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Hexadecagon facts for kids

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Regular hexadecagon
Szesnastokąt foremny.svg
A regular hexadecagon
Type Regular polygon
Edges and vertices 16
Schläfli symbol {16}, t{8}, tt{4}
Coxeter diagram CDel node 1.pngCDel 16.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.png
Symmetry group Dihedral (D16), order 2×16
Internal angle (degrees) 157.5°
Dual polygon Self
Properties Convex, cyclic, equilateral, isogonal, isotoxal

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

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 twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.

Area

The area of a regular hexadecagon with edge length t is

\begin{align}
A = 4t^2 \cot \frac{\pi}{16} =& 4t^2 \left(1+\sqrt{2}+\sqrt{ 4+2\sqrt{2} }\right)\\
=& 4t^2 (\sqrt{2}+1)(\sqrt{4-2\sqrt{2}}+1)
.\end{align}

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}}.

Since the area of the circumcircle is \pi R^2, the regular hexadecagon fills approximately 97.45% of its circumcircle.

Dissection

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. 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
8-cube.svg 16-gon-dissection.svg

Skew hexadecagon

3 regular skew zig-zag hexadecagon
{8}#{ } { 83 }#{ } { 85 }#{ }
8 antiprism skew 16-gon.png 8-3 antiprism skew 16-gon.png 8-5 antiprism skew 16-gon.png
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 art

Raffaello Sanzio - Spozalizio (detail) - WGA18625
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.

ReconquistaAlhambra
A hexadecagrammic pattern from the Alhambra

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

Irregular hexadecagons

An octagonal star can be seen as a concave hexadecagon:

Octagonal star.pngSquared octagonal star.png


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