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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 class="mw-selflink selflink">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 href="/wiki/Dihedral_symmetry_in_three_dimensions" title="Dihedral symmetry in three dimensions">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 three dimensional <a href="/wiki/Geometry" title="Geometry">geometry</a>, there are four infinite series of <a href="/wiki/Point_groups_in_three_dimensions" title="Point groups in three dimensions">point groups in three dimensions</a> (<i>n</i>≥1) with <i>n</i>-fold rotational or reflectional symmetry about one axis (by an angle of 360°/<i>n</i>) that does not change the object. </p><p>They are the finite <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry groups</a> on a <a href="/wiki/Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">cone</a>. For <i>n</i> = ∞ they correspond to four <a href="/wiki/Frieze_group" title="Frieze group">frieze groups</a>. <a href="/wiki/Arthur_Moritz_Sch%C3%B6nflies" class="mw-redirect" title="Arthur Moritz Schönflies">Schönflies</a> notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are <a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter notation</a> in brackets, and, in parentheses, <a href="/wiki/Orbifold_notation" title="Orbifold notation">orbifold notation</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><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/220px-Order_4_dihedral_symmetry_subgroup_tree.png" decoding="async" width="220" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Order_4_dihedral_symmetry_subgroup_tree.png/330px-Order_4_dihedral_symmetry_subgroup_tree.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Order_4_dihedral_symmetry_subgroup_tree.png/440px-Order_4_dihedral_symmetry_subgroup_tree.png 2x" data-file-width="523" data-file-height="606" /></a><figcaption>Example symmetry subgroup tree for dihedral symmetry: <i>D<sub>4h</sub></i>, [4,2], (*224)</figcaption></figure> <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=Cyclic_symmetry_in_three_dimensions&action=edit&section=1" title="Edit section: Types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Chiral</dt> <dd></dd></dl> <ul><li><b><i>C<sub>n</sub></i>, [n]<sup>+</sup>, (<i>nn</i>)</b> of order <i>n</i> - <i>n</i>-fold rotational symmetry - <b>acro-n-gonal group</b> (abstract group <a href="/wiki/Cyclic_group" title="Cyclic group"><i>Z<sub>n</sub></i></a>); for <i>n</i>=1: <b>no symmetry</b> (<a href="/wiki/Trivial_group" title="Trivial group">trivial group</a>)</li></ul> <dl><dt>Achiral</dt> <dd></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:S_shaped_packing.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/S_shaped_packing.jpeg/220px-S_shaped_packing.jpeg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/S_shaped_packing.jpeg/330px-S_shaped_packing.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/S_shaped_packing.jpeg/440px-S_shaped_packing.jpeg 2x" data-file-width="2592" data-file-height="1944" /></a><figcaption>Piece of loose-fill <a href="/wiki/Cushioning" class="mw-redirect" title="Cushioning">cushioning</a> with <i>C<sub>2h</sub></i> symmetry</figcaption></figure> <ul><li><b><i>C<sub>nh</sub></i>, [n<sup>+</sup>,2], (<i>n</i>*)</b> of order 2<i>n</i> - <b>prismatic symmetry</b> or <b>ortho-n-gonal group</b> (abstract group <a href="/wiki/Cyclic_group" title="Cyclic group"><i>Z<sub>n</sub></i></a> × <i>Dih<sub>1</sub></i>); for <i>n</i>=1 this is denoted by <b><i>C<sub>s</sub></i> (1*)</b> and called <b><a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a></b>, also <b><a href="/wiki/Symmetry_(biology)#Bilateral_symmetry" class="mw-redirect" title="Symmetry (biology)">bilateral symmetry</a></b>. It has <a href="/wiki/Reflection_symmetry" title="Reflection symmetry">reflection symmetry</a> with respect to a plane perpendicular to the <i>n</i>-fold rotation axis.</li> <li><b><i>C<sub>nv</sub></i>, [n], (*<i>nn</i>)</b> of order 2<i>n</i> - <b>pyramidal symmetry</b> or <b>full acro-n-gonal group</b> (abstract group <i>Dih<sub>n</sub></i>); in biology <i>C<sub>2v</sub></i> is called <b>biradial symmetry</b>. For <i>n</i>=1 we have again <i>C<sub>s</sub></i> (1*). It has vertical mirror planes. This is the symmetry group for a regular <i>n</i>-sided <a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">pyramid</a>.</li> <li><b><i>S<sub>2n</sub></i>, [2<sup>+</sup>,2n<sup>+</sup>], (<i>n</i>×)</b> of order 2<i>n</i> - <b>gyro-n-gonal group</b> (not to be confused with <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric groups</a>, for which the same notation is used; abstract group <i>Z<sub>2n</sub></i>); It has a 2<i>n</i>-fold <a href="/wiki/Improper_rotation" title="Improper rotation">rotoreflection</a> axis, also called 2<i>n</i>-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like <i>D<sub>nd</sub></i>, it contains a number of improper rotations without containing the corresponding rotations. <ul><li>for <i>n</i>=1 we have <i>S<sub>2</sub></i> (<b>1×</b>), also denoted by <i><b>C<sub>i</sub></b></i>; this is <a href="/wiki/Inversion_in_a_point" class="mw-redirect" title="Inversion in a point">inversion symmetry</a>.