Type Regular polygon
Edges and vertices 16
Schläfli symbol {16}
Coxeter diagram
Symmetry group D16, 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) is a polygon with 16 sides and 16 vertices.[1]

A regular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent.

## Construction

As the ancient Greek mathematicians already knew,[2] a regular hexadecagon is constructible using compass and straightedge:

## 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 is: (with t = edge length)

$A = 4t^2 \cot \frac{\pi}{16} = 4t^2 (\sqrt{2}+1)(\sqrt{4-2\sqrt{2}}+1)$

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:

$A=r^2\cdot\frac{2}{1}\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}=4r^2\sqrt{2-\sqrt{2}}.$

### Petrie polygons

The regular hexadecagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including:

 A15 B8 D9 15-simplex 8-orthoplex Rectified 8-orthoplex Birectified 8-orthoplex Trirectified 8-orthoplex Trirectified 8-cube Birectified 8-cube Rectified 8-cube 8-cube t7(161) t6(161) t5(161) t4(161) t3(161) t2(161) t1(161) 9-demicube (161)

## In art

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.[3]

A hexadecagram pattern from the Alhambra

Hexadecagrams, 16-pointed star polygons, are included in the Girih patterns in the Alhambra.[4]

## References

1. ^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1365. ISBN 9781420035223.
2. ^ Koshy, Thomas (2007), Elementary Number Theory with Applications (2nd ed.), Academic Press, p. 142, ISBN 9780080547091.
3. ^ Speiser, David (2011), "Architecture, mathematics and theology in Raphael’s paintings", 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.
4. ^ Hankin, E. Hanbury (May 1925), "Examples of methods of drawing geometrical arabesque patterns", The Mathematical Gazette 12 (176): 370–373, doi:10.2307/3604213.