Hexadecagon

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Regular hexadecagon
Regular polygon 16 annotated.svg
A regular hexadecagon
Type Regular polygon
Edges and vertices 16
Schläfli symbol {16}
Coxeter diagram CDel node 1.pngCDel 16.pngCDel node.png
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[edit]

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

Regular Hexadecagon Inscribed in a Circle.gif
Construction of a regular hexadecagon

Measurements[edit]

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[edit]

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

A15 15-simplex t0.svg
15-simplex
B8 8-cube t7.svg
8-orthoplex
8-cube t6.svg
Rectified 8-orthoplex
8-cube t5.svg
Birectified 8-orthoplex
8-cube t4.svg
Trirectified 8-orthoplex
8-cube t3.svg
Trirectified 8-cube
8-cube t2.svg
Birectified 8-cube
8-cube t1.svg
Rectified 8-cube
8-cube t0.svg
8-cube
D9 9-cube t8 B8.svg
t7(161)
9-cube t7 B8.svg
t6(161)
9-cube t6 B8.svg
t5(161)
9-cube t5 B8.svg
t4(161)
9-cube t4 B8.svg
t3(161)
9-cube t3 B8.svg
t2(161)
9-cube t2 B8.svg
t1(161)
9-demicube.svg
9-demicube
(161)

In art[edit]

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[edit]

  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 .

External links[edit]