Myriagon

<h2 id="regular-myriagon">Regular myriagon</h2>
<p>A <a href="/facts/Regular_polygon/ws89Nhpt">regular</a> myriagon is represented by <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> {10,000} and can be constructed as a <a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> 5000-gon, t{5000}, or a twice-truncated 2500-gon, tt{2500}, or a thrice-truncated 1250-gon, ttt{1250}, or a four-fold-truncated 625-gon, tttt{625}.
</p><p>The measure of each <a href="/facts/Internal_angle/jd78Qkbt">internal angle</a> in a regular myriagon is 179.964°. The <a href="/facts/Area/KsNG1G9t">area</a> of a <a href="/facts/Regular_polygon/ws89Nhpt">regular</a> myriagon with sides of length <i>a</i> is given by
</p>

A
        =
        2500
        
          a
          
            2
          
        
        cot
        ⁡
        
          
            π
            10000
          
        
      
    
    {\displaystyle A=2500a^{2}\cot {\frac {\pi }{10000}}}

<p>The result differs from the area of its <a href="/facts/Circumscribed_circle/RafYT4am">circumscribed circle</a> by up to 400 <a href="/facts/Parts_per_billion/PozRuoLM">parts per billion</a>.
</p><p>Because 10,000 = 24 × 54, the number of sides is neither a product of distinct <a href="/facts/Fermat_prime/SJT8fN9m">Fermat primes</a> nor a power of two. Thus the regular myriagon is not a <a href="/facts/Constructible_polygon/ATBRc55D">constructible polygon</a>. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct <a href="/facts/Pierpont_prime/MGNnTvpJ">Pierpont primes</a>, nor a product of powers of two and three.
</p>
<h2 id="symmetry">Symmetry</h2>

<p>The <i>regular myriagon</i> has Dih10000 <a href="/facts/Dihedral_symmetry/1PUlRh4M">dihedral symmetry</a>, order 20000, represented by 10000 lines of reflection. Dih10000 has 24 dihedral subgroups: (Dih5000, Dih2500, Dih1250, Dih625), (Dih2000, Dih1000, Dih500, Dih250, Dih125), (Dih400, Dih200, Dih100, Dih50, Dih25), (Dih80, Dih40, Dih20, Dih10, Dih5), and (Dih16, Dih8, Dih4, Dih2, Dih1). It also has 25 more <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a> symmetries as subgroups: (Z10000, Z5000, Z2500, Z1250, Z625), (Z2000, Z1000, Z500, Z250, Z125), (Z400, Z200, Z100, Z50, Z25), (Z80, Z40, Z20, Z10), and (Z16, Z8, Z4, Z2, Z1), with Z<i>n</i> representing <i>π</i>/<i>n</i> radian rotational symmetry.
</p><p><a href="/facts/John_Horton_Conway/g3PA1zoK">John Conway</a> labels these lower symmetries with a letter and order of the symmetry follows the letter.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> r20000 represents full symmetry, and a1 labels no symmetry.  He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. 
</p><p>These lower symmetries allows degrees of freedom in defining irregular myriagons. Only the g10000 subgroup has no degrees of freedom but can be seen as <a href="/facts/Directed_edge/YBdfQ0um">directed edges</a>.
</p>
<h2 id="myriagram">Myriagram</h2>
<p>A myriagram is a 10,000-sided <a href="/facts/Star_polygon/mDfuIe81">star polygon</a>. There are 1999 regular forms<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> given by <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbols</a> of the form {10000/<i>n</i>}, where <i>n</i> is an integer between 2 and 5,000 that is <a href="/facts/Coprime/bnCxCXxs">coprime</a> to 10,000. There are also 3000 regular <a href="/facts/Star_figure/I4naM8nS">star figures</a> in the remaining cases.
</p>
<h2 id="in-popular-culture">In popular culture</h2>
<p>In the novella <i><a href="/facts/Flatland/QN8eyl1p">Flatland</a></i>, the Chief Circle is assumed to have ten thousand sides, making him a myriagon.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Chiliagon/ynv7L6w2">Chiliagon</a></li>
<li><a href="/facts/Megagon/p5k4h60A">Megagon</a></li></ul>
<h2 id="notes">Notes</h2>

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

<ol>
<li id="fn:1"><p>Meditation VI by Descartes (English translation). <a href="https://en.wikisource.org/wiki/Meditations_on_First_Philosophy/Meditation_VI" target="_blank">https://en.wikisource.org/wiki/Meditations_on_First_Philosophy/Meditation_VI</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Hippolyte Taine, On Intelligence: pp. 9–10. <a href="https://books.google.com/books?id=za5-AAAAMAAJ&dq=myriagon&pg=PA9" target="_blank">https://books.google.com/books?id=za5-AAAAMAAJ&dq=myriagon&pg=PA9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Jacques Maritain, An Introduction to Philosophy: p. 108. <a href="https://books.google.com/books?id=YWw40mKMLAsC&dq=myriagon&pg=PA108" target="_blank">https://books.google.com/books?id=YWw40mKMLAsC&dq=myriagon&pg=PA108</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Alan Nelson (ed.), A Companion to Rationalism: p. 285. <a href="https://books.google.com/books?id=Awc5wgbeeW0C&dq=myriagon&pg=PT285" target="_blank">https://books.google.com/books?id=Awc5wgbeeW0C&dq=myriagon&pg=PT285</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Paolo Fabiani, The Philosophy of the Imagination in Vico and Malebranche: p. 222. <a href="https://books.google.com/books?id=JPhx55N9r64C&dq=myriagon&pg=PA222" target="_blank">https://books.google.com/books?id=JPhx55N9r64C&dq=myriagon&pg=PA222</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>The Symmetries of Things, Chapter 20 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>5000 cases − 1 (convex) − 1,000 (multiples of 5) − 2,500 (multiples of 2) + 500 (multiples of 2 and 5) <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Myriagon open-in-new

Myriagon