| Regular hexadecagon | |
|---|---|
 A regular hexadecagon | |
| Type | Regular polygon |
| Edges andvertices | 16 |
| Schläfli symbol | {16}, t{8}, tt{4} |
| Coxeter–Dynkin diagrams |     |
| Symmetry group | Dihedral (D16), order 2×16 |
| Internal angle (degrees) | 157.5° |
| Properties | Convex,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | Self |
In mathematics, ahexadecagon (sometimes called ahexakaidecagon or16-gon) is a sixteen-sidedpolygon.[1]
Aregular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent. ItsSchläfli symbol is {16} and can be constructed as atruncatedoctagon, t{8}, and a twice-truncatedsquare tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.
As 16 = 24 (apower of two), a regular hexadecagon isconstructible usingcompass and straightedge: this was already known to ancient Greek mathematicians.[2]
Each angle of a regular hexadecagon is 157.5degrees, and the total angle measure of any hexadecagon is 2520 degrees.
Thearea of a regular hexadecagon with edge lengtht is
Because the hexadecagon has a number of sides that is apower of two, its area can be computed in terms of thecircumradiusR by truncatingViète's formula:
Since the area of the circumcircle is the regular hexadecagon fills approximately 97.45% of its circumcircle.
 | 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. |
Theregular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups: Dih8, Dih4, Dih2, and Dih1, and 5cyclic 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 asr32 and no symmetry is labeleda1. 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 asg for their central gyration orders.[3]
The most common high symmetry hexadecagons ared16, anisogonal hexadecagon constructed by eight mirrors can alternate long and short edges, andp16, anisotoxal hexadecagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms areduals 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 theg16 subgroup has no degrees of freedom but can be seen asdirected edges.
| 16-cube projection | 112 rhomb dissection | |
|---|---|---|
 |  Regular |  Isotoxal |
Coxeter states that everyzonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected intom(m-1)/2 parallelograms.[4]In particular this is true forregular polygons with evenly many sides, in which case the parallelograms are all rhombi. For theregular hexadecagon,m=8, and it can be divided into 28: 4 squares and 3 sets of 8 rhombs. This decomposition is based on aPetrie polygon projection of an8-cube, with 28 of 1792 faces. The listOEIS: A006245 enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection.
 8-cube |  |  |  |  |
| {8}#{ } | {8⁄3}#{ } | {8⁄5}#{ } |
|---|---|---|
 |  |  |
| A regular skew hexadecagon is seen as zig-zagging edges of anoctagonal antiprism, anoctagrammic antiprism, and anoctagrammic crossed-antiprism. | ||
Askew hexadecagon is askew polygon with 24 vertices and edges but not existing on the same plane. The interior of such a hexadecagon is not generally defined. Askew zig-zag hexadecagon has vertices alternating between two parallel planes.
Aregular skew hexadecagon isvertex-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 anoctagonal antiprism with the same D8d, [2+,16] symmetry, order 32. Theoctagrammic antiprism, s{2,16/3} andoctagrammic crossed-antiprism, s{2,16/5} also have regular skew octagons.
The regular hexadecagon is thePetrie polygon for many higher-dimensional polytopes, shown in these skeworthogonal projections, including:
| A15 | B8 | D9 | 2B2 (4D) | |||
|---|---|---|---|---|---|---|
 15-simplex |  8-orthoplex |  8-cube |  611 |  161 |  8-8 duopyramid |  8-8 duoprism |
Ahexadecagram is a 16-sided star polygon, represented by symbol {16/n}. There are three regularstar 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 twooctagons, {16/4} is reduced to 4{4} as four squares and {16/6} reduces to 2{8/3} as twooctagrams, and finally {16/8} is reduced to 8{2} as eightdigons.
| Compound and star hexadecagons | ||||
|---|---|---|---|---|
| Form | Convex polygon | Compound | Star polygon | Compound |
| Image |  {16/1} or {16} | .svg) {16/2} or 2{8} |  {16/3} | .svg) {16/4} or 4{4} |
| Interior angle | 157.5° | 135° | 112.5° | 90° |
| Form | Star polygon | Compound | Star polygon | Compound |
| Image |  {16/5} | .svg) {16/6} or 2{8/3} |  {16/7} | .svg) {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.[5]
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 | ||||
|---|---|---|---|---|
| Quasiregular | Isogonal | Quasiregular | ||
 t{8}={16} |  |  |  |  t{8/7}={16/7} |
 t{8/3}={16/3} |  |  |  |  t{8/5}={16/5} |
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In the early 16th century,Raphael was the first to construct aperspective image of a regular hexadecagon: the tower in his paintingThe Marriage of the Virgin has 16 sides, elaborating on an eight-sided tower in a previous painting byPietro Perugino.[6]
Hexadecagrams (16-sidedstar polygons) are included in theGirih patterns in theAlhambra.[7]
Anoctagonal star can be seen as a concave hexadecagon:
The latter one is seen in many architectures from Christian to Islamic, and also in the logo ofIRIB TV4.
