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Regular tetracontadigon | |
---|---|

A regular tetracontadigon | |

Type | Regular polygon |

Edges and vertices | 42 |

Schläfli symbol | {42}, t{21} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{42}), order 2×42 |

Internal angle (degrees) | ≈171.429° |

Dual polygon | Self |

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

In geometry, a **tetracontadigon** (or **tetracontakaidigon**) or **42-gon** is a forty-two-sided polygon. (In Greek, the prefix tetraconta- means 40 and di- means 2.) The sum of any tetracontadigon's interior angles is 7200 degrees.

The *regular tetracontadigon* can be constructed as a truncated icosihenagon, t{21}.

One interior angle in a regular tetracontadigon is 171^{3}⁄_{7}°, meaning that one exterior angle would be 8^{4}⁄_{7}°.

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

and its inradius is

The circumradius of a regular tetracontadigon is

Since 42 = 2 × 3 × 7, a regular tetracontadigon is not constructible using a compass and straightedge,^{[1]} but is constructible if the use of an angle trisector is allowed.^{[2]}

The *regular tetracontadigon* has Dih_{42} dihedral symmetry, order 84, represented by 42 lines of reflection. Dih_{42} has 7 dihedral subgroups: Dih_{21}, (Dih_{14}, Dih_{7}), (Dih_{6}, Dih_{3}), and (Dih_{2}, Dih_{1}) and 8 more cyclic symmetries: (Z_{42}, Z_{21}), (Z_{14}, Z_{7}), (Z_{6}, Z_{3}), and (Z_{2}, Z_{1}), with Z_{n} representing π/*n* radian rotational symmetry.

These 16 symmetries generate 20 unique symmetries on the regular tetracontadigon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[3]} He gives **r84** for the full reflective symmetry, Dih_{42}, 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 tetracontadigons. Only the **g42** subgroup has no degrees of freedom but can seen as directed edges.

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.^{[4]}
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the *regular tetracontatetragon*, *m*=21, it can be divided into 210: 10 sets of 21 rhombs. This decomposition is based on a Petrie polygon projection of a 21-cube.

An equilateral triangle, a regular heptagon, and a regular tetracontadigon can completely fill a plane vertex. However, the entire plane cannot be tiled with regular polygons while including this vertex figure,^{[5]} although it can be used in a tiling with equilateral polygons and rhombi.^{[6]}

A tetracontadigram is a 42-sided star polygon. There are five regular forms given by Schläfli symbols {42/5}, {42/11}, {42/13}, {42/17}, and {42/19}, as well as 15 compound star figures with the same vertex configuration.

Picture | {42/5} |
{42/11} |
{42/13} |
{42/17} |
{42/19} |
---|---|---|---|---|---|

Interior angle | ≈137.143° | ≈85.7143° | ≈68.5714° | ≈34.2857° | ≈17.1429° |

**^**Constructible Polygon**^**"Archived copy" (PDF). Archived from the original (PDF) on 2015-07-14. Retrieved 2015-02-19.CS1 maint: Archived copy as title (link)**^**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**^**[1] Topics in Mathematics for Elementary Teachers: A Technology-enhanced ... By Sergei Abramovich**^**Shield - a 3.7.42 tiling