Myriagon

Regular myriagon

A regular myriagon
Type Regular polygon
Edges and vertices 10000
Schläfli symbol {10000}
t{5000}
Coxeter diagram
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 is a polygon with 10000 sides. Several philosophers have used it to illustrate issues regarding thought.[1][2][3][4][5]

A regular myriagon is represented by Schläfli symbol {10000} and can be constructed as a quasiregular truncated 5000-gon, t{5000}, which alternates two types of edges.

Properties

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

A = 2500a^2 \cot \frac{\pi}{10000}

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 power of two, three, or six.

Myriagram

A myriagram is an 10000-sided star polygon. There are 1999 regular forms[6] 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

References

  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. ↑ 5000 cases - 1 (convex) - 1000 (multiples of 5) - 2500 (multiples of 2)+ 500 (multiples of 2 and 5)