Megagon (original) (raw)

From Wikipedia, the free encyclopedia

Polygon with 1 million edges

This article is about a polygon. For megaton(ne), see Ton and Tonne.

Not to be confused with Megatron.

Regular megagon
A regular megagon
Type Regular polygon
Edges and vertices 1000000
Schläfli symbol {1000000}, t{500000}, tt{250000}, ttt{125000}, tttt{62500}, ttttt{31250}, tttttt{15625}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D1000000), order 2×1000000
Internal angle (degrees) 179.99964°
Properties Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon Self

A megagon or 1,000,000-gon (million-gon) is a circle-like polygon with one million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix denoting a factor of one million).[1][2]

A regular megagon is represented by the Schläfli symbol {1,000,000} and can be constructed as a truncated 500,000-gon, t{500,000}, a twice-truncated 250,000-gon, tt{250,000}, a thrice-truncated 125,000-gon, ttt{125,000}, or a four-fold-truncated 62,500-gon, tttt{62,500}, a five-fold-truncated 31,250-gon, ttttt{31,250}, or a six-fold-truncated 15,625-gon, tttttt{15,625}.

A regular megagon has an interior angle of 179°59'58.704" or 3.14158637 radians.[1] The area of a regular megagon with sides of length a is given by

A = 250 , 000 a 2 cot ⁡ π 1 , 000 , 000 . {\displaystyle A=250,000\ a^{2}\cot {\frac {\pi }{1,000,000}}.} {\displaystyle A=250,000\ a^{2}\cot {\frac {\pi }{1,000,000}}.}

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

2 , 000 , 000 sin ⁡ π 1 , 000 , 000 , {\displaystyle 2,000,000\ \sin {\frac {\pi }{1,000,000}},} {\displaystyle 2,000,000\ \sin {\frac {\pi }{1,000,000}},}

which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 40,075 kilometres, one edge of a megagon inscribed in such a circle would be slightly over 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters.[3]

Because 1,000,000 = 26 × 56, the number of sides is not a product of distinct Fermat primes and a power of two. Thus the regular megagon is not a constructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.

Philosophical application

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Like René Descartes's 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]

The regular megagon has Dih1,000,000 dihedral symmetry, order 2,000,000, represented by 1,000,000 lines of reflection. Dih1,000,000 has 48 dihedral subgroups: (Dih500,000, Dih250,000, Dih125,000, Dih62,500, Dih31,250, Dih15,625), (Dih200,000, Dih100,000, Dih50,000, Dih25,000, Dih12,500, Dih6,250, Dih3,125), (Dih40,000, Dih20,000, Dih10,000, Dih5,000, Dih2,500, Dih1,250, Dih625), (Dih8,000, Dih4,000, Dih2,000, Dih1,000, Dih500, Dih250, Dih125, Dih1,600, Dih800, Dih400, Dih200, Dih100, Dih50, Dih25), (Dih320, Dih160, Dih80, Dih40, Dih20, Dih10, Dih5), and (Dih64, Dih32, Dih16, Dih8, Dih4, Dih2, Dih1). It also has 49 more cyclic symmetries as subgroups: (Z1,000,000, Z500,000, Z250,000, Z125,000, Z62,500, Z31,250, Z15,625), (Z200,000, Z100,000, Z50,000, Z25,000, Z12,500, Z6,250, Z3,125), (Z40,000, Z20,000, Z10,000, Z5,000, Z2,500, Z1,250, Z625), (Z8,000, Z4,000, Z2,000, Z1,000, Z500, Z250, Z125), (Z1,600, Z800, Z400, Z200, Z100, Z50, Z25), (Z320, Z160, Z80, Z40, Z20, Z10, Z5), and (Z64, Z32, Z16, Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labeled these lower symmetries with a letter and order of the symmetry follows the letter.[12] r2000000 represents full symmetry and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry.

These lower symmetries allows degrees of freedom in defining irregular megagons. Only the g1000000 subgroup has no degrees of freedom but can be seen as directed edges.

A megagram is a million-sided star polygon. There are 199,999 regular forms[a] 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.

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

  2. ^ a b Darling, David (2004-10-28). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. Wiley. p. 249. ISBN 978-0-471-66700-1.

  3. ^ Dugopolski, Mark (1999). College Algebra and Trigonometry. Addison-Wesley. p. 505. ISBN 978-0-201-34712-8.

  4. ^ An Elementary Treatise on the Differential Calculus. Forgotten Books. p. 45. ISBN 978-1-4400-6681-8.

  5. ^ McCormick, John Francis (1928). Scholastic Metaphysics: Being, its division and causes. Loyola University Press. p. 18.

  6. ^ Merrill, John Calhoun; Odell, S. Jack (1983). Philosophy and Journalism. Longman. p. 47. ISBN 978-0-582-28157-8.

  7. ^ Hospers, John (1997). An Introduction to Philosophical Analysis (4 ed.). Psychology Press. p. 56. ISBN 978-0-415-15792-6.

  8. ^ Mandik, Pete (2010-05-13). Key Terms in Philosophy of Mind. A&C Black. p. 26. ISBN 978-1-84706-349-6.

  9. ^ Kenny, Anthony (2006-06-29). The Rise of Modern Philosophy: A New History of Western Philosophy. OUP Oxford. p. 124. ISBN 978-0-19-875277-6.

  10. ^ Balmes, Jaime Luciano (1856). Fundamental Philosophy. Sadlier. p. 27.

  11. ^ Potter, Vincent G. (1994). On Understanding Understanding: A Philosophy of Knowledge. Fordham University Press. p. 86. ISBN 978-0-8232-1486-0.

  12. ^ Russell, Bertrand (2004). History of Western Philosophy. Psychology Press. p. 202. ISBN 978-0-415-32505-9.

  13. ^ The Symmetries of Things, Chapter 20