Hexadecagon

In mathematics, a hexadecagon (sometimes called a hexakaidecagon) is a 16-sided polygon or 16-gon.

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}.

Construction

As 16 = 2⁴ (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians.

Construction of a regular hexadecagon

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

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

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:

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

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

Symmetry

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 Dih₁₆ symmetry, order 32. There are 4 dihedral subgroups: Dih₈, Dih₄, Dih₂, and Dih₁, and 5 cyclic subgroups: Z₁₆, Z₈, Z₄, Z₂, and Z₁, 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.

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.

Related figures

A hexadecagram is an 16-sided 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.

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 (vertex-transitive) intermediate hexadecagram forms with equally spaced vertices and two edge lengths.

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: {8}, {8/3}

Quasiregular	Isogonal			Quasiregular
t{8}={16}				t{8/7}={16/7}
t{8/3}={16/3}				t{8/5}={16/5}

Petrie polygons

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

A15	B8		D9		2B2 (4D)
15-simplex	8-orthoplex	8-cube	611	161	8-8 duopyramid	8-8 duoprism

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.

A hexadecagrammic pattern from the Alhambra

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

Irregular hexadecagons

An octagonal star can be seen as a concave hexadecagon:

External links

• Weisstein, Eric W., "Hexadecagon", MathWorld.