Myriagon
| Regular myriagon | |
|---|---|
A regular myriagon
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| Type | Regular polygon |
| Edges and vertices | 10000 |
| Schläfli symbol | {10000}, t{5000}, tt{2500}, ttt{1250}, tttt{625} |
| Coxeter diagram |
    
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| Symmetry group | Dihedral (D10000), order 2×10000 |
| Internal angle (degrees) | 179.964° |
| Dual polygon | Self |
| Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
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.
A regular myriagon is represented by Schläfli symbol {10000} and can be constructed as a truncated 5000-gon, t{5000}, or a twice-truncated 2500-gon, tt{2500}, or a thrice-truncaed 1250-gon, ttt{1250), or a four-fold-truncated 625-gon, 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 = 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 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 Dih10000dihedral symmetry, order 20000, represented by 10000 lines of reflection. Dih100 has 24 dihedral subgroups: (Dih5000, Dih2500, Dih1250, Dih625), (Dih2000, Dih1000, Dih500, Dih250, Dih125), (Dih400, Dih200, Dih100, Dih50, Dih25), (Dih80, Dih40, Dih20, Dih10, Dih5), and (Dih16, Dih8, Dih4, Dih2, Dih1). It also has 25 more cyclic symmetries as subgroups: (Z10000, Z5000, Z2500, Z1250, Z625), (Z2000, Z1000, Z500, Z250, Z125), (Z400, Z200, Z100, Z50, Z25), (Z80, Z40, Z20, Z10), and (Z16, Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.
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