Pentadecagon#Pentadecagram

A regular pentadecagon is represented by Schläfli symbol {15}.

A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by

:$$

\begin{align} 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{align}

As 15 = 3 × 5, a product of distinct Fermat primes, a regular pentadecagon is constructible using compass and straightedge:

The following constructions of regular pentadecagons with given circumcircle are similar to the illustration of the proposition XVI in Book IV of Euclid's Elements.{{cite book |last=Dunham |first=William |date=1991 |title=Journey through Genius - The Great Theorems of Mathematics |url=https://www.ms.uky.edu/~sohum/ma330/files/william-dunham-journey-through-genius-the-great-theorems-of-mathematics-penguin-non-classics1991014014739x.pdf#80 |publisher=Penguin |access-date=2015-11-12|page=65|via=the University of Kentucky College of Arts & Sciences Mathematics}}

File:Regular_Pentadecagon_Inscribed_in_a_Circle.gif

Compare the construction according to Euclid in this image: [https://commons.wikimedia.org/wiki/File:01-Pentadecagon.svg Pentadecagon]

In the construction for given circumcircle: $\backslash overline\{FG\}\; =\; \backslash overline\{CF\}\backslash text\{,\}\; \backslash ;\; \backslash overline\{AH\}\; =\; \backslash overline\{GM\}\backslash text\{,\}\; \backslash ;\; |E\_1E\_6|$ is a side of equilateral triangle and $|E\_2E\_5|$ is a side of a regular pentagon.{{cite book|last=Kepler |first=Johannes, translated and initiated by MAX CASPAR 1939 |title=WELT-HARMONIK |date=2006 |url=https://books.google.com/books?id=Dggv8PGKS8IC&q=%22XLIV.+Satz.%22+%22Verbindet+man+also+C+mit+F+so+wird+die+Linie+CF+die+Seite+sein%22&pg=RA1-PA44|access-date=2015-12-07 |page=44 |publisher=Oldenbourg Verlag |isbn=978-3-486-58046-4 |via=Google Books|language=de}} Retrieved on June 5, 2017

The point $H$ divides the radius $\backslash overline\{AM\}$ in golden ratio: $\backslash frac\{\backslash overline\{AH\}\}\{\backslash overline\{HM\}\}\; =\; \backslash frac\{\backslash overline\{AM\}\}\{\backslash overline\{AH\}\}\; =\; \backslash frac\{1+\; \backslash sqrt\{5\}\}\{2\}\; =\; \backslash Phi\; \backslash approx\; 1.618\; \backslash text\{.\}$

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 $\backslash overline\{CG\}$, but rather they use segment $\backslash overline\{MG\}$ as radius $\backslash overline\{AH\}$ for the second circular arc (angle 36°).

File:01-Fünfzehneck.svgFile:01-FünfzehneckAnimation.gif

A compass and straightedge construction for a given side length. The construction is nearly equal to that of the pentagon at a given side, then also the presentation is succeed by extension one side and it generates a segment, here $\backslash overline\{FE\_2\}\backslash text\{,\}$ which is divided according to the golden ratio:

$\backslash frac\{\backslash overline\{E\_1\; E\_2\}\}\{\backslash overline\{E\_1\; F\}\}\; =\; \backslash frac\{\backslash overline\{E\_2\; F\}\}\{\backslash overline\{E\_1\; E\_2\}\}\; =\; \backslash frac\{1+\; \backslash sqrt\{5\}\}\{2\}\; =\; \backslash Phi\; \backslash approx\; 1.618\; \backslash text\{.\}$

Circumradius $\backslash overline\{E\_2\; M\}\; =\; R\backslash ;;\backslash ;\backslash ;$ Side length $\backslash overline\{E\_1\; E\_2\}\; =\; a\backslash ;;\backslash ;\backslash ;$ Angle $D\; E\_1M\; =\; ME\_2D\; =\; 78^\backslash circ$

$\backslash begin\{align\}$

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{align}

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| image2 = 01-FünfzehneckSeite-Animation.gif

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File:Symmetries_of_pentadecagon.png

The regular pentadecagon has Dih₁₅ dihedral symmetry, order 30, represented by 15 lines of reflection. Dih₁₅ has 3 dihedral subgroups: Dih₅, Dih₃, and Dih₁. And four more cyclic 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.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) He gives r30 for the full reflective symmetry, Dih₁₅. He gives d (diagonal) with reflection lines through vertices, p with reflection lines through edges (perpendicular), and for the odd-sided pentadecagon i with mirror lines through both vertices and edges, and g for cyclic symmetry. a1 labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular pentadecagons. Only the g15 subgroup has no degrees of freedom but can be seen as directed edges.

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There are three regular star 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 regular star figures: {15/3}, {15/5}, {15/6}, the first being a compound of three pentagons, the second a compound of five equilateral triangles, and the third a compound of three pentagrams.

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

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.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

The regular pentadecagon is the Petrie polygon for some higher-dimensional polytopes, projected in a skew orthogonal projection:

160px
A regular triangle, decagon, and pentadecagon can completely fill 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 a semiregular tiling.

• Construction of the pentadecagon at given side length, calculation of the circumradius $R$ (German)
• [https://commons.wikimedia.org/wiki/File:01-Fünfzehneck-Seite-Radius_R.svg Construction of the pentadecagon at given side length, exemplification: circumradius $\backslash overline\{CG\}\; =\; R$]

{{reflist}}

• {{MathWorld|title=Pentadecagon|urlname=Pentadecagon}}

{{Polygons}}

Category:Constructible polygons

Category:Polygons by the number of sides