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Regular chiliagon
Polygon 1000.svg
A regular chiliagon
Type Regular polygon
Edges and vertices 1000
Schläfli symbol {1000}
Coxeter diagram CDel node 1.pngCDel 10.pngCDel 0x.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel 0x.pngCDel 0x.pngCDel node 1.png
Symmetry group Dihedral (D1000), order 2×1000
Internal angle (degrees) 179.64°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal
A whole regular chiliagon is not visually discernible from a circle. The lower section is a portion of a regular chiliagon, 200 times as large as the smaller one, with the vertices highlighted.

In geometry, a chiliagon (pronounced /ˈkɪli.əɡɒn/) is a polygon with 1000 sides. Several philosophers have used it to illustrate issues regarding thought.

A regular chiliagon is represented by Schläfli symbol {1000} and can be constructed as a quasiregular truncated 500-gon, t{500}, which alternates two types of edges.


The measure of each internal angle in a regular chiliagon is 179.64°. The area of a regular chiliagon with sides of length a is given by

A = 250a^2 \cot \frac{\pi}{1000} \simeq 79577.2\,a^2

This result differs from the area of its circumscribed circle by less than 4 parts per million.

Because 1000 = 23 × 53, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular chiliagon 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 power of two, three, or six.

Philosophical application[edit]

René Descartes 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.[1] Philosopher Pierre Gassendi, 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 [but] that 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."[2]

The example of a chiliagon is also referenced by other philosophers, such as Immanuel Kant.[3] David Hume points out that it is "impossible for the eye to determine the angles of a chiliagon to be equal to 1996 right angles, or make any conjecture, that approaches this proportion."[4] Gottfried Leibniz comments on a use of the chiliagon by John Locke, noting that one can have an idea of the polygon without having an image of it, and thus distinguishing ideas from images.[5]

Henri Poincaré 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."[6]

Inspired by Descartes's chiliagon example, Roderick Chisholm 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.[7]


A chiliagram is a 1000-sided star polygon. There are 199 regular forms[8] given by Schläfli symbols of the form {1000/n}, where n is an integer between 2 and 500 that is coprime to 1000. There are also 300 regular star figures in the remaining cases.

For example, the regular {1000/499} star polygon is constructed by 1000 nearly radial edges. Each star vertex has an internal angle of 0.36 degrees.[9]

Star polygon 1000-499.svg Star polygon 1000-499 center.png
Central area with Moiré patterns

See also[edit]


  1. ^ Meditation VI by Descartes (English translation).
  2. ^ Sepkoski, David (2005). "Nominalism and constructivism in seventeenth-century mathematical philosophy". Historia Mathematica 32: 33–59. doi:10.1016/ Retrieved 9 February 2014. 
  3. ^ Immanuel Kant, "On a Discovery," trans. Henry Allison, in Theoretical Philosophy After 1791, ed. Henry Allison and Peter Heath, Cambridge UP, 2002 [Akademie 8:121]. Kant does not actually use a chiliagon as his example, instead using a 96-sided figure, but he is responding to the same question raised by Descartes.
  4. ^ David Hume, The Philosophical Works of David Hume, Volume 1, Black and Tait, 1826, p. 101.
  5. ^ Jonathan Francis Bennett (2001), Learning from Six Philosophers: Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume, Volume 2, Oxford University Press, ISBN 0198250924, p. 53.
  6. ^ Henri Poincaré (1900) "Intuition and Logic in Mathematics" in William Bragg Ewald (ed) From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume 2, Oxford University Press, 2007, ISBN 0198505361, p. 1015.
  7. ^ Roderick Chisholm, "The Problem of the Speckled Hen", Mind 51 (1942): pp. 368–373. "These problems are all descendants of Descartes's 'chiliagon' argument in the sixth of his Meditations" (Joseph Heath, Following the Rules: Practical Reasoning and Deontic Constraint, Oxford: OUP, 2008, p. 305, note 15).
  8. ^ 199 = 500 cases − 1 (convex) − 100 (multiples of 5) − 250 (multiples of 2) + 50 (multiples of 2 and 5)
  9. ^ 0.36=180(1-2/(1000/499))=180(1-998/1000)=180(2/1000)=180/500