Megagon

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Regular megagon
Circle - black simple.svg
A regular megagon
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), order 2×1000000
Internal angle (degrees) 179.99964°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

A megagon is a polygon with 1 million sides (mega-, from the Greek μέγας megas, meaning "great").[1][2] Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle.

A regular megagon is represented by Schläfli symbol {1000000} and can be constructed as a quasiregular truncated 500000-gon, t{500000}, which alternates two types of edges.

Properties[edit]

A regular megagon has an interior angle of 179.99964°.[1] The area of a regular megagon with sides of length a is given by

A = 250000a^2 \cot \frac{\pi}{1000000}

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, so one edge of a megagon the size of the earth would be about 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of the circle comes to less than 1/16 millimeters.[3]

Because 1000000 = 26 × 56, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular megagon 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 power of two, three, or six.

Philosophical application[edit]

Like 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]

Megagram[edit]

A megagram is an million-sided star polygon. There are 199,999 regular forms[12] given by Schläfli symbols of the form {1000000/n}, where n is an integer between 2 and 500,000 that is coprime to 1,000,000. There are also 300,000 regular star figures in the remaining cases.

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.
  12. ^ 199,999 = 500,000 cases - 1 (convex) - 100,000 (multiples of 5) - 250,000 (multiples of 2) + 50,000 (multiples of 2 and 5)