<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="//commons.wikimedia.org/wiki/File:Coxeter_graph_related_to_Klein_quartic.svg" class="image"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Coxeter_graph_related_to_Klein_quartic.svg/220px-Coxeter_graph_related_to_Klein_quartic.svg.png" decoding="async" width="220" height="211" class="thumbimage" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Coxeter_graph_related_to_Klein_quartic.svg/330px-Coxeter_graph_related_to_Klein_quartic.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Coxeter_graph_related_to_Klein_quartic.svg/440px-Coxeter_graph_related_to_Klein_quartic.svg.png 2x" data-file-width="626" data-file-height="599"></a>  <div class="thumbcaption"><div class="magnify"><a href="//commons.wikimedia.org/wiki/File:Coxeter_graph_related_to_Klein_quartic.svg" class="internal" title="Enlarge"></a></div>Coxeter graph</div></div></div>
<p>The <a href="https://en.wikipedia.org/wiki/fundamental_domain" class="extiw" title="w:fundamental domain">fundamental domain</a> of the <a href="https://en.wikipedia.org/wiki/Klein_quartic" class="extiw" title="w:Klein quartic">Klein quartic</a> has 336 right triangles.<br>
They have three types of sides, hereafter called <i>segments</i>: 
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<ul><li>short <a href="https://en.wikipedia.org/wiki/Cathetus" class="extiw" title="w:Cathetus">leg</a> (S) between 3-fold and 4-fold points</li>
<li>long leg (L) between 7-fold and 4-fold points</li>
<li><a href="https://en.wikipedia.org/wiki/hypotenuse" class="extiw" title="w:hypotenuse">hypotenuse</a> (<b>H</b>) between 3-fold and 7-fold points</li></ul><p>The 504 segments form 28 closed geodesics, each consisting of 18 segments of these types:<br>
S<b>H</b>L L<b>H</b>S S<b>H</b>L L<b>H</b>S S<b>H</b>L L<b>H</b>S 
</p><p>They are marked with 7 colors and 4 line styles.<br>
Geodesics with less dominant line styles appear more separated in the 14-gon:
</p>
<ul><li>Thick are cut in 2 parts: LL<b>H</b>SS<b>H</b>LL<b>H</b>SS<b>H</b> (12), LL<b>H</b>SS<b>H</b> (6)   <small>(These are the mirror axis and the 14-gon edge, which appears twice.)</small></li>
<li>Thin are cut in 3 parts: LL<b>H</b>SS<b>H</b>LL (8), <b>H</b>SS<b>H</b>L (5), L<b>H</b>SS<b>H</b> (5)</li>
<li>Dashed are cut in 4 parts: <b>H</b>SS<b>H</b>LL<b>H</b> (7), SS (2), <b>H</b>LL<b>H</b>SS<b>H</b> (7), LL (2)</li>
<li>Dotted are cut in 5 parts: SS<b>H</b>LL<b>H</b>SS (8), <b>H</b>LL (3), <b>H</b>S (2), S<b>H</b> (2), LL<b>H</b> (3)</li></ul><p>The geodesics correspond to the vertices of the <a href="https://en.wikipedia.org/wiki/Coxeter_graph" class="extiw" title="w:Coxeter graph">Coxeter graph</a>.<br>
42 pairs of geodesics have two rectangular crossings <small>(each in the middle of one of the 84 rhombi)</small>.<br></p>
These pairs correspond to the edges of the Coxeter graph.
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