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Pentadecagon

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Polygon with 15 edges

Regular pentadecagon
A regular pentadecagon
Type	Regular polygon
Edges andvertices	15
Schläfli symbol	{15}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D₁₅), order 2×15
Internal angle (degrees)	156°
Properties	Convex,cyclic,equilateral,isogonal,isotoxal
Dual polygon	Self

Ingeometry, apentadecagon orpentakaidecagon or15-gon is a fifteen-sidedpolygon.

Regular pentadecagon

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Aregular pentadecagon is represented bySchläfli symbol {15}.

Aregular pentadecagon has interior angles of 156°, and with a side lengtha, has an area given by

{\begin{aligned}A={\frac {15}{4}}a^{2}\cot {\frac {\pi }{15}}&={\frac {15}{4}}{\sqrt {7+2{\sqrt {5}}+2{\sqrt {15+6{\sqrt {5}}}}}}a^{2}\\&={\frac {15a^{2}}{8}}\left({\sqrt {3}}+{\sqrt {15}}+{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}\right)\\&\simeq 17.6424\,a^{2}.\end{aligned}}

Construction

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As 15 = 3 × 5, a product of distinctFermat primes, a regular pentadecagon isconstructible usingcompass and straightedge: The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV ofEuclid'sElements.^[1]

Compare the construction according to Euclid in this image:Pentadecagon

In the construction for given circumcircle: ${\overline {FG}}={\overline {CF}}{\text{,}}\;{\overline {AH}}={\overline {GM}}{\text{,}}\;|E_{1}E_{6}|$ is a side of equilateral triangle and $|E_{2}E_{5}|$ is a side of a regular pentagon.^[2]The point $H {\displaystyle H}$ divides the radius ${\overline {AM}}$ ingolden ratio: ${\frac {\overline {AH}}{\overline {HM}}}={\frac {\overline {AM}}{\overline {AH}}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}$

Compared with the first animation (with green lines) are in the following two images the two circular arcs (for angles 36° and 24°) rotated 90° counterclockwise shown. They do not use the segment ${\overline {CG}}$ , but rather they use segment ${\overline {MG}}$ as radius ${\overline {AH}}$ for the second circular arc (angle 36°).

A compass and straightedge construction for a given side length. The construction is nearly equal to that of thepentagon at a given side, then also the presentation is succeed by extension one side and it generates a segment, here ${\overline {FE_{2}}}{\text{,}}$ which is divided according to the golden ratio:

${\frac {\overline {E_{1}E_{2}}}{\overline {E_{1}F}}}={\frac {\overline {E_{2}F}}{\overline {E_{1}E_{2}}}}={\frac {1+{\sqrt {5}}}{2}}=\Phi \approx 1.618{\text{.}}$

Circumradius ${\overline {E_{2}M}}=R\;;\;\;$ Side length ${\overline {E_{1}E_{2}}}=a\;;\;\;$ Angle $DE_{1}M=ME_{2}D=78^{\circ }$

${\begin{aligned}R&=a\cdot {\frac {1}{2}}\cdot \left({\sqrt {5+2\cdot {\sqrt {5}}}}+{\sqrt {3}}\right)={\frac {1}{2}}\cdot {\sqrt {8+2\cdot {\sqrt {5}}+2{\sqrt {15+6\cdot {\sqrt {5}}}}}}\cdot a\\&={\frac {\sin(78^{\circ })}{\sin(24^{\circ })}}\cdot a\approx 2.40486\cdot a\end{aligned}}$

Construction for a given side length

Construction for a given side length as animation,

{\overline {E_{2}M}}={\overline {CG}}

Symmetry

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The symmetries of a regular pentadecagon as shown with colors on edges and vertices. Lines of reflections are blue. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions.

Theregular pentadecagon has Dih₁₅dihedral symmetry, order 30, represented by 15 lines of reflection. Dih₁₅ has 3 dihedral subgroups: Dih₅, Dih₃, and Dih₁. And four morecyclic symmetries: Z₁₅, Z₅, Z₃, and Z₁, with Z_n representing π/n radian rotational symmetry.

On the pentadecagon, there are 8 distinct symmetries.John Conway labels these symmetries with a letter and order of the symmetry follows the letter.^[3] He givesr30 for the full reflective symmetry, Dih₁₅. He givesd (diagonal) with reflection lines through vertices,p with reflection lines through edges (perpendicular), and for the odd-sided pentadecagoni with mirror lines through both vertices and edges, andg for cyclic symmetry.a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular pentadecagons. Only theg15 subgroup has no degrees of freedom but can be seen asdirected edges.

Pentadecagrams

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There are three regularstar polygons: {15/2}, {15/4}, {15/7}, constructed from the same 15 vertices of a regular pentadecagon, but connected by skipping every second, fourth, or seventh vertex respectively.

There are also three regularstar figures: {15/3}, {15/5}, {15/6}, the first being a compound of threepentagons, the second a compound of fiveequilateral triangles, and the third a compound of threepentagrams.

The compound figure {15/3} can be loosely seen as the two-dimensional equivalent of the 3Dcompound of five tetrahedra.

Picture	{15/2}	{15/3} or 3{5}	{15/4}	{15/5} or 5{3}	{15/6} or 3{5/2}	{15/7}
Interior angle	132°	108°	84°	60°	36°	12°

Isogonal pentadecagons

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Deeper truncations of the regular pentadecagon and pentadecagrams can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths.^[4]

Vertex-transitive truncations of the pentadecagon
Quasiregular	Isogonal	Quasiregular
t{15/2}={30/2}		t{15/13}={30/13}
t{15/7} = {30/7}		t{15/8}={30/8}
t{15/11}={30/22}		t{15/4}={30/4}

Petrie polygons

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The regular pentadecagon is thePetrie polygon for some higher-dimensional polytopes, projected in a skeworthogonal projection:

14-simplex (14D)

Uses

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A regular triangle, decagon, and pentadecagon can completelyfill a plane vertex. However, due to the triangle's odd number of sides, the figures cannot alternate around the triangle, so the vertex cannot produce asemiregular tiling.

References

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^Dunham, William (1991).Journey through Genius - The Great Theorems of Mathematics(PDF). Penguin. p. 65. Retrieved2015-11-12 – via the University of Kentucky College of Arts & Sciences Mathematics.
^Kepler, Johannes, translated and initiated by MAX CASPAR 1939 (2006).WELT-HARMONIK (in German). Oldenbourg Verlag. p. 44.ISBN 978-3-486-58046-4. Retrieved2015-12-07 – via Google Books.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) Retrieved on June 5, 2017
^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)
^The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994),Metamorphoses of polygons,Branko Grünbaum

External links

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Weisstein, Eric W."Pentadecagon".MathWorld.

Polygons (List)

Triangles

Quadrilaterals

By number
of sides

1–10 sides	Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8) Nonagon/Enneagon (9) Decagon (10)
11–20 sides	Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Icosagon (20)
>20 sides	Icositrigon (23) Icositetragon (24) Triacontagon (30) 257-gon Chiliagon (1000) Myriagon (10,000) 65537-gon Megagon (1,000,000) Apeirogon (∞)