# Triacontadigon

In geometry, a **triacontadigon** (or **triacontakaidigon**) or **32-gon** is a thirty-two-sided polygon. In Greek, the prefix triaconta- means 30 and di- means 2. The sum of any triacontadigon's interior angles is 5400 degrees.

Regular triacontadigon | |
---|---|

A regular triacontadigon | |

Type | Regular polygon |

Edges and vertices | 32 |

Schläfli symbol | {32}, t{16}, tt{8}, ttt{4} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{32}), order 2×32 |

Internal angle (degrees) | 168.75° |

Dual polygon | Self |

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

An older name is **tricontadoagon**.[1] Another name is **icosidodecagon**, suggesting a (20 and 12)-gon, in parallel to the 32-faced icosidodecahedron, which has 20 triangles and 12 pentagons.[2]

## Regular triacontadigon

The *regular triacontadigon* can be constructed as a truncated hexadecagon, t{16}, a twice-truncated octagon, tt{8}, and a thrice-truncated square. A truncated triacontadigon, t{32}, is a hexacontatetragon, {64}.

One interior angle in a regular triacontadigon is 168^{1}⁄_{4}°, meaning that one exterior angle would be 11^{1}⁄_{4}°.

The area of a regular triacontadigon is (with *t* = edge length)

and its inradius is

The circumradius of a regular triacontadigon is

### Construction

As 32 = 2^{5} (a power of two), the regular triacontadigon is a constructible polygon. It can be constructed by an edge-bisection of a regular hexadecagon.[3]

## Symmetry

The symmetries of a regular triacontadigon. Lines of reflections are blue through vertices, and purple through edges. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions. |

The *regular triacontadigon* has Dih_{32} dihedral symmetry, order 64, represented by 32 lines of reflection. Dih_{32} has 5 dihedral subgroups: Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2} and Dih_{1} and 6 more cyclic symmetries: Z_{32}, Z_{16}, Z_{8}, Z_{4}, Z_{2}, and Z_{1}, with Z_{n} representing π/*n* radian rotational symmetry.

On the regular triacontadigon, there are 17 distinct symmetries. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[4] He gives **r64** for the full reflective symmetry, Dih_{16}, and **a1** for no symmetry. He gives **d** (diagonal) with mirror lines through vertices, **p** with mirror lines through edges (perpendicular), **i** with mirror lines through both vertices and edges, and **g** for rotational symmetry. **a1** labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular triacontadigons. Only the **g32** subgroup has no degrees of freedom but can seen as directed edges.

## Dissection

regular |
Isotoxal |

Coxeter states that every zonogon (a 2*m*-gon whose opposite sides are parallel and of equal length) can be dissected into *m*(*m*-1)/2 parallelograms.[5]
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the *regular triacontadigon*, *m*=16, and it can be divided into 120: 8 squares and 7 sets of 16 rhombs. This decomposition is based on a Petrie polygon projection of a 16-cube.

## Triacontadigram

A triacontadigram is a 32-sided star polygon. There are seven regular forms given by Schläfli symbols {32/3}, {32/5}, {32/7}, {32/9}, {32/11}, {32/13}, and {32/15}, and eight compound star figures with the same vertex configuration.

Regular star polygons {32/k} | |||||||
---|---|---|---|---|---|---|---|

Picture | {32/3} |
{32/5} |
{32/7} |
{32/9} |
{32/11} |
{32/13} |
{32/15} |

Interior angle | 146.25° | 123.75° | 101.25° | 78.75° | 56.25° | 33.75° | 11.25° |

Many isogonal triacontadigrams can also be constructed as deeper truncations of the regular hexadecagon {16} and hexadecagrams {16/3}, {16/5}, and {16/7}. These also create four quasitruncations: t{16/9} = {32/9}, t{16/11} = {32/11}, t{16/13} = {32/13}, and t{16/15} = {32/15}. Some of the isogonal triacontadigrams are depicted below as part of the aforementioned truncation sequences.[6]

isogonal triacontadigrams | ||||||||
---|---|---|---|---|---|---|---|---|

t{16} = {32} |
t{16/15}={32/15} | |||||||

t{16/3} = {32/3} |
t{16/13}={32/13} | |||||||

t{16/5} = {32/5} |
t{16/11}={32/11} | |||||||

t{16/7} = {32/7} |
t{16/9}={32/9} |

## References

- A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems by Benjamin Franklin Finkel
- Weisstein, Eric W. "Icosidodecagon".
*MathWorld*. - Constructible Polygon
- 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)
- Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
- The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994),
*Metamorphoses of polygons*, Branko Grünbaum

- Naming Polygons and Polyhedra
- CRC Concise Encyclopedia of Mathematics, Second Edition, Eric W. Weisstein icosidodecagon