Myriagon

Regular myriagon
Regular myriagon
	cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

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.^[1]^[2]^[3]^[4]^[5]

Regular myriagon

A regular myriagon is represented by Schläfli symbol {10,000} and can be constructed as a truncated 5000-gon, t{5000}, or a twice-truncated 2500-gon, tt{2500}, or a thrice-truncated 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

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: (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 be seen as

directed edges

.

Myriagram

A myriagram is a 10,000-sided

star figures

in the remaining cases.

In popular culture

In the novella Flatland, the Chief Circle is assumed to have ten thousand sides, making him a myriagon.

Notes

^ 5000 cases − 1 (convex) − 1,000 (multiples of 5) − 2,500 (multiples of 2) + 500 (multiples of 2 and 5)

References

^ Meditation VI by Descartes (English translation).
^ Hippolyte Taine, On Intelligence: pp. 9–10
^ Jacques Maritain, An Introduction to Philosophy: p. 108
^ Alan Nelson (ed.), A Companion to Rationalism: p. 285
^ Paolo Fabiani, The philosophy of the imagination in Vico and Malebranche: p. 222
^ The Symmetries of Things, Chapter 20

[7] 5000 cases − 1 (convex) − 1,000 (multiples of 5) − 2,500 (multiples of 2) + 500 (multiples of 2 and 5)

[meditations-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

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