</li></ul></li></ul> <p><b><i>C<sub>2h</sub></i>, [2,2<sup>+</sup>] (2*)</b> and <b><i>C<sub>2v</sub></i>, [2], (*22)</b> of order 4 are two of the three 3D symmetry group types with the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a> as abstract group. <i>C<sub>2v</sub></i> applies e.g. for a rectangular tile with its top side different from its bottom side. </p> <div class="mw-heading mw-heading2"><h2 id="Frieze_groups">Frieze groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cyclic_symmetry_in_three_dimensions&action=edit&section=2" title="Edit section: Frieze groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the limit these four groups represent Euclidean plane <a href="/wiki/Frieze_group" title="Frieze group">frieze groups</a> as C<sub>∞</sub>, C<sub>∞h</sub>, C<sub>∞v</sub>, and S<sub>∞</sub>. Rotations become translations in the limit. Portions of the infinite plane can also be cut and connected into an infinite cylinder. </p> <table class="wikitable"> <caption><a href="/wiki/Frieze_group" title="Frieze group">Frieze groups</a> </caption> <tbody><tr> <th colspan="4">Notations </th> <th colspan="2">Examples </th></tr> <tr> <th><a href="/wiki/IUC_notation" class="mw-redirect" title="IUC notation">IUC</a> </th> <th><a href="/wiki/Orbifold_notation" title="Orbifold notation">Orbifold</a> </th> <th><a href="/wiki/Coxeter_notation" title="Coxeter notation">Coxeter</a> </th> <th><a href="/wiki/Schoenflies_notation" title="Schoenflies notation">Schönflies</a><sup>*</sup> </th> <th>Euclidean plane </th> <th>Cylindrical (n=6) </th></tr> <tr> <th>p1</th> <th>∞∞</th> <th>[∞]<sup>+</sup></th> <th>C<sub>∞</sub> </th> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_example_p1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Frieze_example_p1.png/150px-Frieze_example_p1.png" decoding="async" width="150" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/10/Frieze_example_p1.png 1.5x" data-file-width="151" data-file-height="49" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniaxial_c6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Uniaxial_c6.png/150px-Uniaxial_c6.png" decoding="async" width="150" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Uniaxial_c6.png/225px-Uniaxial_c6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Uniaxial_c6.png/300px-Uniaxial_c6.png 2x" data-file-width="600" data-file-height="249" /></a></span> </td></tr> <tr> <th>p1m1</th> <th>*∞∞</th> <th>[∞]</th> <th>C<sub>∞v</sub> </th> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_example_p1m1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Frieze_example_p1m1.png/150px-Frieze_example_p1m1.png" decoding="async" width="150" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/5c/Frieze_example_p1m1.png 1.5x" data-file-width="151" data-file-height="49" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniaxial_c6v.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Uniaxial_c6v.png/150px-Uniaxial_c6v.png" decoding="async" width="150" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Uniaxial_c6v.png/225px-Uniaxial_c6v.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Uniaxial_c6v.png/300px-Uniaxial_c6v.png 2x" data-file-width="600" data-file-height="249" /></a></span> </td></tr> <tr> <th>p11m</th> <th>∞*</th> <th>[∞<sup>+</sup>,2]</th> <th>C<sub>∞h</sub> </th> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_example_p11m.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Frieze_example_p11m.png/150px-Frieze_example_p11m.png" decoding="async" width="150" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a4/Frieze_example_p11m.png 1.5x" data-file-width="151" data-file-height="49" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniaxial_c6h.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Uniaxial_c6h.png/150px-Uniaxial_c6h.png" decoding="async" width="150" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Uniaxial_c6h.png/225px-Uniaxial_c6h.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Uniaxial_c6h.png/300px-Uniaxial_c6h.png 2x" data-file-width="600" data-file-height="249" /></a></span> </td></tr> <tr> <th>p11g</th> <th>∞×</th> <th>[∞<sup>+</sup>,2<sup>+</sup>]</th> <th>S<sub>∞</sub> </th> <td><span typeof="mw:File"><a href="/wiki/File:Frieze_example_p11g.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Frieze_example_p11g.png/150px-Frieze_example_p11g.png" decoding="async" width="150" height="49" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/76/Frieze_example_p11g.png 1.5x" data-file-width="151" data-file-height="49" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Uniaxial_s6.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Uniaxial_s6.png/150px-Uniaxial_s6.png" decoding="async" width="150" height="63" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Uniaxial_s6.png/225px-Uniaxial_s6.