Myriagon
In geometry, a myriagon or 10000-gon is a polygon with 10000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.
Regular myriagon
A regular myriagon is represented by Schläfli symbol and can be constructed as a truncated 5000-gon, t, or a twice-truncated 2500-gon, tt, or a thrice-truncated 1250-gon, ttt, or a four-fold-truncated 625-gon, tttt.The measure of each internal angle in a regular myriagon is 179.964°. The area of a regular myriagon with sides of length a is given by
The result differs from the area of its circumscribed circle by up to 400 parts per billion.
Because 10,000 = 24 × 54, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular myriagon 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.
Symmetry
The regular myriagon has Dih10000 dihedral symmetry, order 20000, represented by 10000 lines of reflection. Dih10000 has 24 dihedral subgroups:,,,, and. It also has 25 more cyclic symmetries as subgroups:,,,, and, with Zn representing π/''n'' radian rotational symmetry.John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter. r20000 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 myriagons. Only the g10000 subgroup has no degrees of freedom but can be seen as directed edges.