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 (D1000), order 2×1000
Internal angle (degrees)	179.64°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a chiliagon (pronounced /ˈkɪli.əɡɒn/) or 1000-gon is a polygon with 1000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation.

A regular chiliagon is represented by Schläfli symbol {1000} 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 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 product of powers of two and three.

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. 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."