# Megagon

A **megagon** or **1,000,000-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]

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 |

## 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"
3.14158637 rad.[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.

## 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.[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]

## 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.[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.

## Megagram

A megagram is a million-sided star polygon. There are 199,999 regular forms[lower-alpha 1] 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.

## Notes

- 199,999 = 500,000 cases - 1 (convex) - 100,000 (multiples of 5) - 250,000 (multiples of 2) + 50,000 (multiples of 2 and 5)

## References

- Darling, David J.,
*The Universal Book of Mathematics: from Abracadabra to Zeno's Paradoxes*, John Wiley & Sons, 2004. Page 249. ISBN 0-471-27047-4. - Dugopolski, Mark,
*College AbrakaDABbra and Trigonometry*, 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1. - Williamson, Benjamin,
*An Elementary Treatise on the Differential Calculus*, Longmans, Green, and Co., 1899. Page 45. - McCormick, John Francis,
*Scholastic Metaphysics*, Loyola University Press, 1928, p. 18. - Merrill, John Calhoun and Odell, S. Jack,
*Philosophy and Journalism*, Longman, 1983, p. 47, ISBN 0-582-28157-1. - Hospers, John,
*An Introduction to Philosophical Analysis*, 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7. - Mandik, Pete,
*Key Terms in Philosophy of Mind*, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7. - Kenny, Anthony,
*The Rise of Modern Philosophy*, Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6. - Balmes, James,
*Fundamental Philosophy, Vol II*, Sadlier and Co., Boston, 1856, p. 27. - Potter, Vincent G.,
*On Understanding Understanding: A Philosophy of Knowledge*, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0-8232-1486-9. - Russell, Bertrand,
*History of Western Philosophy*, reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6. -
**The Symmetries of Things**, Chapter 20