Megagon
| Regular megagon | |
|---|---|
A regular megagon
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| Type | Regular polygon |
| Edges and vertices | 1000000 |
| Schläfli symbol | {1000000}, t{500000}, tt{250000}, ttt{125000}, tttt{62500}, ttttt{31250}, tttttt{15625} |
| Coxeter diagram |
    
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| Symmetry group | Dihedral (D1000000), order 2×1000000 |
| Internal angle (degrees) | 179.99964° |
| Dual polygon | Self |
| Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
A megagon or 1000000-gon is a polygon with 1 million sides (mega-, from the Greek μέγας megas, meaning "great"). Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle.
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.
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.
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Wikipedia