Hexadecagon

Regular 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 (D16), order 2×16
Internal angle (degrees)	157.5°
Dual polygon	Self
Properties	Convex, cyclic, equilateral, isogonal, isotoxal

An hexadecagon or hexakaidecagon 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{aligned}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{aligned}}

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. ^[1] 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

Skew hexadecagon

3 regular skew zig-zag hexadecagon
{8}#{ }	{ ⁸⁄₃ }#{ }	{ ⁸⁄₅ }#{ }

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 D_8d, [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

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.^[2]

A hexadecagrammic pattern from the Alhambra

Hexadecagrams (16-sided star polygons) are included in the Girih patterns in the Alhambra.^[3]

Irregular hexadecagons

An octagonal star can be seen as a concave hexadecagon:

References

↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
↑ Speiser, David (2011), "Architecture, mathematics and theology in Raphael's paintings", in Williams, Kim (ed.), 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.
↑ Hankin, E. Hanbury (May 1925), "Examples of methods of drawing geometrical arabesque patterns", The Mathematical Gazette, 12 (176): 370–373, doi:10.2307/3604213, JSTOR 3604213, S2CID 250431566.

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[1] Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141

[2] Speiser, David (2011), "Architecture, mathematics and theology in Raphael's paintings", in Williams, Kim (ed.), 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.

[3] Hankin, E. Hanbury (May 1925), "Examples of methods of drawing geometrical arabesque patterns", The Mathematical Gazette, 12 (176): 370–373, doi:10.2307/3604213, JSTOR 3604213, S2CID 250431566.

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