geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a myriagon or 10000-gon is a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

with

10,000 10,000 (ten thousand) is the natural number following 9,999 and preceding 10,001. Name Many languages have a specific word for this number: in Ancient Greek it is (the etymological root of the word myriad in English), in Aramaic , in Hebrew ...

sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought. Meditation VI by Descartes (English translation).

Regular myriagon

A regular myriagon is represented by

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

and can be constructed as a truncated 5000-gon, t, or a twice-truncated 2500-gon, tt, or a thrice-truncated 1250-gon, ttt{1250), or a four-fold-truncated 625-gon, tttt{625}. The measure of each

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

in a regular myriagon is 179.964°. The

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

by up to 40

parts per billion In science and engineering, the parts-per notation is a set of pseudo-units to describe small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity measures, they ...

. Because 10,000 = 2⁴ × 5⁴, the number of sides is neither a product of distinct

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 4294967 ...

s nor a power of two. Thus the regular myriagon is not a

constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...

. 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 prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...

s, nor a product of powers of two and three.

Symmetry

The ''regular myriagon'' has Dih₁₀₀₀₀

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...

, 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 Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in soc ...

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 John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

labels these lower symmetries with a letter and order of the symmetry follows the letter.The Symmetries of Things, Chapter 20 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 edge In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...

Myriagram

A myriagram is a 10,000-sided

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

. There are 1999 regular forms given by

s of the form {10000/''n''}, where ''n'' is an integer between 2 and 5,000 that is

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

to 10,000. There are also 3000 regular

star figure In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but can also include disconnected sets of edges, called a compound polygon. For example, a regular pentagram, ...

s in the remaining cases.

In popular culture

In the novella ''

Flatland ''Flatland: A Romance of Many Dimensions'' is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dime ...

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

Notes

References