Ditrigonal polyhedron

<h2 id="ditrigonal-vertex-figures">Ditrigonal vertex figures</h2>
<p>There are five uniform ditrigonal polyhedra, all with icosahedral symmetry.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>The three <a href="/facts/Uniform_star_polyhedron/rSv9XpHQ">uniform star polyhedron</a> with <a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbol</a> of the form 3 | <i>p</i> <i>q</i> or ⁠3/2⁠ | <i>p</i> <i>q</i> are ditrigonal, at least if <i>p</i> and <i>q</i> are not 2. Each polyhedron includes two types of faces, being of <a href="/facts/Triangle/cRXUwsMn">triangles</a>, <a href="/facts/Pentagon/xoq5oJeQ">pentagons</a>, or <a href="/facts/Pentagram/cN0ke8tp">pentagrams</a>. Their <a href="/facts/Vertex_configuration/wWaBS6t0">vertex configurations</a> are of the form <i>p</i>.<i>q</i>.<i>p</i>.<i>q</i>.<i>p</i>.<i>q</i> or (<i>p</i>.<i>q</i>)3 with a symmetry of order 3. Here, term ditrigonal refers to a <a href="/facts/Hexagon/DPzuCzWE">hexagon</a> having a symmetry of order 3 (triangular symmetry) acting with 2 rotational orbits on the 6 angles of the vertex figure (the word <i>ditrigonal</i> means "having two sets of 3 angles").<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<table><tbody><tr><th>Type</th><th><a href="/facts/Small_ditrigonal_icosidodecahedron/m01432JP">Small ditrigonal icosidodecahedron</a></th><th><a href="/facts/Ditrigonal_dodecadodecahedron/wCBPXeon">Ditrigonal dodecadodecahedron</a></th><th><a href="/facts/Great_ditrigonal_icosidodecahedron/mlWlwj8h">Great ditrigonal icosidodecahedron</a></th></tr><tr><th>Image</th><td></td><td></td><td></td></tr><tr><th><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></th><td></td><td></td><td></td></tr><tr><th><a href="/facts/Vertex_configuration/wWaBS6t0">Vertex configuration</a></th><td>3.5⁄2.3.5⁄2.3.5⁄2</td><td>5.5⁄3.5.5⁄3.5.5⁄3</td><td>(3.5.3.5.3.5)/2</td></tr><tr><th>Faces</th><td>3220 {3}, 12 { 5⁄2 }</td><td>2412 {5}, 12 { 5⁄2 }</td><td>3220 {3}, 12 {5}</td></tr><tr><th><a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbol</a></th><td>3 | 5/2 3</td><td>3 | 5/3 5</td><td>3 | 3/2 5</td></tr><tr><th><a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagram</a></th><td></td><td></td><td></td></tr></tbody></table>
<h2 id="other-uniform-ditrigonal-polyhedra">Other uniform ditrigonal polyhedra</h2>
<p>The <a href="/facts/Small_ditrigonal_dodecicosidodecahedron/5pUKdXeS">small ditrigonal dodecicosidodecahedron</a> and the <a href="/facts/Great_ditrigonal_dodecicosidodecahedron/69qaP8Jv">great ditrigonal dodecicosidodecahedron</a> are also uniform.
</p><p>Their duals are respectively the <a href="/facts/Small_ditrigonal_dodecacronic_hexecontahedron/H7y56xBk">small ditrigonal dodecacronic hexecontahedron</a> and <a href="/facts/Great_ditrigonal_dodecacronic_hexecontahedron/XCmdPAnS">great ditrigonal dodecacronic hexecontahedron</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Small_complex_icosidodecahedron/c4zVP1dT">Small complex icosidodecahedron</a></li>
<li><a href="/facts/Great_complex_icosidodecahedron/pEPjw0sf">Great complex icosidodecahedron</a></li></ul>

<h3>Notes</h3>

<h3>Bibliography</h3>
<ul><li><a href="/facts/Coxeter/D2l5jWGG">Coxeter, H.S.M.</a>, <a href="/facts/Michael_S._Longuet-Higgins/abuxImrj">M.S. Longuet-Higgins</a> and J.C.P Miller, Uniform Polyhedra, <i>Phil. Trans.</i> 246 A (1954) pp. 401–450.</li>
<li>Har'El, Z. <a href="https://web.archive.org/web/20090715034226/http://www.math.technion.ac.il/~rl/docs/uniform.pdf"><i>Uniform Solution for Uniform Polyhedra.</i></a>, Geometriae Dedicata 47, 57–110, 1993. <a href="https://web.archive.org/web/20090727182130/http://www.math.technion.ac.il/~rl">Zvi Har'El</a>, <a href="https://web.archive.org/web/20110520092545/http://www.math.technion.ac.il/~rl/kaleido/">Kaleido software</a>, <a href="https://web.archive.org/web/20110520080303/http://www.math.technion.ac.il/~rl/kaleido/poly.html">Images</a>, <a href="https://web.archive.org/web/20110520080425/http://www.math.technion.ac.il/~rl/kaleido/dual.html">dual images</a></li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Johnson, N.; <i>The Theory of Uniform Polytopes and Honeycombs</i>, Ph.D. Dissertation, University of Toronto, 1966 <a href="https://web.archive.org/web/20131014175200/https://getinfo.de/app/The-theory-of-uniform-polytopes-and-honeycombs/id/TIBKAT%3A22693604X">[1]</a></li>
<li>Skilling, J. (1975), "The complete set of uniform polyhedra", <i>Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences</i>, 278 (1278): 111–135, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1975RSPTA.278..111S">1975RSPTA.278..111S</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1098%2Frsta.1975.0022">10.1098/rsta.1975.0022</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0080-4614">0080-4614</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/74475">74475</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0365333">0365333</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122634260">122634260</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Har'El, 1993 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Har'El, 1993 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Uniform Polyhedron, Mathworld (retrieved 10 June 2016) <a href="http://mathworld.wolfram.com/UniformPolyhedron.html" target="_blank">http://mathworld.wolfram.com/UniformPolyhedron.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Har'El, 1993 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Ditrigonal polyhedron open-in-new

Ditrigonal polyhedron