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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 1000gon 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 500gon, t{500}, or a twicetruncated 250gon, tt{250}, or a thricetruncated 125gon, 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 = 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.
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 20thcentury 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]}
The regular chiliagon has Dih_{1000} dihedral symmetry, order 2000, represented by 1000 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 allows degrees of freedom in defining irregular chiliagons. Only the g1000 subgroup has no degrees of freedom but can seen as directed edges.
A chiliagram is a 1000sided 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 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.^{[10]}
Central area with moiré patterns 