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Hexagonal antiprism
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<h2 id="crossed-antiprism">Crossed antiprism</h2> <p>A crossed hexagonal antiprism is a <a href="/facts/Star_polyhedron/XyeJWj3d">star polyhedron</a>, topologically identical to the convex <i>hexagonal antiprism</i> with the same <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a>, but it can't be made uniform; the sides are <a href="/facts/Isosceles_triangle/rvLe3yqK">isosceles triangles</a>. Its <a href="/facts/Vertex_configuration/wWaBS6t0">vertex configuration</a> is 3.3/2.3.6, with one triangle retrograde. It has D6d symmetry, order 24. </p> <h2 id="related-polyhedra">Related polyhedra</h2> <p>The hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles. </p> <table><tbody><tr><th colspan="15">Uniform hexagonal dihedral spherical polyhedra</th></tr><tr><th colspan="7"><a href="/facts/List_of_spherical_symmetry_groups/3X3YcHoQ">Symmetry</a>: <a href="/facts/Dihedral_symmetry/1PUlRh4M">[6,2]</a>, (*622)</th><th>[6,2]+, (622)</th><th>[6,2+], (2*3)</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr><tr><td><a href="/facts/Dihedron/YkNgovlv">{6,2}</a></td><td><a href="/facts/Dihedron/YkNgovlv">t{6,2}</a></td><td><a href="/facts/Dihedron/YkNgovlv">r{6,2}</a></td><td><a href="/facts/Hexagonal_prism/oxYc0nF2">t{2,6}</a></td><td><a href="/facts/Hosohedron/t2dyV0bW">{2,6}</a></td><td><a href="/facts/Hexagonal_prism/oxYc0nF2">rr{6,2}</a></td><td><a href="/facts/Dodecagonal_prism/50AfF1sz">tr{6,2}</a></td><td>sr{6,2}</td><td><a href="/facts/Octahedron/G38q92xW">s{2,6}</a></td></tr><tr><th colspan="10">Duals to uniforms</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td><a href="/facts/Hosohedron/t2dyV0bW">V62</a></td><td><a href="/facts/Hosohedron/t2dyV0bW">V122</a></td><td><a href="/facts/Hosohedron/t2dyV0bW">V62</a></td><td><a href="/facts/Hexagonal_bipyramid/LXcnCg3M">V4.4.6</a></td><td><a href="/facts/Dihedron/YkNgovlv">V26</a></td><td><a href="/facts/Hexagonal_bipyramid/LXcnCg3M">V4.4.6</a></td><td><a href="/facts/Bipyramid/MkzrCA0W">V4.4.12</a></td><td><a href="/facts/Hexagonal_trapezohedron/oR1jFD6P">V3.3.3.6</a></td><td><a href="/facts/Cube/65lUaAqN">V3.3.3.3</a></td></tr></tbody></table> Family of <a href="/facts/Uniform_polyhedron/PHnprqRC">uniform</a><i>n</i>-gonal <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a><ul><li>v</li><li>t</li><li>e</li></ul><table> <tbody><tr><th>Antiprism name</th><th><a href="/facts/Disphenoid/7rU7jc7U">Digonal antiprism</a></th><th>(Trigonal)<a href="/facts/Octahedron/G38q92xW">Triangular antiprism</a></th><th>(Tetragonal)<a href="/facts/Square_antiprism/4514Q02s">Square antiprism</a></th><th><a href="/facts/Pentagonal_antiprism/gR8aMBKN">Pentagonal antiprism</a></th><th>Hexagonal antiprism</th><th><a href="/facts/Heptagonal_antiprism/gh9eMN8a">Heptagonal antiprism</a></th><th>...</th><th><a href="/facts/Apeirogonal_antiprism/JJ49cEzm">Apeirogonal antiprism</a></th></tr><tr><th>Polyhedron image</th><td></td><td></td><td></td><td></td><td></td><td></td><td>...</td><td></td></tr><tr><th>Spherical tiling image</th><td></td><td></td><td></td><td></td><td></td><td></td><th>Plane tiling image</th><td></td></tr><tr><th><a href="/facts/Vertex_configuration/wWaBS6t0">Vertex config.</a></th><td>2.3.3.3</td><td>3.3.3.3</td><td>4.3.3.3</td><td>5.3.3.3</td><td>6.3.3.3</td><td>7.3.3.3</td><td>...</td><td>∞.3.3.3</td></tr></tbody></table> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Antiprism.html">"Antiprism"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="https://web.archive.org/web/20080616071808/http://polyhedra.org/poly/show/29/hexagonal_antiprism">Hexagonal Antiprism: Interactive Polyhedron model</a></li> <li><a href="http://www.georgehart.com/virtual-polyhedra/vp.html">Virtual Reality Polyhedra</a> www.georgehart.com: The Encyclopedia of Polyhedra <ul><li><a href="/facts/VRML/EkNXun7G">VRML</a> <a href="http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrl">model</a> <a href="https://web.archive.org/web/20070207112806/http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrl">Archived</a> 2007-02-07 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li></ul></li> <li><a href="http://levskaya.github.io/polyhedronisme/?recipe=C100A6">polyhedronisme</a> A6</li></ul>