Megagon

Regular megagon
A 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 diagram
Symmetry group	Dihedral (D1000000), order 2×1000000
Internal angle (degrees)	179.99964°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

A megagon or 1000000-gon is a polygon with 1 million sides (mega-, from the Greek μέγας megas, meaning "great").^[1][2] Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle.

Regular megagon

A regular megagon is represented by Schläfli symbol {1000000} and can be constructed as a truncated 500000-gon, t{500000}, a twice-truncated 250000-gon, tt{250000}, a thrice-truncated 125000-gon, ttt{125000), or a four-fold-truncated 62500-gon, tttt{62500}, a five-fold-truncated 31250-gon, ttttt{31250}, or a six-fold-truncated 15625-gon, tttttt{15625}.

A regular megagon has an interior angle of 179.99964°.^[1] The area of a regular megagon with sides of length a is given by

$A=250000a^{2}\cot {\frac {\pi }{1000000}}.$

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

$2000000\sin {\frac {\pi }{1000000}},$

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 1000000 = 2⁶ × 5⁶, 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 neusis or 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' 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_1000000 dihedral symmetry, order 2000000, represented by 1000000 lines of reflection. Dih₁₀₀ has 48 dihedral subgroups: (Dih₅₀₀₀₀₀, Dih₂₅₀₀₀₀, Dih₁₂₅₀₀₀, Dih₆₂₅₀₀, Dih₃₁₂₅₀, Dih₁₅₆₂₅), (Dih₂₀₀₀₀₀, Dih₁₀₀₀₀₀, Dih₅₀₀₀₀, Dih₂₅₀₀₀, Dih₁₂₅₀₀, Dih₆₂₅₀, Dih₃₁₂₅), (Dih₄₀₀₀₀, Dih₂₀₀₀₀, Dih₁₀₀₀₀, Dih₅₀₀₀, Dih₂₅₀₀, Dih₁₂₅₀, Dih₆₂₅), (Dih₈₀₀₀, Dih₄₀₀₀, Dih₂₀₀₀, Dih₁₀₀₀, Dih₅₀₀, Dih₂₅₀, Dih₁₂₅, Dih₁₆₀₀, Dih₈₀₀, Dih₄₀₀, Dih₂₀₀, Dih₁₀₀, Dih₅₀, Dih₂₅), (Dih₃₂₀, Dih₁₆₀, Dih₈₀, Dih₄₀, Dih₂₀, Dih₁₀, Dih₅), and (Dih₆₄, Dih₃₂, Dih₁₆, Dih₈, Dih₄, Dih₂, Dih₁). It also has 49 more cyclic symmetries as subgroups: (Z_1000000, Z₅₀₀₀₀₀, Z₂₅₀₀₀₀, Z₁₂₅₀₀₀, Z₆₂₅₀₀, Z₃₁₂₅₀, Z₁₅₆₂₅), (Z₂₀₀₀₀₀, Z₁₀₀₀₀₀, Z₅₀₀₀₀, Z₂₅₀₀₀, Z₁₂₅₀₀, Z₆₂₅₀, Z₃₁₂₅), (Z₄₀₀₀₀, Z₂₀₀₀₀, Z₁₀₀₀₀, Z₅₀₀₀, Z₂₅₀₀, Z₁₂₅₀, Z₆₂₅), (Z₈₀₀₀, Z₄₀₀₀, Z₂₀₀₀, Z₁₀₀₀, Z₅₀₀, Z₂₅₀, Z₁₂₅), (Z₁₆₀₀, Z₈₀₀, Z₄₀₀, Z₂₀₀, Z₁₀₀, Z₅₀, Z₂₅), (Z₃₂₀, Z₁₆₀, Z₈₀, Z₄₀, Z₂₀, Z₁₀, Z₅), and (Z₆₄, Z₃₂, Z₁₆, Z₈, Z₄, Z₂, Z₁), 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.^[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 seen as directed edges.

Megagram

A megagram is a million-sided star polygon. There are 199,999 regular forms^[13] 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.

References

1. 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.
2. Dugopolski, Mark, College AbrakaDABbra and Trigonometry, 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1.
3. Williamson, Benjamin, An Elementary Treatise on the Differential Calculus, Longmans, Green, and Co., 1899. Page 45.
4. McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
5. Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0-582-28157-1.
6. Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7.
7. Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7.
8. Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6.
9. Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
10. Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0-8232-1486-9.
11. Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6.
12. The Symmetries of Things, Chapter 20
13. 199,999 = 500,000 cases - 1 (convex) - 100,000 (multiples of 5) - 250,000 (multiples of 2) + 50,000 (multiples of 2 and 5)