View source for Hexadecagon

{{short description|Polygon with 16 edges}}
{{Regular polygon db|Regular polygon stat table|p16}}
In mathematics, a '''hexadecagon''' (sometimes called a '''hexakaidecagon''' or '''16-gon''') is a sixteen-sided [[Polygon|polygon]].<ref name=Weisstein2002>{{cite book|last=Weisstein|first=Eric W.|title=CRC Concise Encyclopedia of Mathematics, Second Edition|year=2002|publisher=CRC Press|isbn=9781420035223|page=1365}}</ref>

== Regular hexadecagon==
A ''[[Regular polygon|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 [[Truncation (geometry)|truncated]] [[Octagon|octagon]], t{8}, and a twice-truncated [[Square|square]] tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.

===Construction===
As 16 = 2<sup>4</sup> (a [[Power of two|power of two]]), a regular hexadecagon is [[Constructible polygon|constructible]] using compass and straightedge: this was already known to ancient Greek mathematicians.<ref>{{citation|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|edition=2nd|publisher=Academic Press|year=2007|isbn=9780080547091|page=142|url=https://books.google.com/books?id=d5Z5I3gnFh0C&pg=PA142}}.</ref>
{{multiple image
| align = left
| image1 = Regular_Hexadecagon_Inscribed_in_a_Circle.gif
| width1 = 330 
| alt1 = 
| caption1 = Construction of a regular hexadecagon<br />
at a given circumcircle
| image2 = 01-Sechzehneck-Seite-gegeben Animation.gif
| width2 = 281
| alt2 = 
| caption2 = Construction of a regular hexadecagon<br />
at a given side length, animation. (The construction is very similar to that of [[Octagon#Construction and elementary properties|octagon at a given side length]].)
| footer = 
}}
{{clear}}

==Measurements==
Each angle of a regular hexadecagon is 157.5 [[Degree (angle)|degrees]], and the total angle measure of any hexadecagon is 2520 degrees.

The [[Area|area]] of a regular hexadecagon with edge length ''t'' is
:<math>\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}</math>

Because the hexadecagon has a number of sides that is a [[Power of two|power of two]], its area can be computed in terms of the circumradius ''R'' by truncating [[Viète's formula]]:
:<math>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}}.</math>

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

==Symmetry==
{| class=wikitable width=320 align=right
|+ Symmetry
|- valign=top
|[[File:Symmetries_of_hexadecagon.png|160px]]
|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 Dih<sub>16</sub> symmetry, order 32. There are 4 dihedral subgroups: Dih<sub>8</sub>, Dih<sub>4</sub>, Dih<sub>2</sub>, and Dih<sub>1</sub>, and 5 [[Cyclic group|cyclic subgroups]]: Z<sub>16</sub>, Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>, and  Z<sub>1</sub>, 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.<ref>John H. Conway, Heidi Burgiel, [[Biography:Chaim Goodman-Strauss|Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{isbn|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)</ref>

The most common high symmetry hexadecagons are '''d16''', an [[Isogonal figure|isogonal]] hexadecagon constructed by eight mirrors can alternate long and short edges, and '''p16''', an [[Isotoxal figure|isotoxal]] hexadecagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [[Dual polygon|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 be seen as directed edges.
{{Clear}}

== Dissection==
{| class=wikitable align=right
!16-cube projection
!colspan=2|112 rhomb dissection
|- align=center
|[[File:16-cube t0 A15.svg|160px]]
|[[File:16-gon rhombic dissection-size2.svg|160px]]<BR>Regular
|[[File:Isotoxal 20-gon rhombic dissection-size2.svg|160px]]<BR>Isotoxal
|}

Coxeter states that every [[Zonogon|zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms.
<ref>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular this is true for [[Regular polygon|regular polygon]]s with evenly many sides, in which case the parallelograms are all rhombi. 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 an 8-cube, with 28 of 1792 faces.  The list {{OEIS2C|1=A006245}} enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection.
{| class=wikitable
|+ Dissection into 28 rhombs
|- align=center valign=top
|[[File:8-cube.svg|150px]]<BR>8-cube
|[[File:16-gon-dissection.svg|160px]]
|[[File:16-gon rhombic dissection2.svg|160px]]
|[[File:16-gon rhombic dissectionx.svg|160px]]
|[[File:16-gon-dissection-random.svg|160px]]
|}

== Skew hexadecagon ==
{| class=wikitable align=right width=300
|+ 3 regular skew zig-zag hexadecagon
!{8}#{ }
! {<nowiki/>{{frac|8|3}}}#{ }
! {<nowiki/>{{frac|8|5}}}#{ }
|-
|[[File:8 antiprism_skew_16-gon.png|120px]]
|[[File:8-3_antiprism_skew_16-gon.png|80px]]
|[[File:8-5_antiprism_skew_16-gon.png|120px]]
|-
|colspan=3|A regular skew hexadecagon is seen as zig-zagging edges of an [[Octagonal antiprism|octagonal antiprism]], an [[Octagrammic antiprism|octagrammic antiprism]], and an [[Octagrammic crossed-antiprism|octagrammic crossed-antiprism]].
|}
A '''skew hexadecagon''' is a [[Skew polygon|skew polygon]] with 24 vertices and edges but not existing on the same plane. The interior of such a 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 an [[Octagonal antiprism|octagonal antiprism]] with the same D<sub>8d</sub>, [2<sup>+</sup>,16] symmetry, order 32. The [[Octagrammic antiprism|octagrammic antiprism]], s{2,16/3} and [[Octagrammic crossed-antiprism|octagrammic crossed-antiprism]], s{2,16/5} also have regular skew octagons.

