A megagon or 1,000,000-gon (million-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]
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" or3.14158637 radians.[2] The area of a regular megagon with sides of length a is given by
A=250,000 a2\cot
| \pi | |
| 1,000,000 |
.
The perimeter of a regular megagon inscribed in the unit circle is:
2,000,000 \sin
| \pi | |
| 1,000,000 |
,
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 Dih1,000,000 dihedral symmetry, order 2,000,000, represented by 1,000,000 lines of reflection. Dih1,000,000 has 48 dihedral subgroups: (Dih500,000, Dih250,000, Dih125,000, Dih62,500, Dih31,250, Dih15,625), (Dih200,000, Dih100,000, Dih50,000, Dih25,000, Dih12,500, Dih6,250, Dih3,125), (Dih40,000, Dih20,000, Dih10,000, Dih5,000, Dih2,500, Dih1,250, Dih625), (Dih8,000, Dih4,000, Dih2,000, Dih1,000, Dih500, Dih250, Dih125, Dih1,600, 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: (Z1,000,000, Z500,000, Z250,000, Z125,000, Z62,500, Z31,250, Z15,625), (Z200,000, Z100,000, Z50,000, Z25,000, Z12,500, Z6,250, Z3,125), (Z40,000, Z20,000, Z10,000, Z5,000, Z2,500, Z1,250, Z625), (Z8,000, Z4,000, Z2,000, Z1,000, Z500, Z250, Z125), (Z1,600, Z800, Z400, Z200, Z100, Z50, Z25), (Z320, Z160, Z80, Z40, Z20, Z10, Z5), and (Z64, Z32, Z16, Z8, Z4, Z2, Z1), with Zn 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 given by Schläfli symbols of the form, 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.