# Chiliagon

Regular chiliagon | |
---|---|

A regular chiliagon | |

Type | Regular polygon |

Edges and vertices | 1000 |

Schläfli symbol | {1000}, t{500}, tt{250}, ttt{125} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{1000}), order 2×1000 |

Internal angle (degrees) | 179.64° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

In geometry, a **chiliagon** (/ˈkɪliəɡɒn/) or 1000-gon is a 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 {1,000} and can be constructed as a truncated 500-gon, t{500}, or a twice-truncated 250-gon, tt{250}, or a thrice-truncated 125-gon, ttt{125}.

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

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

Because 1,000 = 2^{3} × 5^{3}, 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 product of powers of two and three. Therefore, construction of a chiliagon requires other techniques such as the quadratrix of Hippias, Archimedean spiral, 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 0.36° internal angle required.

## Philosophical application

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]}

## Symmetry

The *regular chiliagon* has Dih_{1000} dihedral symmetry, order 2000, represented by 1,000 lines of reflection. Dih_{100} 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.^{[8]} 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. There are 199 regular forms^{[9]} given by Schläfli symbols of the form {1000/*n*}, where *n* is an integer between 2 and 500 that is coprime to 1,000. 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.^{[10]}

Central area with moiré patterns |

## See also

## References

**^**Meditation VI by Descartes (English translation).**^**Sepkoski, David (2005). "Nominalism and constructivism in seventeenth-century mathematical philosophy".*Historia Mathematica*.**32**: 33–59. doi:10.1016/j.hm.2003.09.002.**^**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.**^**David Hume,*The Philosophical Works of David Hume*, Volume 1, Black and Tait, 1826, p. 101.**^**Jonathan Francis Bennett (2001),*Learning from Six Philosophers: Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume*, Volume 2, Oxford University Press, ISBN 0198250924, p. 53.**^**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.**^**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).**^****The Symmetries of Things**, Chapter 20**^**199 = 500 cases − 1 (convex) − 100 (multiples of 5) − 250 (multiples of 2) + 50 (multiples of 2 and 5)**^**0.36=180(1-2/(1000/499))=180(1-998/1000)=180(2/1000)=180/500