
In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a chiliagon () or 1,000-gon is a
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
with
1,000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation.
Regular chiliagon
A ''
regular chiliagon'' is represented by
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
and can be constructed as a
truncated 500-gon, t, or a twice-truncated 250-gon, tt, or a thrice-truncated 125-gon, ttt.
The measure of each
internal angle
In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...
in a regular chiliagon is 179°38'24" or
rad. The
area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of a
regular chiliagon with sides of length ''a'' is given by
:
This result differs from the area of its
circumscribed circle In geometry, a circumscribed circle for a set of points is a circle passing through each of them. Such a circle is said to ''circumscribe'' the points or a polygon formed from them; such a polygon is said to be ''inscribed'' in the circle.
* Circu ...
by less than 4
parts per million
In science and engineering, the parts-per notation is a set of pseudo-units to describe the small values of miscellaneous dimensionless quantity, dimensionless quantities, e.g. mole fraction or mass fraction (chemistry), mass fraction.
Since t ...
.
Because 1,000 = 2
3 × 5
3, the number of sides is neither a product of distinct
Fermat prime
In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have 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, ...
s nor a power of two. Thus the regular chiliagon is not a
constructible polygon
In mathematics, a constructible polygon is a regular polygon that can be Compass and straightedge constructions, constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regu ...
. 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. Therefore, construction of a chiliagon requires other techniques such as the
quadratrix of Hippias
The quadratrix or trisectrix of Hippias (also called the quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is traced out by the crossing point of two Line (geometry), lines, one moving by translation (geometry), tran ...
,
Archimedean spiral
The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Ancient Greece, Greek mathematician Archimedes. The term ''Archimedean spiral'' is sometimes used to refer to the more gene ...
, or other auxiliary curves. For example, a 9° angle can first be constructed with compass and straightedge, which can then be quintisected (divided into five equal parts) twice using an auxiliary curve to produce the 21'36" internal angle required.
Philosophical application
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
uses the chiliagon as an example in his
Sixth Meditation to demonstrate the difference between pure intellection and imagination. He says that, when one thinks of a chiliagon, he "does not imagine the thousand sides or see them as if they were present" before him – as he does when one imagines a triangle, for example. The imagination constructs a "confused representation," which is no different from that which it constructs of a
myriagon (a polygon with ten thousand sides). However, he does clearly understand what a chiliagon is, just as he understands what a triangle is, and he is able to distinguish it from a myriagon. Therefore, the intellect is not dependent on imagination, Descartes claims, as it is able to entertain clear and distinct ideas when imagination is unable to.
[ Meditation VI by Descartes (English translation).] Philosopher
Pierre Gassendi
Pierre Gassendi (; also Pierre Gassend, Petrus Gassendi, Petrus Gassendus; 22 January 1592 – 24 October 1655) was a French philosopher, Catholic priest, astronomer, and mathematician. While he held a church position in south-east France, he a ...
, a contemporary of Descartes, was critical of this interpretation, believing that while Descartes could imagine a chiliagon, he could not understand it: one could "perceive that the word 'chiliagon' signifies a figure with a thousand angles
utthat is just the meaning of the term, and it does not follow that you understand the thousand angles of the figure any better than you imagine them."
The example of a chiliagon is also referenced by other philosophers.
David Hume
David Hume (; born David Home; – 25 August 1776) was a Scottish philosopher, historian, economist, and essayist who was best known for his highly influential system of empiricism, philosophical scepticism and metaphysical naturalism. Beg ...
points out that it is "impossible for the eye to determine the angles of a chiliagon to be equal to 1.996 right angles, or make any conjecture, that approaches this proportion."
Gottfried Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
comments on a use of the chiliagon by
John Locke
John Locke (; 29 August 1632 (Old Style and New Style dates, O.S.) – 28 October 1704 (Old Style and New Style dates, O.S.)) was an English philosopher and physician, widely regarded as one of the most influential of the Enlightenment thi ...
, noting that one can have an idea of the polygon without having an image of it, and thus distinguishing ideas from images.
Immanuel Kant
Immanuel Kant (born Emanuel Kant; 22 April 1724 – 12 February 1804) was a German Philosophy, philosopher and one of the central Age of Enlightenment, Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works ...
refers instead to the enneacontahexagon (96-gon), but responds to the same question raised by Descartes.
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
uses the chiliagon as evidence that "intuition is not necessarily founded on the evidence of the senses" because "we can not represent to ourselves a chiliagon, and yet we reason by intuition on polygons in general, which include the chiliagon as a particular case."
Inspired by Descartes's chiliagon example,
Roderick Chisholm
Roderick Milton Chisholm ( ; November 27, 1916 – January 19, 1999) was an American philosopher known for his work on epistemology, metaphysics, free will, value theory, deontology, deontic logic and the philosophy of perception.
Richard and ...
and other 20th-century philosophers have used similar examples to make similar points. Chisholm's "
speckled hen", which need not have a determinate number of speckles to be successfully imagined, is perhaps the most famous of these.
Symmetry

The ''regular chiliagon'' has Dih
1000 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, g ...
, order 2000, represented by 1,000 lines of reflection. Dih
1000 has 15 dihedral subgroups: Dih
500, Dih
250, Dih
125, Dih
200, Dih
100, Dih
50, Dih
25, Dih
40, Dih
20, Dih
10, Dih
5, Dih
8, Dih
4, Dih
2, and Dih
1. It also has 16 more
cyclic symmetries as subgroups: Z
1000, Z
500, Z
250, Z
125, Z
200, Z
100, Z
50, Z
25, Z
40, Z
20, Z
10, Z
5, Z
8, Z
4, Z
2, and 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.
[The Symmetries of Things, Chapter 20] 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. a1 labels no symmetry.
These lower symmetries allow degrees of freedom in defining irregular chiliagons. Only the g1000 subgroup has no degrees of freedom but can be seen as
directed edges.
Chiliagram
A chiliagram is a 1,000-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, Decagram (geometry)#Related figures, certain notable ones can ...
. There are 199 regular forms given by
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
s of the form , where ''n'' is an integer between 2 and 500 that is
coprime
In number theory, 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 equiv ...
to 1,000. There are also 300 regular
star figures in the remaining cases.
For example, the regular star polygon is constructed by 1000 nearly radial edges. Each star vertex has an
internal angle
In geometry, an angle of a polygon is formed by two adjacent edge (geometry), sides. For a simple polygon (non-self-intersecting), regardless of whether it is Polygon#Convexity and non-convexity, convex or non-convex, this angle is called an ...
of 0.36 degrees.
See also
*
Myriagon
*
Megagon
*
Philosophy of Mind
Philosophy of mind is a branch of philosophy that deals with the nature of the mind and its relation to the Body (biology), body and the Reality, external world.
The mind–body problem is a paradigmatic issue in philosophy of mind, although a ...
*
Philosophy of Language
Philosophy of language refers to the philosophical study of the nature of language. It investigates the relationship between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy), me ...
Notes
References
chiliagon
{{Polygons
Polygons by the number of sides
1000 (number)