Myriagon
| Regular myriagon | |
|---|---|
 A regular myriagon | |
| Type | Regular polygon |
| Edgesandvertices | 10000 |
| Schläfli symbol | {10000}, t{5000}, tt{2500}, ttt{1250}, tttt{625} |
| Coxeter–Dynkin diagrams |      |
| Symmetry group | Dihedral(D10000), order 2×10000 |
| Internal angle(degrees) | 179.964° |
| Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | Self |
Ingeometry,amyriagonor 10000-gon is apolygonwith 10000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.[1][2][3][4][5]
Regular myriagon[edit]
Aregularmyriagon is represented bySchläfli symbol{10,000} and can be constructed as atruncated5000-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 eachinternal anglein a regular myriagon is 179.964°. Theareaof aregularmyriagon with sides of lengthais given by
The result differs from the area of itscircumscribed circleby up to 40parts per billion.
Because 10,000 = 24× 54,the number of sides is neither a product of distinctFermat primesnor a power of two. Thus the regular myriagon is not aconstructible polygon.Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinctPierpont primes,nor a product of powers of two and three.
Symmetry[edit]
Theregular myriagonhas Dih10000dihedral symmetry,order 20000, represented by 10000 lines of reflection. Dih10000has 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 morecyclicsymmetries 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 Znrepresentingπ/nradian rotational symmetry.
John Conwaylabels these lower symmetries with a letter and order of the symmetry follows the letter.[6]r20000represents full symmetry, anda1labels no symmetry. He givesd(diagonal) with mirror lines through vertices,pwith mirror lines through edges (perpendicular),iwith mirror lines through both vertices and edges, andgfor rotational symmetry.
These lower symmetries allows degrees of freedom in defining irregular myriagons. Only theg10000subgroup has no degrees of freedom but can be seen asdirected edges.
Myriagram[edit]
A myriagram is a 10,000-sidedstar polygon.There are 1999 regular forms[a]given bySchläfli symbolsof the form {10000/n}, wherenis an integer between 2 and 5,000 that iscoprimeto 10,000. There are also 3000 regularstar figuresin the remaining cases.
In popular culture[edit]
In the novellaFlatland,the Chief Circle is assumed to have ten thousand sides, making him a myriagon.
See also[edit]
Notes[edit]
- ^5000 cases − 1 (convex) − 1,000 (multiples of 5) − 2,500 (multiples of 2) + 500 (multiples of 2 and 5)
References[edit]
- ^Meditation VIby Descartes (English translation).
- ^Hippolyte Taine,On Intelligence:pp. 9–10.
- ^Jacques Maritain,An Introduction to Philosophy:p. 108.
- ^Alan Nelson (ed.),A Companion to Rationalism:p. 285.
- ^Paolo Fabiani,The Philosophy of the Imagination in Vico and Malebranche:p. 222.
- ^The Symmetries of Things,Chapter 20