Regular megagon | |
---|---|

Type | Regular polygon |

Edges and vertices | 1000000 |

Schläfli symbol | {1000000}, t{500000}, tt{250000}, ttt{125000}, tttt{62500}, ttttt{31250}, tttttt{15625} |

Coxeter–Dynkin diagrams | |

Symmetry group | Dihedral (D_{1000000}), order 2×1000000 |

Internal angle (degrees) | 179.99964° |

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

Dual polygon | Self |

A **megagon** or **1,000,000-gon** (million-gon) is a polygon with one million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix denoting a factor of one million).^{[1]}^{[2]}

A regular megagon is represented by the Schläfli symbol {1,000,000} and can be constructed as a truncated 500,000-gon, t{500,000}, a twice-truncated 250,000-gon, tt{250,000}, a thrice-truncated 125,000-gon, ttt{125,000}, or a four-fold-truncated 62,500-gon, tttt{62,500}, a five-fold-truncated 31,250-gon, ttttt{31,250}, or a six-fold-truncated 15,625-gon, tttttt{15,625}.

A regular megagon has an interior angle of 179°59'58.704" or
3.14158637 radians.^{[1]} The area of a regular megagon with sides of length *a* is given by

The perimeter of a regular megagon inscribed in the unit circle is:

which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 40,075 kilometres, one edge of a megagon inscribed in such a circle would be slightly over 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters.^{[3]}

Because 1,000,000 = 2^{6} × 5^{6}, the number of sides is not a product of distinct Fermat primes and a power of two. Thus the regular megagon 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.

Like René Descartes's example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}

The megagon is also used as an illustration of the convergence of regular polygons to a circle.^{[11]}

The *regular megagon* has Dih_{1,000,000} dihedral symmetry, order 2,000,000, represented by 1,000,000 lines of reflection. Dih_{1,000,000} has 48 dihedral subgroups: (Dih_{500,000}, Dih_{250,000}, Dih_{125,000}, Dih_{62,500}, Dih_{31,250}, Dih_{15,625}), (Dih_{200,000}, Dih_{100,000}, Dih_{50,000}, Dih_{25,000}, Dih_{12,500}, Dih_{6,250}, Dih_{3,125}), (Dih_{40,000}, Dih_{20,000}, Dih_{10,000}, Dih_{5,000}, Dih_{2,500}, Dih_{1,250}, Dih_{625}), (Dih_{8,000}, Dih_{4,000}, Dih_{2,000}, Dih_{1,000}, Dih_{500}, Dih_{250}, Dih_{125}, Dih_{1,600}, Dih_{800}, Dih_{400}, Dih_{200}, Dih_{100}, Dih_{50}, Dih_{25}), (Dih_{320}, Dih_{160}, Dih_{80}, Dih_{40}, Dih_{20}, Dih_{10}, Dih_{5}), and (Dih_{64}, Dih_{32}, Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). It also has 49 more cyclic symmetries as subgroups: (Z_{1,000,000}, Z_{500,000}, Z_{250,000}, Z_{125,000}, Z_{62,500}, Z_{31,250}, Z_{15,625}), (Z_{200,000}, Z_{100,000}, Z_{50,000}, Z_{25,000}, Z_{12,500}, Z_{6,250}, Z_{3,125}), (Z_{40,000}, Z_{20,000}, Z_{10,000}, Z_{5,000}, Z_{2,500}, Z_{1,250}, Z_{625}), (Z_{8,000}, Z_{4,000}, Z_{2,000}, Z_{1,000}, Z_{500}, Z_{250}, Z_{125}), (Z_{1,600}, Z_{800}, Z_{400}, Z_{200}, Z_{100}, Z_{50}, Z_{25}), (Z_{320}, Z_{160}, Z_{80}, Z_{40}, Z_{20}, Z_{10}, Z_{5}), and (Z_{64}, Z_{32}, Z_{16}, Z_{8}, Z_{4}, Z_{2}, Z_{1}), with Z_{n} representing π/*n* radian rotational symmetry.

John Conway labeled these lower symmetries with a letter and order of the symmetry follows the letter.^{[12]} **r2000000** 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 megagons. Only the **g1000000** subgroup has no degrees of freedom but can be seen as directed edges.

A megagram is a million-sided star polygon. There are 199,999 regular forms^{[a]} given by Schläfli symbols of the form {1000000/*n*}, where *n* is an integer between 2 and 500,000 that is coprime to 1,000,000. There are also 300,000 regular star figures in the remaining cases.