Myriagon

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Regular myriagon
Polygon 10000.svg
A regular myriagon
Type Regular polygon
Edges and vertices 10000
Schläfli symbol {10000}, t{5000}, tt{2500}, ttt{1250}, tttt{625}
Coxeter diagram CDel node 1.pngCDel 10.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel 0x.pngCDel 0x.pngCDel 0x.pngCDel node 1.png
Symmetry group Dihedral (D10000), order 2×10000
Internal angle (degrees) 179.964°
Dual polygon Self
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a myriagon or 10000-gon is a polygon with 10000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.[1][2][3][4][5]

Regular myriagon[edit]

A regular myriagon is represented by Schläfli symbol {10000} and can be constructed as a truncated 5000-gon, t{5000}, or a twice-truncated 2500-gon, tt{2500}, or a thrice-truncated 1250-gon, ttt{1250), or a four-fold-truncated 625-gon, tttt{625}.

The measure of each internal angle in a regular myriagon is 179.964°. The area of a regular myriagon with sides of length a is given by

The result differs from the area of its circumscribed circle by up to 40 parts per billion.

Because 10000 = 24 × 54, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular myriagon 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 product of powers of two and three.

Symmetry[edit]

The symmetries of a regular myriagon. Light blue lines show subgroups of index 2, the 5 boxed subgraphs are positionally related by index 5 subgroups.

The regular myriagon has Dih10000 dihedral symmetry, order 20000, represented by 10000 lines of reflection. Dih100 has 24 dihedral subgroups: (Dih5000, Dih2500, Dih1250, Dih625), (Dih2000, Dih1000, Dih500, Dih250, Dih125), (Dih400, Dih200, Dih100, Dih50, Dih25), (Dih80, Dih40, Dih20, Dih10, Dih5), and (Dih16, Dih8, Dih4, Dih2, Dih1). It also has 25 more cyclic symmetries as subgroups: (Z10000, Z5000, Z2500, Z1250, Z625), (Z2000, Z1000, Z500, Z250, Z125), (Z400, Z200, Z100, Z50, Z25), (Z80, Z40, Z20, Z10), and (Z16, Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.

John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[6] r20000 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 myriagons. Only the g10000 subgroup has no degrees of freedom but can seen as directed edges.

Myriagram[edit]

A myriagram is an 10000-sided star polygon. There are 1999 regular forms[7] given by Schläfli symbols of the form {10000/n}, where n is an integer between 2 and 5000 that is coprime to 10000. There are also 3000 regular star figures in the remaining cases.

See also[edit]

References[edit]

  1. ^ Meditation VI by Descartes (English translation).
  2. ^ Hippolyte Taine, On Intelligence: pp. 9–10
  3. ^ Jacques Maritain, An Introduction to Philosophy: p. 108
  4. ^ Alan Nelson (ed.), A Companion to Rationalism: p. 285
  5. ^ Paolo Fabiani, The philosophy of the imagination in Vico and Malebranche: p. 222
  6. ^ The Symmetries of Things, Chapter 20
  7. ^ 5000 cases - 1 (convex) - 1000 (multiples of 5) - 2500 (multiples of 2)+ 500 (multiples of 2 and 5)