Exponentially equivalent measures: Difference between revisions

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{{Infobox polygon
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| name      = Megagon
| image      = Circle - black simple.svg
| caption    = Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a [[circle]].
| type      = [[Regular polygon]]
| euler      =
| edges      = 1000000
| schläfli  = {1000000}<br>t{500000}
| wythoff    =
| coxeter    = {{CDD|node_1|10|0x|0x|0x|0x|0x|node}}<br>{{CDD|node_1|5|0x|0x|0x|0x|0x|node_1}}
| symmetry  = [[Dihedral symmetry|Dihedral]] (D<sub>1000000</sub>)
| area      =
| angle      = 179.99964°
| dual      =
| properties = [[Convex polygon|convex]], [[Cyclic polygon|cyclic]], [[equilateral]], [[Isogonal figure|isogonal]], [[isotoxal figure|isotoxal]]}}
 
A '''megagon''' is a [[polygon]] with 1 million sides (from [[mega-]]).<ref name=Darling/><ref>Dugopolski, Mark, ''[http://books.google.com/books?id=l8tWAAAAYAAJ College Algebra and Trigonometry]'', 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1.</ref>
 
==Properties==
A [[regular polygon|regular]] megagon has an interior angle of 179.99964°.<ref name=Darling>Darling, David J., ''[http://books.google.com/books?id=0YiXM-x--4wC&pg=PA249&dq=polygon+megagon&hl=en&sa=X&ei=0TE4T7jOMc-G0QGH1ezGAg&ved=0CDgQ6AEwAA#v=onepage&q=polygon%20megagon&f=false The universal book of mathematics: from Abracadabra to Zeno's paradoxes]'', John Wiley & Sons, 2004. Page 249. ISBN 0-471-27047-4.</ref>
 
The [[perimeter]] of a regular megagon inscribed in the unit circle is:
:<math>2000000 \sin\frac{\pi}{1000000}</math>
which is very close to 2π. In fact, for a circle the size of the [[Earth]], with a [[circumference]] of 40,075 kilometres, the difference between the perimeter of the megagon and the circumference of the circle comes to less than 1/16 millimeters.<ref>Williamson, Benjamin, ''[http://books.google.com.au/books?id=E72yNz5sBY0C&pg=PA45 An Elementary Treatise on the Differential Calculus]'', Longmans, Green, and Co., 1899. Page 45.</ref>
 
==Philosophical application==
As with [[René Descartes]]' example of the [[chiliagon]], the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised.<ref>McCormick, John Francis, ''[http://books.google.com/books?id=KyFHAAAAIAAJ&q=%22million-sided+polygon%22&dq=%22million-sided+polygon%22&hl=en&sa=X&ei=gl06T6CeAcGjiQeO3qCNCg&ved=0CEkQ6AEwBA Scholastic Metaphysics]'', Loyola University Press, 1928, p. 18.</ref><ref>Merrill, John Calhoun and Odell, S. Jack, ''[http://books.google.com/books?id=_aNZAAAAMAAJ&q=%22million-sided+polygon%22&dq=%22million-sided+polygon%22&hl=en&sa=X&ei=gl06T6CeAcGjiQeO3qCNCg&ved=0CD0Q6AEwAg Philosophy and Journalism]'', Longman, 1983, p. 47, ISBN 0-582-28157-1.</ref><ref>Hospers, John, ''[http://books.google.com/books?id=OVu0CORmhL4C&pg=PA56&lpg=PA56 An Introduction to Philosophical Analysis]'', 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7.</ref><ref>Mandik, Pete, ''[http://books.google.com/books?id=5yHtsM-NToYC&pg=PA26 Key Terms in Philosophy of Mind]'', Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7.</ref><ref>Kenny, Anthony, ''[http://books.google.com/books?id=ehZGIy_ZYTgC&pg=PA124 The Rise of Modern Philosophy]'', Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6.</ref><ref>Balmes, James, ''[http://books.google.com/books?id=MrwKHqw06hMC&pg=PA27 Fundamental Philosophy, Vol II]'', Sadlier and Co., Boston, 1856, p. 27.</ref><ref>Potter, Vincent G., ''[http://books.google.com/books?id=SnO1FKnJui4C&pg=PA86 On Understanding Understanding: A Philosophy of Knowledge]'', 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0-8232-1486-9.</ref>
 
The megagon is also used as an illustration of the convergence of regular polygons to a circle.<ref>Russell, Bertrand, ''[http://books.google.com/books?id=Ey94E3sOMA0C&pg=PA202 History of Western Philosophy]'', reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6.</ref>
 
==References==
{{reflist}}
 
{{Polygons}}
 
[[Category:Polygons]]

Latest revision as of 09:40, 3 December 2014

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