# Pentadecagon

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

Regular pentadecagon | |
---|---|

A regular pentadecagon | |

Type | Regular polygon |

Edges and vertices | 15 |

Schläfli symbol | {15} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{15}), order 2×15 |

Internal angle (degrees) | 156° |

Dual polygon | Self |

Properties | Convex, cyclic, equilateral, isogonal, isotoxal |

## Regular pentadecagon

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

## Uses

A regular triangle, decagon, and pentadecagon can completely fill a plane vertex.

### Construction

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*.[1]

Compare the construction according Euclid in this image: Pentadecagon

In the construction for given circumcircle: is a side of equilateral triangle and is a side of a regular pentagon.[2] The point divides the radius in golden ratio:

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 , but rather they use segment as radius 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 the pentagon at a given side, then also the presentation is succeed by extension one side and it generates a segment, here which is divided according to the golden ratio:

Circumradius Side length Angle

## Symmetry

The *regular pentadecagon* has Dih_{15} dihedral symmetry, order 30, represented by 15 lines of reflection. Dih_{15} has 3 dihedral subgroups: Dih_{5}, Dih_{3}, and Dih_{1}. And four more cyclic symmetries: Z_{15}, Z_{5}, Z_{3}, and Z_{1}, 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 gives **r30** for the full reflective symmetry, Dih_{15}. 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 seen as directed edges.

### Pentadecagrams

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.

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° |

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

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

14-simplex (14D) |

It is also the Petrie polygon for the great 120-cell and grand stellated 120-cell.

## See also

## References

- Dunham, William (1991).
*Journey through Genius - The Great Theorems of Mathematics*(PDF). Penguin. p. 65. Retrieved 2015-11-12 – via the University of Kentucky College of Arts & Sciences Mathematics. - Kepler, Johannes, translated and initiated by MAX CASPAR 1939.
*WELT-HARMONIK*(in German). p. 44. Retrieved 2015-12-07 – via Google Books. 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