# Megagon

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).

## Regular megagon

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. The area of a regular megagon with sides of length a is given by
    $A=250,000\ a^{2}\cot {\frac {\pi }{1,000,000}}.$
The perimeter of a regular megagon inscribed in the unit circle is:
    $2,000,000\ \sin {\frac {\pi }{1,000,000}},$
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.
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.

## Philosophical application

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.
The megagon is also used as an illustration of the convergence of regular polygons to a circle.

## Symmetry

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. 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.

## Megagram

A megagram is a million-sided star polygon. There are 199,999 regular forms 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.