View source for Myriagon

{{short description|Polygon with 10000 edges}}
{{Regular polygon db|Regular polygon stat table|p10000}}
In [[geometry]], a '''myriagon''' or 10000-gon is a [[polygon]] with 10000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.<ref name="meditations">[[wikisource:Meditations on First Philosophy/Meditation VI|Meditation VI]] by Descartes (English translation).</ref><ref>[https://books.google.com/books?id=za5-AAAAMAAJ&dq=myriagon&pg=PA9 Hippolyte Taine, ''On Intelligence'']: pp.&nbsp;9–10.</ref><ref>[https://books.google.com/books?id=YWw40mKMLAsC&dq=myriagon&pg=PA108 Jacques Maritain, ''An Introduction to Philosophy'']: p.&nbsp;108.</ref><ref>[https://books.google.com/books?id=Awc5wgbeeW0C&dq=myriagon&pg=PT285 Alan Nelson (ed.), ''A Companion to Rationalism'']: p.&nbsp;285.</ref><ref>[https://books.google.com/books?id=JPhx55N9r64C&dq=myriagon&pg=PA222 Paolo Fabiani, ''The Philosophy of the Imagination in Vico and Malebranche'']: p.&nbsp;222.</ref>

== Regular myriagon==
A [[regular polygon|regular]] myriagon is represented by [[Schläfli symbol]] {10,000} and can be constructed as a [[Truncation (geometry)|truncated]] 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}.

The measure of each [[internal angle]] in a regular myriagon is 179.964°. The [[area]] of a [[regular polygon|regular]] myriagon with sides of length ''a'' is given by
:<math>A = 2500a^2 \cot \frac{\pi}{10000}</math>

The result differs from the area of its [[circumscribed circle]] by up to 40 [[parts per billion]].

Because 10,000 = 2<sup>4</sup> &times; 5<sup>4</sup>, the number of sides is neither a product of distinct [[Fermat prime]]s nor a power of two. Thus the regular myriagon is not a [[constructible polygon]]. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct [[Pierpont prime]]s, nor a product of powers of two and three.

== Symmetry==
[[File:Symmetries of myriagon.png|thumb|240px|The symmetries of a regular myriagon. Light blue lines show subgroups of index 2. The 5 boxed subgraphs are positionally related by index 5 subgroups.]]

The ''regular myriagon'' has Dih<sub>10000</sub> [[dihedral symmetry]], order 20000, represented by 10000 lines of reflection. Dih<sub>10000</sub> has 24 dihedral subgroups: (Dih<sub>5000</sub>, Dih<sub>2500</sub>, Dih<sub>1250</sub>, Dih<sub>625</sub>), (Dih<sub>2000</sub>, Dih<sub>1000</sub>, Dih<sub>500</sub>, Dih<sub>250</sub>, Dih<sub>125</sub>), (Dih<sub>400</sub>, Dih<sub>200</sub>, Dih<sub>100</sub>, Dih<sub>50</sub>, Dih<sub>25</sub>), (Dih<sub>80</sub>, Dih<sub>40</sub>, Dih<sub>20</sub>, Dih<sub>10</sub>, Dih<sub>5</sub>), and (Dih<sub>16</sub>, Dih<sub>8</sub>, Dih<sub>4</sub>, Dih<sub>2</sub>, Dih<sub>1</sub>). It also has 25 more [[cyclic group|cyclic]] symmetries as subgroups: (Z<sub>10000</sub>, Z<sub>5000</sub>, Z<sub>2500</sub>, Z<sub>1250</sub>, Z<sub>625</sub>), (Z<sub>2000</sub>, Z<sub>1000</sub>, Z<sub>500</sub>, Z<sub>250</sub>, Z<sub>125</sub>), (Z<sub>400</sub>, Z<sub>200</sub>, Z<sub>100</sub>, Z<sub>50</sub>, Z<sub>25</sub>), (Z<sub>80</sub>, Z<sub>40</sub>, Z<sub>20</sub>, Z<sub>10</sub>), and (Z<sub>16</sub>, Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>, Z<sub>1</sub>), with Z<sub>''n''</sub> representing ''&pi;''/''n'' radian rotational symmetry.

[[John Horton Conway|John Conway]] labels these lower symmetries with a letter and order of the symmetry follows the letter.<ref>'''The Symmetries of Things''', Chapter 20</ref> '''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. 

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 [[directed edge]]s.

==Myriagram==
A myriagram is a 10,000-sided [[star polygon]]. There are 1999 regular forms{{efn|content=5000 cases − 1 (convex) − 1,000 (multiples of 5) − 2,500 (multiples of 2) + 500 (multiples of 2 and 5)}} given by [[Schläfli symbol]]s of the form {10000/''n''}, where ''n'' is an integer between 2 and 5,000 that is [[coprime]] to 10,000. There are also 3000 regular [[star figure]]s in the remaining cases.

==In popular culture==
In the novella ''[[Flatland]]'', the Chief Circle is assumed to have ten thousand sides, making him a myriagon.

==See also==
* [[Chiliagon]]
* [[Megagon]]

==Notes==
{{notelist}}

==References==
{{reflist}}

{{Polygons}}

[[Category:Polygons by the number of sides]]