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

${\displaystyle A=250,000\ a^{2}\cot {\frac {\pi }{1,000,000}}.}$

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

${\displaystyle 2,000,000\ \sin {\frac {\pi }{1,000,000}},}$

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.[3]

Because 1,000,000 = 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 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, Dih₆₂₅), (Dih_8,000, Dih_4,000, Dih_2,000, Dih_1,000, Dih₅₀₀, Dih₂₅₀, Dih₁₂₅, Dih_1,600, 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_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, Z₆₂₅), (Z_8,000, Z_4,000, Z_2,000, Z_1,000, Z₅₀₀, Z₂₅₀, Z₁₂₅), (Z_1,600, 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 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[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.

• Chiliagon
• Myriagon