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Uniaxial_s6.png/300px-Uniaxial_s6.png 2x" data-file-width="600" data-file-height="250" /></a></span> </td></tr></tbody></table> <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=Cyclic_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="400"> <tbody><tr> <th><i>S<sub>2</sub></i>/<i>C<sub>i</sub></i> (1x): </th> <th colspan="2"><i>C<sub>4v</sub></i> (*44): </th> <th><i>C<sub>5v</sub></i> (*55): </th></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/File:Parallelepiped.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Parallelepiped.svg/100px-Parallelepiped.svg.png" decoding="async" width="100" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Parallelepiped.svg/150px-Parallelepiped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Parallelepiped.svg/200px-Parallelepiped.svg.png 2x" data-file-width="1156" data-file-height="888" /></a></span><br /><a href="/wiki/Parallelepiped" title="Parallelepiped">Parallelepiped</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Square_pyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Square_pyramid.png/100px-Square_pyramid.png" decoding="async" width="100" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Square_pyramid.png/150px-Square_pyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Square_pyramid.png/200px-Square_pyramid.png 2x" data-file-width="420" data-file-height="259" /></a></span><br /><a href="/wiki/Square_pyramid" title="Square pyramid">Square pyramid</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Elongated_square_pyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Elongated_square_pyramid.png/100px-Elongated_square_pyramid.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Elongated_square_pyramid.png/150px-Elongated_square_pyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Elongated_square_pyramid.png/200px-Elongated_square_pyramid.png 2x" data-file-width="512" data-file-height="512" /></a></span><br /><a href="/wiki/Elongated_square_pyramid" title="Elongated square pyramid">Elongated square pyramid</a> </td> <td><span typeof="mw:File"><a href="/wiki/File:Pentagonal_pyramid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Pentagonal_pyramid.png/100px-Pentagonal_pyramid.png" decoding="async" width="100" height="62" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Pentagonal_pyramid.png/150px-Pentagonal_pyramid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/Pentagonal_pyramid.png/200px-Pentagonal_pyramid.png 2x" data-file-width="413" data-file-height="255" /></a></span><br /><a href="/wiki/Pentagonal_pyramid" title="Pentagonal pyramid">Pentagonal pyramid</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=Cyclic_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/Dihedral_symmetry_in_three_dimensions" title="Dihedral symmetry in three dimensions">Dihedral 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=Cyclic_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="CITEREFSands1993" class="citation book cs1">Sands, Donald E. (1993). "Crystal Systems and Geometry". <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontocr00desa"><i>Introduction to Crystallography</i></a></span>. Mineola, New York: Dover Publications, Inc. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontocr00desa/page/n173">165</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-67839-3" title="Special:BookSources/0-486-67839-3"><bdi>0-486-67839-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Crystal+Systems+and+Geometry&rft.btitle=Introduction+to+Crystallography&rft.place=Mineola%2C+New+York&rft.pages=165&rft.pub=Dover+Publications%2C+Inc.&rft.date=1993&rft.isbn=0-486-67839-3&rft.aulast=Sands&rft.aufirst=Donald+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontocr00desa&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACyclic+symmetry+in+three+dimensions" class="Z3988"></span></li> <li><i>On Quaternions and Octonions</i>, 2003, John Horton Conway and Derek A. Smith <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/978-1-56881-134-5" title="Special:BookSources/978-1-56881-134-5">978-1-56881-134-5</a></li> <li><i>The Symmetries of Things</i> 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, <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/978-1-56881-220-5" title="Special:BookSources/978-1-56881-220-5">978-1-56881-220-5</a></li> <li><b>Kaleidoscopes: Selected Writings of <a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">H.S.M. Coxeter</a></b>, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, <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/978-0-471-01003-6" title="Special:BookSources/978-0-471-01003-6">978-0-471-01003-6</a> <a rel="nofollow" class="external autonumber" href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html">[1]</a></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></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=Cyclic_symmetry_in_three_dimensions&oldid=1189601599">https://en.wikipedia.org/w/index.php?title=Cyclic_symmetry_in_three_dimensions&oldid=1189601599</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">Category</a>: <ul><li><a href="/wiki/Category:Symmetry" title="Category:Symmetry">Symmetry</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 12 December 2023, at 21:55<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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