Megagon

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°. 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.

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.

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

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. 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 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.