Pentacontagon

In geometry, a pentacontagon or pentecontagon or 50-gon is a fifty-sided polygon.[1][2] The sum of any pentacontagon's interior angles is 8640 degrees.

Regular pentacontagon
A regular pentacontagon
TypeRegular polygon
Edges and vertices50
Schläfli symbol{50}, t{25}
Coxeter diagram
Symmetry groupDihedral (D50), order 2×50
Internal angle (degrees)172.8°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

A regular pentacontagon is represented by Schläfli symbol {50} and can be constructed as a quasiregular truncated icosipentagon, t{25}, which alternates two types of edges.

Regular pentacontagon properties

One interior angle in a regular pentacontagon is 172​4⁄5°, meaning that one exterior angle would be 7​1⁄5°.

The area of a regular pentacontagon is (with t = edge length)

and its inradius is

The circumradius of a regular pentacontagon is

Since 50 = 2 × 52, a regular pentacontagon is not constructible using a compass and straightedge,[3] and is not constructible even if the use of an angle trisector is allowed.[4] However, it is constructible using an auxiliary curve (such as the quadratrix of Hippias or an Archimedean spiral), as such curves can be used to divide angles into any number of equal parts. For example, one can construct a 36° angle using compass and straightedge and proceed to quintisect it (divide it into five equal parts) using an Archimedean spiral, giving the 7.2° angle required to construct a pentacontagon.

Symmetry
The symmetries of a regular pentacontagon. Light blue lines show subgroups of index 2. The 3 boxed subgraphs are positionally related by index 5 subgroups.

The regular pentacontagon has Dih50 dihedral symmetry, order 100, represented by 50 lines of reflection. Dih50 has 5 dihedral subgroups: Dih25, (Dih10, Dih5), and (Dih2, Dih1). It also has 6 more cyclic symmetries as subgroups: (Z50, Z25), (Z10, Z5), and (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.[5] 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. a1 labels no symmetry.

These lower symmetries allows degrees of freedom in defining irregular pentacontagons. Only the g50 subgroup has no degrees of freedom but can seen as directed edges.

Dissection
50-gon with 1200 rhombs

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[6] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular pentacontagon, m=25, it can be divided into 300: 12 sets of 25 rhombs. This decomposition is based on a Petrie polygon projection of a 25-cube.

Examples

Pentacontagram

A pentacontagram is a 50-sided star polygon. There are 9 regular forms given by Schläfli symbols {50/3}, {50/7}, {50/9}, {50/11}, {50/13}, {50/17}, {50/19}, {50/21}, and {50/23}, as well as 16 compound star figures with the same vertex configuration.

Regular star polygons {50/k}
Picture
{​50⁄3}
{​50⁄7}
{​50⁄9}
{​50⁄11}
​50⁄13
Interior angle 158.4° 129.6° 115.2° 100.8° 86.4°
Picture
{​50⁄17 }
{​50⁄19 }
{​50⁄21 }
{​50⁄23 }		  
Interior angle 57.6° 43.2° 28.8° 14.4°  

References
  1. Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 120, ISBN 9781438109572.
  2. The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
  3. Constructible Polygon
  4. "Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.CS1 maint: archived copy as title (link)
  5. The Symmetries of Things, Chapter 20
  6. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
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