Enneacontahexagon
In geometry, an enneacontahexagon or enneacontakaihexagon or 96gon is a ninetysixsided polygon. The sum of any enneacontahexagon's interior angles is 16920 degrees.
Regular enneacontahexagon  

A regular enneacontahexagon  
Type  Regular polygon 
Edges and vertices  96 
Schläfli symbol  {96}, t{48}, tt{24}, ttt{12}, tttt{6}, ttttt{3} 
Coxeter diagram  
Symmetry group  Dihedral (D_{96}), order 2×96 
Internal angle (degrees)  176.25° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
Regular enneacontahexagon
The regular enneacontahexagon is represented by Schläfli symbol {96} and can also be constructed as a truncated tetracontaoctagon, t{48}, or a twicetruncated icositetragon, tt{24}, or a thricetruncated dodecagon, ttt{12}, or a fourfoldtruncated hexagon, tttt{6}, or a fivefoldtruncated triangle, ttttt{3}.
One interior angle in a regular enneacontahexagon is 176^{1}⁄_{4}°, meaning that one exterior angle would be 3^{3}⁄_{4}°.
The area of a regular enneacontahexagon is: (with t = edge length)
The enneacontahexagon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6gon), dodecagon (12gon), icositetragon (24gon), and tetracontaoctagon (48gon).
Construction
Since 96 = 2^{5} × 3, a regular enneacontahexagon is constructible using a compass and straightedge.[1] As a truncated tetracontaoctagon, it can be constructed by an edgebisection of a regular tetracontaoctagon.
Symmetry
The regular enneacontahexagon has Dih_{96} symmetry, order 192. There are 11 subgroup dihedral symmetries: (Dih_{48}, Dih_{24}, Dih_{12}, Dih_{6}, Dih_{3}), (Dih_{32}, Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2} and Dih_{1}), and 12 cyclic group symmetries: (Z_{96}, Z_{48}, Z_{24}, Z_{12}, Z_{6}, Z_{3}), (Z_{32}, Z_{16}, Z_{8}, Z_{4}, Z_{2}, and Z_{1}).
These 24 symmetries can be seen in 34 distinct symmetries on the enneacontahexagon. John Conway labels these by a letter and group order.[2] The full symmetry of the regular form is r192 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 g96 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
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 enneacontahexagon, m=48, and it can be divided into 1128: 24 squares and 23 sets of 48 rhombs. This decomposition is based on a Petrie polygon projection of a 48cube.
Enneacontahexagram
An enneacontahexagram is a 96sided star polygon. There are 15 regular forms given by Schläfli symbols {96/5}, {96/7}, {96/11}, {96/13}, {96/17}, {96/19}, {96/23}, {96/25}, {96/29}, {96/31}, {96/35}, {96/37}, {96/41}, {96/43}, and {96/47}, as well as 32 compound star figures with the same vertex configuration.
Picture  {96/5} 
{96/7} 
{96/11} 
{96/13} 
{96/17} 
{96/19} 
{96/23} 
{96/25} 

Interior angle  161.25°  153.75°  138.75°  131.25°  116.25°  108.75°  93.75°  86.25° 
Picture  {96/29} 
{96/31} 
{96/35} 
{96/37} 
{96/41} 
{96/43} 
{96/47} 

Interior angle  71.25°  63.75°  48.75°  41.25°  26.25°  18.75°  3.75° 
References
 Constructible Polygon
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275278)
 Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141