+three+dimensions" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></li> <li><a href="/wiki/Norman_Johnson_(mathematician)" title="Norman Johnson (mathematician)">N.W. Johnson</a>: <i>Geometries and Transformations</i>, (2018) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-10340-5" title="Special:BookSources/978-1-107-10340-5">978-1-107-10340-5</a> Chapter 11: <i>Finite symmetry groups</i>, 11.5 Spherical Coxeter groups</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwayHuson2002" class="citation cs2"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John Horton</a>; Huson, Daniel H. (2002), "The Orbifold Notation for Two-Dimensional Groups", <i>Structural Chemistry</i>, <b>13</b> (3), Springer Netherlands: 247–257, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1015851621002">10.1023/A:1015851621002</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:33947139">33947139</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Structural+Chemistry&rft.atitle=The+Orbifold+Notation+for+Two-Dimensional+Groups&rft.volume=13&rft.issue=3&rft.pages=247-257&rft.date=2002&rft_id=info%3Adoi%2F10.1023%2FA%3A1015851621002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A33947139%23id-name%3DS2CID&rft.aulast=Conway&rft.aufirst=John+Horton&rft.au=Huson%2C+Daniel+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADihedral+symmetry+in+three+dimensions" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dihedral_symmetry_in_three_dimensions&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html">Graphic overview of the 32 crystallographic point groups</a> – form the first parts (apart from skipping <i>n</i>=5) of the 7 infinite series and 5 of the 7 separate 3D point groups</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Dihedral_symmetry_in_three_dimensions&oldid=1142739758">https://en.wikipedia.org/w/index.php?title=Dihedral_symmetry_in_three_dimensions&oldid=1142739758</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Symmetry" title="Category:Symmetry">Symmetry</a></li><li><a href="/wiki/Category:Euclidean_symmetries" title="Category:Euclidean symmetries">Euclidean symmetries</a></li><li><a href="/wiki/Category:Group_theory" title="Category:Group theory">Group theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 March 2023, at 03:07<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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