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{{Infobox polygon | |||
| name = Regular chiliagon | |||
| image = Chiliagon.png | |||
| caption = 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]] | |||
| euler = | |||
| edges = 1000 | |||
| schläfli = {1000}<br>t{500} | |||
| wythoff = | |||
| coxeter = {{CDD|node_1|10|0x|0x|node}}<br>{{CDD|node_1|5|0x|0x|node_1}} | |||
| symmetry = [[Dihedral symmetry|Dihedral]] (D<sub>1000</sub>) | |||
| area = | |||
| angle = 179.64° | |||
| dual = | |||
| properties = [[Convex polygon|convex]], [[Cyclic polygon|cyclic]], [[equilateral]], [[Isogonal figure|isogonal]], [[isotoxal figure|isotoxal]]}} | |||
In [[geometry]], a '''chiliagon''' (pronounced {{IPAc-en|ˈ|k|ɪ|l|i|.|ə|ɡ|ɒ|n}}) is a [[polygon]] with [[Thousand|1000]] sides. Several philosophers have used it to illustrate issues regarding thought. | |||
==Properties== | |||
The measure of each [[internal angle]] in a regular chiliagon is 179.64°. The [[area]] of a [[regular polygon|regular]] chiliagon with sides of length ''a'' is given by | |||
:<math>A = 250a^2 \cot \frac{\pi}{1000} \simeq 79577.2\,a^2</math> | |||
This result differs from the area of its [[circumscribed circle]] by less than 0.0004%. | |||
Because 1000 = 2<sup>3</sup> × 5<sup>3</sup>, the number of sides is neither a product of distinct [[Fermat prime]]s nor a power of two. Thus the regular chiliagon is not a [[constructible polygon]]. | |||
==Philosophical application== | |||
[[René Descartes]] uses the chiliagon as an example in his [[Meditations on First Philosophy#Meditation VI|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.<ref name="meditations">[[wikisource:Meditations on First Philosophy/Meditation VI|Meditation VI]] by Descartes (English translation).</ref> 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."<ref name=sepkoski>{{cite journal|last=Sepkoski|first=David|title=Nominalism and constructivism in seventeenth-century mathematical philosophy|journal=Historia Mathematica|year=2005|volume=32|pages=33–59|url=http://people.uncw.edu/sepkoskid/docs/nomcon.pdf|accessdate=18 April 2012}}</ref> | |||
The example of a chiliagon is also referenced by other philosophers, such as [[Immanuel Kant]].<ref>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.</ref> [[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."<ref>David Hume, ''The Philosophical Works of David Hume'', Volume 1, Black and Tait, 1826, [http://books.google.com/books?id=4uQBAAAAcAAJ&pg=PA101 p. 101.]</ref> [[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.<ref>Jonathan Francis Bennett (2001), ''Learning from Six Philosophers: Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume'', Volume 2, Oxford University Press, ISBN 0198250924, [http://books.google.com/books?id=jzIK2dgKLPMC&pg=PA53 p. 53.]</ref> | |||
[[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."<ref>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, [http://books.google.com/books?id=BtZM33PzV9UC&pg=PA1015 p. 1015.]</ref> | |||
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.<ref>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).</ref> | |||
==See also== | |||
* [[Megagon]] | |||
==References== | |||
{{reflist}} | |||
{{Polygons}} | |||
[[Category:Polygons]] | |||
Revision as of 21:21, 4 November 2013
Template:Infobox polygon In geometry, a chiliagon (pronounced Template:IPAc-en) is a polygon with 1000 sides. Several philosophers have used it to illustrate issues regarding thought.
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
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
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.[7]
See also
References
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- ↑ Meditation VI by Descartes (English translation).
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There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 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).