Hexadecagon

polygon with 16 sides

An hexadecagon or hekkaidecagon is a shape with 16 sides and 16 corners.

Regular hexadecagon
Szesnastokąt foremny.svg
A regular hexadecagon
TypeRegular polygon
Edges and vertices16
Schläfli symbol{16}, t{8}, tt{4}
Coxeter diagramCDel node 1.pngCDel 16.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.png
Symmetry groupDihedral (D16), order 2×16
Internal angle (degrees)157.5°
Dual polygonSelf
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal

Regular hexadecagonEdit

A regular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.

AreaEdit

The area of a regular hexadecagon with edge length t is

 

Because the hexadecagon has a number of sides that is a power of two, its area can be computed in terms of the circumradius R by truncating Viète's formula:

 

Since the area of the circumcircle is   the regular hexadecagon fills approximately 97.45% of its circumcircle.

DissectionEdit

Coxeter states that every parallel-sided 2m-gon can be divided into m(m-1)/2 rhombs. For the regular hexadecagon, m=8, and it can be divided into 28: 4 squares and 3 sets of 8 rhombs. This decomposition is based on a Petrie polygon projection of a 8-cube, with 28 of 1792 faces. [1] The list A006245 enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection.

Dissection into 28 rhombs
   

Skew hexadecagonEdit

3 regular skew zig-zag hexadecagon
{8}#{ } { ​83 }#{ } { ​85 }#{ }
     
A regular skew hexadecagon is seen as zig-zagging edges of a octagonal antiprism, a octagrammic antiprism, and a octagrammic crossed-antiprism.

A skew hexadecagon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an hexadecagon is not generally defined. A skew zig-zag hexadecagon has vertices alternating between two parallel planes.

A regular skew hexadecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D8d, [2+,16] symmetry, order 32. The octagrammic antiprism, s{2,16/3} and octagrammic crossed-antiprism, s{2,16/5} also have regular skew octagons.

In artEdit

 
The hexadecagonal tower from Raphael's The Marriage of the Virgin

In the early 16th century, Raphael was the first to construct a perspective image of a regular hexadecagon: the tower in his painting The Marriage of the Virgin has 16 sides, elaborating on an eight-sided tower in a previous painting by Pietro Perugino.[2]

 
A hexadecagrammic pattern from the Alhambra

Hexadecagrams (16-sided star polygons) are included in the Girih patterns in the Alhambra.[3]


Irregular hexadecagonsEdit

An octagonal star can be seen as a concave hexadecagon:

  

ReferencesEdit

  1. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  2. Speiser, David (2011), "Architecture, mathematics and theology in Raphael's paintings", in Williams, Kim (ed.), Crossroads: History of Science, History of Art. Essays by David Speiser, vol. II, Springer, pp. 29–39, doi:10.1007/978-3-0348-0139-3_3. Originally published in Nexus III: Architecture and Mathematics, Kim Williams, ed. (Ospedaletto, Pisa: Pacini Editore, 2000), pp. 147–156.
  3. Hankin, E. Hanbury (May 1925), "Examples of methods of drawing geometrical arabesque patterns", The Mathematical Gazette, 12 (176): 370–373, doi:10.2307/3604213.