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Regular tetracontaoctagon | |
---|---|

A regular tetracontaoctagon | |

Type | Regular polygon |

Edges and vertices | 48 |

Schläfli symbol | {48}, t{24}, tt{12}, ttt{6}, tttt{3} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{48}), order 2×48 |

Internal angle (degrees) | 172.5° |

Dual polygon | Self |

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

In geometry, a **tetracontaoctagon** (or **tetracontakaioctagon**) or **48-gon** is a forty-eight sided polygon. The sum of any tetracontaoctagon's interior angles is 8280 degrees.

The *regular tetracontaoctagon* is represented by Schläfli symbol {48} and can also be constructed as a truncated icositetragon, t{24}, or a twice-truncated dodecagon, tt{12}, or a thrice-truncated hexagon, ttt{6}, or a fourfold-truncated triangle, tttt{3}.

One interior angle in a regular tetracontaoctagon is 172^{1}⁄_{2}°, meaning that one exterior angle would be 7^{1}⁄_{2}°.

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

The tetracontaoctagon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), icositetragon (24-gon), and enneacontahexagon (96-gon).

Since 48 = 2^{4} × 3, a regular tetracontaoctagon is constructible using a compass and straightedge.^{[1]} As a truncated icositetragon, it can be constructed by an edge-bisection of a regular icositetragon.

The *regular tetracontaoctagon* has Dih_{48} symmetry, order 96. There are nine subgroup dihedral symmetries: (Dih_{24}, Dih_{12}, Dih_{6}, Dih_{3}), and (Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2} Dih_{1}), and 10 cyclic group symmetries: (Z_{48}, Z_{24}, Z_{12}, Z_{6}, Z_{3}), and (Z_{16}, Z_{8}, Z_{4}, Z_{2}, Z_{1}).

These 20 symmetries can be seen in 28 distinct symmetries on the tetracontaoctagon. John Conway labels these by a letter and group order.^{[2]} The full symmetry of the regular form is **r96** and no symmetry is labeled **a1**. The dihedral symmetries are divided depending on whether they pass through vertices (**d** for diagonal) or edges (**p** for perpendiculars), and **i** when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as **g** for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the **g48** subgroup has no degrees of freedom but can seen as directed edges.

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.^{[3]}
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the *regular tetracontaoctagon*, *m*=24, and it can be divided into 276: 12 squares and 11 sets of 24 rhombs. This decomposition is based on a Petrie polygon projection of a 24-cube.

A tetracontaoctagram is a 48-sided star polygon. There are seven regular forms given by Schläfli symbols {48/5}, {48/7}, {48/11}, {48/13}, {48/17}, {48/19}, and {48/23}, as well as 16 compound star figures with the same vertex configuration.

Picture | {48/5} |
{48/7} |
{48/11} |
{48/13} |
{48/17} |
{48/19} |
{48/23} |
---|---|---|---|---|---|---|---|

Interior angle | 142.5° | 127.5° | 97.5° | 82.5° | 52.5° | 37.5° | 7.5° |

**^**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