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

{| class=wikitable width=80
!A<sub>15</sub>
!colspan=2|B<sub>8</sub>
!colspan=2|D<sub>9</sub>
!colspan=2|2B<sub>2</sub> (4D)
|- align=center valign=top
|[[File:15-simplex_t0.svg|100px]]<br>15-simplex
|[[File:8-cube_t7.svg|100px]]<br>8-orthoplex
|[[File:8-cube_t0.svg|100px]]<br>8-cube
|[[File:9-cube_t8_B8.svg|100px]]<br>[[9-orthoplex|6<sub>11</sub>]]
|[[File:9-demicube.svg|100px]]<br>[[9-demicube|1<sub>61</sub>]]
|[[File:8-8 duoprism ortho3.png|100px]]<BR>8-8 duopyramid
|[[File:8-8 duoprism ortho-3.png|100px]]<BR>[[8-8 duoprism|8-8 duoprism]]
|}

==Related figures==
A '''hexadecagram''' is a 16-sided star polygon, represented by symbol {16/n}. There are three regular [[Star polygon|star polygon]]s, {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 [[Octagon|octagon]]s, {16/4} is reduced to 4{4} as four squares and {16/6} reduces to 2{8/3} as two [[Octagram|octagram]]s, and finally {16/8} is reduced to 8{2} as eight [[Digon|digon]]s.
{| class="wikitable collapsible collapsed"
!colspan=5|Compound and star hexadecagons
|-
!Form
!Convex polygon
!Compound
!Star polygon
!Compound
|- align=center valign=top
!valign=center|Image
|BGCOLOR="#ffe0e0"|[[File:Regular_polygon_16.svg|120px]]<br>{16/1} or {16}
|[[File:Regular_star_figure_2(8,1).svg|120px]]<br>{16/2} or 2{8}
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_16-3.svg|120px]]<br>{16/3}
|[[File:Regular_star_figure_4(4,1).svg|120px]]<br>{16/4} or 4{4}
|- align=center
!Interior angle
|157.5°||135°||112.5°||90°
|-
!Form
!Star polygon
!Compound
!Star polygon
!Compound
|- align=center valign=top
!valign=center|Image
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_16-5.svg|120px]]<br>{16/5}
|[[File:Regular_star_figure_2(8,3).svg|120px]]<br>{16/6} or 2{8/3}
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_16-7.svg|120px]]<br>{16/7}
|[[File:Regular_star_figure_8(2,1).svg|120px]]<br>{16/8} or 8{2}
|- align=center
!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.<ref>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', [[Biography:Branko Grünbaum|Branko Grünbaum]]</ref>

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}.
{| class="wikitable collapsible collapsed"
!colspan=5|Isogonal truncations of octagon and octagram
|-
!Quasiregular
!colspan=3|Isogonal
!Quasiregular
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_polygon_truncation_8_1.svg|100px]]<BR>t{8}={16}
|[[File:regular_polygon_truncation_8_2.svg|100px]]
|[[File:regular_polygon_truncation_8_3.svg|100px]]
|[[File:regular_polygon_truncation_8_4.svg|100px]]
|BGCOLOR="#ffe0e0"|[[File:regular_polygon_truncation_8_5.svg|100px]]<BR>t{8/7}={16/7}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_8-3_1.svg|100px]]<BR>t{8/3}={16/3}
|[[File:regular_star_truncation_8-3_2.svg|100px]]
|[[File:regular_star_truncation_8-3_3.svg|100px]]
|[[File:regular_star_truncation_8-3_4.svg|100px]]
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_8-3_5.svg|100px]]<BR>t{8/5}={16/5}
|}

==In art==
[[File:Raffaello Sanzio - Spozalizio (detail) - WGA18625.jpg|thumb|240px|The hexadecagonal tower from Raphael's ''The Marriage of the Virgin'']]
In the early 16th century, Raphael was the first to construct a [[Perspective (graphical)|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.<ref>{{citation|title=Crossroads: History of Science, History of Art. Essays by David Speiser, vol. II|year=2011|pages=29–39|contribution=Architecture, mathematics and theology in Raphael's paintings|editor-first=Kim|editor-last=Williams|first=David|last=Speiser|publisher=Springer|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.</ref>

[[File:ReconquistaAlhambra.JPG|thumb|A hexadecagrammic pattern from the Alhambra]]
Hexadecagrams (16-sided [[Star polygon|star polygon]]s) are included in the Girih patterns in the Alhambra.<ref>{{citation|title=Examples of methods of drawing geometrical arabesque patterns|first=E. Hanbury|last=Hankin|journal=The Mathematical Gazette|volume=12|issue=176|date=May 1925|pages=370–373|doi=10.2307/3604213}}.</ref>

== Irregular hexadecagons ==
An octagonal star can be seen as a concave hexadecagon:
:[[File:Octagonal star.png|120px]][[File:Squared_octagonal star.png|120px]]

The latter one is seen in many architectures from Christian to Islamic, and also in the logo of IRIB TV4.

== See also ==
* [[Octagram]]
* [[Earth:Rhumbline network|Rhumbline network]]

==References==
{{reflist}}

==External links==
*{{MathWorld|title=Hexadecagon|urlname=Hexadecagon}}

{{Polygons}}

[[Category:Constructible polygons]]

{{Sourceattribution|Hexadecagon}}