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# Chiliagon

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 larger than the smaller one, with the vertices highlighted.
Edges and vertices 1000
Schläfli symbol {1000}
Coxeter–Dynkin diagram
Symmetry group Dihedral (D1000)
Internal angle
(degrees)
179.64°
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a chiliagon (pronounced /ˈkɪli.əˌgɑn/) is a polygon with 1000 sides.

## Properties

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 0.0004%.

Because $1000=2^3 \times 5^3$, it is not a product of distinct Fermat primes and a power of two, thus the regular chiliagon it is not a constructible polygon.

## Philosophical construction

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]

## References

1. ^ Meditation VI by Descartes (English translation).

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### Look at other dictionaries:

• Chiliagon — Chil i*a*gon, n. [Gr. ?; chi lioi a thousand + ? angle.] A plane figure of a thousand angles and sides. Barlow. [1913 Webster] …   The Collaborative International Dictionary of English

• chiliagon — noun /ˈkɪlɪæɡən/ (rare) A polygon with a thousand vertices and a thousand edges. , 1995: If Kants claim is just that it is impossible to have a sensory experience of the absence of space, then by contrast it is not impossible to have a sensory… …   Wiktionary

• chiliagon — (Gk., chilioi, 1,000 + gonia, angle) A closed plane figure with 1,000 internal angles (thus almost indistinguishable from a circle). Chiliasm is the doctrine that Christ will come again and reign for 1,000 years, a form of millenarianism …   Philosophy dictionary

• chiliagon — chilˈiagon noun A plane figure with 1000 angles • • • Main Entry: ↑chiliad …   Useful english dictionary

• Polygon — For other uses, see Polygon (disambiguation). Some polygons of different kinds In geometry a polygon (   …   Wikipedia

• Triangle — This article is about the basic geometric shape. For other uses, see Triangle (disambiguation). Isosceles and Acute Triangle redirect here. For the trapezoid, see Isosceles trapezoid. For The Welcome to Paradox episode, see List of Welcome to… …   Wikipedia

• Pentagram — For other uses, see Pentagram (disambiguation). Regular pentagram A pentagram Type Star polygon Edges and vertices 5 Schläfli symbol {5/2} …   Wikipedia

• List of mathematical shapes — Following is a list of some mathematically well defined shapes. See also list of polygons, polyhedra and polytopes and list of geometric shapes.0D with no surface*point1D with 0D surface*interval *line2D with 1D surface*Bézier curve: ( As + Bt )… …   Wikipedia

• Star polygon — Set of regular star polygons {5/2} {7/2} …   Wikipedia