| Regular megagon | |
|---|---|
 A regular megagon | |
| Type | Regular polygon |
| Edges andvertices | 1000000 |
| Schläfli symbol | {1000000}, t{500000}, tt{250000}, ttt{125000}, tttt{62500}, ttttt{31250}, tttttt{15625} |
| Coxeter–Dynkin diagrams |      |
| Symmetry group | Dihedral (D1000000), order 2×1000000 |
| Internal angle (degrees) | 179.99964° |
| Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | Self |
Amegagon or1,000,000-gon (million-gon) is acircle-likepolygon withone million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix denoting a factor of one million).[1][2]
Aregular megagon is represented by theSchläfli symbol {1,000,000} and can be constructed as atruncated 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}.
Aregular megagon has an interior angle of 179°59'58.704" or3.14158637 radians.[1] Thearea of aregular megagon with sides of lengtha is given by
Theperimeter of a regular megagon inscribed in the unitcircle is:
which is exceedingly close to2π. In fact, for a circle the size of theEarth's equator, with acircumference 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 distinctFermat primes and a power of two. Thus the regular megagon is not aconstructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinctPierpont primes, nor a product of powers of two and three.
LikeRené Descartes's example of thechiliagon, 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]
Theregular megagon has Dih1,000,000dihedral 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 morecyclic 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 anda1 labels no symmetry. He givesd (diagonal) with mirror lines through vertices,p with mirror lines through edges (perpendicular),i with mirror lines through both vertices and edges, andg for rotational symmetry.
These lower symmetries allows degrees of freedom in defining irregular megagons. Only theg1000000 subgroup has no degrees of freedom but can be seen asdirected edges.
A megagram is a million-sidedstar polygon. There are 199,999 regular forms[a] given bySchläfli symbols of the form {1000000/n}, wheren is an integer between 2 and 500,000 that iscoprime to 1,000,000. There are also 300,000 regularstar figures in the remaining cases.