Regular myriagon | |
---|---|

Type | Regular polygon |

Edges and vertices | 10000 |

Schläfli symbol | {10000}, t{5000}, tt{2500}, ttt{1250}, tttt{625} |

Coxeter–Dynkin diagrams | |

Symmetry group | Dihedral (D_{10000}), order 2×10000 |

Internal angle (degrees) | 179.964° |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

Dual polygon | Self |

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.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}

A regular myriagon is represented by Schläfli symbol {10,000} and can be constructed as a 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 myriagon with sides of length *a* is given by

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

Because 10,000 = 2^{4} × 5^{4}, the number of sides is neither a product of distinct Fermat primes 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 primes, nor a product of powers of two and three.

The *regular myriagon* has Dih_{10000} dihedral symmetry, order 20000, represented by 10000 lines of reflection. Dih_{10000} has 24 dihedral subgroups: (Dih_{5000}, Dih_{2500}, Dih_{1250}, Dih_{625}), (Dih_{2000}, Dih_{1000}, Dih_{500}, Dih_{250}, Dih_{125}), (Dih_{400}, Dih_{200}, Dih_{100}, Dih_{50}, Dih_{25}), (Dih_{80}, Dih_{40}, Dih_{20}, Dih_{10}, Dih_{5}), and (Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). It also has 25 more cyclic symmetries as subgroups: (Z_{10000}, Z_{5000}, Z_{2500}, Z_{1250}, Z_{625}), (Z_{2000}, Z_{1000}, Z_{500}, Z_{250}, Z_{125}), (Z_{400}, Z_{200}, Z_{100}, Z_{50}, Z_{25}), (Z_{80}, Z_{40}, Z_{20}, Z_{10}), and (Z_{16}, Z_{8}, Z_{4}, Z_{2}, Z_{1}), with Z_{n} representing *π*/*n* radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[6]} **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 edges.

A myriagram is a 10,000-sided star polygon. There are 1999 regular forms^{[a]} given by Schläfli symbols 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 figures in the remaining cases.

In the novella *Flatland*, the Chief Circle is assumed to have ten thousand sides, making him a myriagon.