Hexadecagon
Regular hexadecagon  

A regular hexadecagon  
Type  Regular polygon 
Edges and vertices  16 
Schläfli symbol  {16}, t{8}, tt{4} 
Coxeter diagram 

Symmetry group  Dihedral (D_{16}), order 2×16 
Internal angle (degrees)  157.5° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In mathematics, a hexadecagon (sometimes called a hexakaidecagon) or 16gon is a sixteensided polygon.^{[1]}
Regular hexadecagon
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 twicetruncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.
Construction
As 16 = 2^{4} (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians.^{[2]}
Measurements
Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any hexadecagon is 2520 degrees.
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.
Symmetry
The 14 symmetries of a regular hexadecagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position. 
The regular hexadecagon has Dih_{16} symmetry, order 32. There are 4 dihedral subgroups: Dih_{8}, Dih_{4}, Dih_{2}, and Dih_{1}, and 5 cyclic subgroups: Z_{16}, Z_{8}, Z_{4}, Z_{2}, and Z_{1}, the last implying no symmetry.
On the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders.^{[3]}
The most common high symmetry hexadecagons are d16, a isogonal hexadecagon constructed by eight mirrors can alternate long and short edges, and p16, an isotoxal hexadecagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexadecagon.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g16 subgroup has no degrees of freedom but can seen as directed edges.
Skew hexadecagon
{8}#{ }  { ^{8}⁄_{3} }#{ }  { ^{8}⁄_{5} }#{ } 

A regular skew hexadecagon is seen as zigzagging edges of a octagonal antiprism, a octagrammic antiprism, and a octagrammic crossedantiprism. 
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 zigzag hexadecagon has vertices alternating between two parallel planes.
A regular skew hexadecagon is vertextransitive with equal edge lengths. In 3dimensions it will be a zigzag skew hexadecagon and can be seen in the vertices and side edges of a octagonal antiprism with the same D_{8d}, [2^{+},16] symmetry, order 32. The octagrammic antiprism, s{2,16/3} and octagrammic crossedantiprism, s{2,16/5} also have regular skew octagons.
Petrie polygons
The regular hexadecagon is the Petrie polygon for many higherdimensional polytopes, shown in these skew orthogonal projections, including:
A_{15}  B_{8}  D_{9}  2B_{2} (4D)  

15simplex 
8orthoplex 
8cube 
6_{11} 
1_{61} 
88 duopyramid 
88 duoprism 
Related figures
A hexadecagram is a 16sided star polygon, represented by symbol {16/n}. There are three regular star polygons, {16/3}, {16/5}, {16/7}, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds: {16/2} is reduced to 2{8} as two octagons, {16/4} is reduced to 4{4} as four squares and {16/6} reduces to 2{8/3} as two octagrams, and finally {16/8} is reduced to 8{2} as eight digons.
Compound and star hexadecagons  

Form  Convex polygon  Compound  Star polygon  Compound 
Image  {16/1} or {16} 
{16/2} or 2{8} 
{16/3} 
{16/4} or 4{4} 
Interior angle  157.5°  135°  112.5°  90° 
Form  Star polygon  Compound  Star polygon  Compound 
Image  {16/5} 
{16/6} or 2{8/3} 
{16/7} 
{16/8} or 8{2} 
Interior angle  67.5°  45°  22.5°  0° 
Deeper truncations of the regular octagon and octagram can produce isogonal (vertextransitive) intermediate hexadecagram forms with equally spaced vertices and two edge lengths.^{[4]}
A truncated octagon is a hexadecagon, t{8}={16}. A quasitruncated octagon, inverted as {8/7}, is a hexadecagram: t{8/7}={16/7}. A truncated octagram {8/3} is a hexadecagram: t{8/3}={16/3} and a quasitruncated octagram, inverted as {8/5}, is a hexadecagram: t{8/5}={16/5}.
Isogonal truncations of octagon and octagram  

Quasiregular  Isogonal  Quasiregular  
t{8}={16} 
t{8/7}={16/7}  
t{8/3}={16/3} 
t{8/5}={16/5} 
In art
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 eightsided tower in a previous painting by Pietro Perugino.^{[5]}
Hexadecagrams (16sided star polygons) are included in the Girih patterns in the Alhambra.^{[6]}
Irregular hexadecagons
An octagonal star can be seen as a concave hexadecagon:
References
 ↑ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1365. ISBN 9781420035223.
 ↑ Koshy, Thomas (2007), Elementary Number Theory with Applications (2nd ed.), Academic Press, p. 142, ISBN 9780080547091.
 ↑ John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278)
 ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
 ↑ Speiser, David (2011), "Architecture, mathematics and theology in Raphael's paintings", in Williams, Kim, Crossroads: History of Science, History of Art. Essays by David Speiser, vol. II, Springer, pp. 29–39, doi:10.1007/9783034801393_3. Originally published in Nexus III: Architecture and Mathematics, Kim Williams, ed. (Ospedaletto, Pisa: Pacini Editore, 2000), pp. 147–156.
 ↑ Hankin, E. Hanbury (May 1925), "Examples of methods of drawing geometrical arabesque patterns", The Mathematical Gazette, 12 (176): 370–373, doi:10.2307/3604213.