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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 48gon is a fortyeight 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 twicetruncated dodecagon, tt{12}, or a thricetruncated hexagon, ttt{6}, or a fourfoldtruncated 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 (6gon), dodecagon (12gon), icositetragon (24gon), and enneacontahexagon (96gon).
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 edgebisection 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 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/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 24cube.
A tetracontaoctagram is a 48sided 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° 