Myriagon

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 (D10000), order 2×10000
Internal angle (degrees)	179.964°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a myriagon is a polygon with 10000 sides or 10000-gon. Several philosophers have used the regular myriagon to illustrate issues regarding thought.^{[1][2][3][4][5]}

Regular myriagon

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

$A = 2500a^2 \cot \frac{\pi}{10000}$

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.

Symmetry

The symmetries of a regular myriagon. Light blue lines show subgroups of index 2. The 5 boxed subgraphs are positionally related by index 5 subgroups.

The regular myriagon has Dih₁₀₀₀₀ dihedral 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 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.

Myriagram

A myriagram is an 10000-sided 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.

See also

• Chiliagon
• Megagon

References

1. Meditation VI by Descartes (English translation).
2. Hippolyte Taine, On Intelligence: pp. 9–10
3. Jacques Maritain, An Introduction to Philosophy: p. 108
4. Alan Nelson (ed.), A Companion to Rationalism: p. 285
5. Paolo Fabiani, The philosophy of the imagination in Vico and Malebranche: p. 222
6. The Symmetries of Things, Chapter 20
7. 5000 cases - 1 (convex) - 1000 (multiples of 5) - 2500 (multiples of 2)+ 500 (multiples of 2 and 5)