Megagon
A megagon or 1,000,000-gon is a circle-like polygon with one million sides.
Regular megagon
A regular megagon is represented by the Schläfli symbol and can be constructed as a truncated 500,000-gon, t, a twice-truncated 250,000-gon, tt, a thrice-truncated 125,000-gon, ttt, or a four-fold-truncated 62,500-gon, tttt, a five-fold-truncated 31,250-gon,, or a six-fold-truncated 15,625-gon,.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
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 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.
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 Dih1,000,000 dihedral symmetry, order 2,000,000, represented by 1,000,000 lines of reflection. Dih1,000,000 has 48 dihedral subgroups:,,,,, and. It also has 49 more cyclic symmetries as subgroups:,,,,,, and, with Zn representing π/n radian rotational symmetry.John [Horton Conway|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 with mirror lines through vertices, p with mirror lines through edges, 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.