| Regular chiliagon | |
|---|---|
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.
| |
| Type | Regular polygon |
| Edges andvertices | 1000 |
| Schläfli symbol | {1000} t{500} |
| Coxeter diagram |      |
| Symmetry group | Dihedral (D1000) |
| Internal angle(degrees) | 179.64° |
| Properties | convex, cyclic, equilateral,isogonal, isotoxal |
In geometry, a chiliagon (pronounced /ˈkɪli.əɡɒn/) is a polygon with 1000 sides. Several philosophers have used it to illustrate issues regarding thought.
Properties[edit]
The measure of each internal angle in a regular chiliagon is 179.64°. The area of aregular chiliagon with sides of length a is given by
This result differs from the area of its circumscribed circle by less than 0.0004%.
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.
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. 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