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Regular myriagon  

A regular myriagon


Type  Regular polygon 
Edges and vertices  10000 
Schläfli symbol  {10000}, t{5000}, tt{2500}, ttt{1250}, tttt{625} 
Coxeter diagram  
Symmetry group  Dihedral (D_{10000}), order 2×10000 
Internal angle (degrees)  179.964° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a myriagon or 10000gon is a polygon with 10000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}
A regular myriagon is represented by Schläfli symbol {10000} and can be constructed as a truncated 5000gon, t{5000}, or a twicetruncated 2500gon, tt{2500}, or a thricetruncated 1250gon, ttt{1250), or a fourfoldtruncated 625gon, tttt{625}.
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 40 parts per billion.
Because 10000 = 2^{4} × 5^{4}, 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 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.
The regular myriagon has Dih_{10000} dihedral symmetry, order 20000, represented by 10000 lines of reflection. Dih_{100} has 24 dihedral subgroups: (Dih_{5000}, Dih_{2500}, Dih_{1250}, Dih_{625}), (Dih_{2000}, Dih_{1000}, Dih_{500}, Dih_{250}, Dih_{125}), (Dih_{400}, Dih_{200}, Dih_{100}, Dih_{50}, Dih_{25}), (Dih_{80}, Dih_{40}, Dih_{20}, Dih_{10}, Dih_{5}), and (Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). It also has 25 more cyclic symmetries as subgroups: (Z_{10000}, Z_{5000}, Z_{2500}, Z_{1250}, Z_{625}), (Z_{2000}, Z_{1000}, Z_{500}, Z_{250}, Z_{125}), (Z_{400}, Z_{200}, Z_{100}, Z_{50}, Z_{25}), (Z_{80}, Z_{40}, Z_{20}, Z_{10}), and (Z_{16}, Z_{8}, Z_{4}, Z_{2}, Z_{1}), with Z_{n} representing π/n radian rotational symmetry.
John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[6]} r20000 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 myriagons. Only the g10000 subgroup has no degrees of freedom but can seen as directed edges.
A myriagram is a 10000sided star polygon. There are 1999 regular forms^{[7]} given by Schläfli symbols of the form {10000/n}, where n is an integer between 2 and 5000 that is coprime to 10000. There are also 3000 regular star figures in the remaining cases.
In the novella Flatland, the Chief Circle is assumed to have tenthousand sides, making him a myriagon.