Hexadecagon

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Regular hexadecagon
Regular polygon 16 annotated.svg
A regular hexadecagon
Type Regular polygon
Edges and vertices 16
Schläfli symbol {16}, t{8}, tt{4}
Coxeter–Dynkin diagrams 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°
Properties Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon Self

In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. [1]

Contents

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}.

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. [2]

Regular Hexadecagon Inscribed in a Circle.gif
Construction of a regular hexadecagon
at a given circumcircle
01-Sechzehneck-Seite-gegeben Animation.gif
Construction of a regular hexadecagon
at a given side length, animation. (The construction is very similar to that of octagon at a given side length.)

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

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:

Since the area of the circumcircle is the regular hexadecagon fills approximately 97.45% of its circumcircle.

Symmetry

Symmetry
Symmetries of hexadecagon.png 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. [3]

The most common high symmetry hexadecagons are d16, an 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.

Dissection

16-cube projection112 rhomb dissection
16-cube t0 A15.svg 16-gon rhombic dissection-size2.svg
Regular
Isotoxal 20-gon rhombic dissection-size2.svg
Isotoxal

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. [4] In particular this is true for regular polygons 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 OEIS:  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
8-cube
16-gon-dissection.svg 16-gon rhombic dissection2.svg 16-gon rhombic dissectionx.svg 16-gon-dissection-random.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 an octagonal antiprism, an octagrammic antiprism, and an 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 an 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.

Petrie polygons

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

A15B8D92B2 (4D)
15-simplex t0.svg
15-simplex
8-cube t7.svg
8-orthoplex
8-cube t0.svg
8-cube
9-cube t8 B8.svg
611
9-demicube.svg
161
8-8 duoprism ortho3.png
8-8 duopyramid
8-8 duoprism ortho-3.png
8-8 duoprism

A hexadecagram is a 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.

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}.

In art

The hexadecagonal tower from Raphael's The Marriage of the Virgin Raffaello Sanzio - Spozalizio (detail) - WGA18625.jpg
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. [6]

A hexadecagrammic pattern from the Alhambra ReconquistaAlhambra.JPG
A hexadecagrammic pattern from the Alhambra

Hexadecagrams (16-sided star polygons) are included in the Girih patterns in the Alhambra. [7]

Irregular hexadecagons

An octagonal star can be seen as a concave hexadecagon:

Octagonal star.png Squared octagonal star.png

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

See also

Related Research Articles

<span class="mw-page-title-main">Antiprism</span> Polyhedron with parallel bases connected by triangles

In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.

<span class="mw-page-title-main">Octahedron</span> Polyhedron with eight triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

<span class="mw-page-title-main">Hexagon</span> Shape with six sides

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<span class="mw-page-title-main">Truncated cube</span>

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.

<span class="mw-page-title-main">Prism (geometry)</span> Solid with 2 parallel n-gonal bases connected by n parallelograms

In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Octagon</span> Polygon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

<span class="mw-page-title-main">Decagon</span> Shape with ten sides

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<span class="mw-page-title-main">Icosagon</span> Polygon with 20 edges

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<span class="mw-page-title-main">Dodecagon</span> Polygon with 12 edges

In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.

<span class="mw-page-title-main">Triacontagon</span> Polygon with 30 edges

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<span class="mw-page-title-main">Pentadecagon</span> Polygon with 15 edges

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

<span class="mw-page-title-main">Equiangular polygon</span> Polygon with equally angled vertices

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<span class="mw-page-title-main">Octagram</span> Star polygon

In geometry, an octagram is an eight-angled star polygon.

<span class="mw-page-title-main">Tetradecagon</span> Polygon with 14 edges

In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.

<span class="mw-page-title-main">Octadecagon</span> Polygon with 18 edges

In geometry, an octadecagon or 18-gon is an eighteen-sided polygon.

<span class="mw-page-title-main">Pentagon</span> Shape with five sides

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In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a cylinder.

References

  1. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1365. ISBN   9781420035223.
  2. Koshy, Thomas (2007), Elementary Number Theory with Applications (2nd ed.), Academic Press, p. 142, ISBN   9780080547091 .
  3. John H. Conway, Heidi Burgiel, 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)
  4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
  5. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
  6. 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.
  7. Hankin, E. Hanbury (May 1925), "Examples of methods of drawing geometrical arabesque patterns", The Mathematical Gazette, 12 (176): 370–373, doi:10.2307/3604213 .