Megagon

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Megagon
Circle - black simple.svg
Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle.
Type Regular polygon
Edges and vertices 1000000
Schläfli symbol {1000000}
t{500000}
Coxeter diagram CDel node 1.pngCDel 10.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel node 1.png
Symmetry group Dihedral (D1000000)
Internal angle (degrees) 179.99964°
Properties convex, cyclic, equilateral, isogonal, isotoxal

A megagon is a polygon with 1 million sides (from mega-).[1][2]

Properties[edit]

A regular megagon has an interior angle of 179.99964°.[1]

The perimeter of a regular megagon inscribed in the unit circle is:

2000000 \sin\frac{\pi}{1000000}

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.[3]

Philosophical application[edit]

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.[4][5][6][7][8][9][10]

The megagon is also used as an illustration of the convergence of regular polygons to a circle.[11]

References[edit]

  1. ^ a b Darling, David J., The universal book of mathematics: from Abracadabra to Zeno's paradoxes, John Wiley & Sons, 2004. Page 249. ISBN 0-471-27047-4.
  2. ^ Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, Addison-Wesley, 1999. Page 505. ISBN 0-201-34712-1.
  3. ^ Williamson, Benjamin, An Elementary Treatise on the Differential Calculus, Longmans, Green, and Co., 1899. Page 45.
  4. ^ McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
  5. ^ Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0-582-28157-1.
  6. ^ Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0-415-15792-7.
  7. ^ Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1-84706-349-7.
  8. ^ Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0-19-875277-6.
  9. ^ Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
  10. ^ Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0-8232-1486-9.
  11. ^ Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0-415-32505-6.