View source for Enneacontahexagon

{{short description|Polygon with 96 edges}}
{{Regular polygon db|Regular polygon stat table|p96}}
In [[Geometry|geometry]], an '''enneacontahexagon''' or '''enneacontakaihexagon''' or '''96-gon''' is a ninety-six-sided [[Polygon|polygon]]. The sum of any enneacontahexagon's interior angles is 16920 degrees.

==Regular enneacontahexagon==

The ''[[Regular polygon|regular]] enneacontahexagon'' is represented by [[Schläfli symbol]] {96} and can also be constructed as a [[Truncation (geometry)|truncated]] [[Tetracontaoctagon|tetracontaoctagon]], t{48}, or a twice-truncated [[Icositetragon|icositetragon]], tt{24}, or a thrice-truncated [[Dodecagon|dodecagon]], ttt{12}, or a fourfold-truncated [[Hexagon|hexagon]], tttt{6}, or a fivefold-truncated [[Triangle|triangle]], {{not a typo|ttttt{3}}}.

{{See|Trigonometric_constants_expressed_in_real_radicals#1.875°: regular enneacontahexagon (96-sided polygon)}}
One interior angle in a [[Regular polygon|regular]] enneacontahexagon is 176{{frac|1|4}}°, meaning that one exterior angle would be 3{{frac|3|4}}°.

The [[Area|area]] of a regular enneacontahexagon is: (with ''t'' = edge length)
: <math>\begin{align}
A = &24t^2 \cot \frac{\pi}{96}\\
= &24t^2 \left(2+ \sqrt{3} + \sqrt{2} + \sqrt{6} + \sqrt{ 16+8 \sqrt{3} +2 \sqrt{ 104+60 \sqrt{3} } } +\sqrt{ 32+ 16\sqrt{3} +4 \sqrt{ 104+60 \sqrt{3} } +2 \sqrt{ 848+488 \sqrt{3} +2(31+16\sqrt{3}) \sqrt{ 104+60 \sqrt{3} } } }\right) \\
= &24t^2 \left(2+ \sqrt{3} + \sqrt{2} + \sqrt{6} + \sqrt{ 16+8 \sqrt{3} +2 \sqrt{ 104+60 \sqrt{3} } } +\sqrt{ 32+ 16\sqrt{3} +4 \sqrt{ 104+60 \sqrt{3} } +2 \sqrt{ 848+488 \sqrt{3} +2\sqrt{ 358376+206908 \sqrt{3} } } }\right)
.\end{align}</math>

The enneacontahexagon appeared in Archimedes' polygon approximation of [[Pi|pi]], along with the [[Hexagon|hexagon]] (6-gon), [[Dodecagon|dodecagon]] (12-gon), [[Icositetragon|icositetragon]] (24-gon), and [[Tetracontaoctagon|tetracontaoctagon]] (48-gon).

===Construction===
Since 96 = 2<sup>5</sup> × 3, a regular enneacontahexagon is [[Constructible polygon|constructible]] using a compass and straightedge.<ref>[http://mathworld.wolfram.com/ConstructiblePolygon.html Constructible Polygon]</ref> As a truncated [[Tetracontaoctagon|tetracontaoctagon]], it can be constructed by an edge-[[Bisection|bisection]] of a regular tetracontaoctagon.

== Symmetry==
[[File:Symmetries of enneacontahexagon.png|thumb|Symmetries of enneacontahexagon. The symmetries in each box are related as index 2 subgroups. The right box groups are related to the left box as index 3 subgroups.]]
The ''regular enneacontahexagon'' has Dih<sub>96</sub> symmetry, order 192. There are 11 subgroup dihedral symmetries: (Dih<sub>48</sub>, Dih<sub>24</sub>, Dih<sub>12</sub>, Dih<sub>6</sub>, Dih<sub>3</sub>), (Dih<sub>32</sub>, Dih<sub>16</sub>, Dih<sub>8</sub>, Dih<sub>4</sub>, Dih<sub>2</sub> and Dih<sub>1</sub>), and 12 [[Cyclic group|cyclic group]] symmetries: (Z<sub>96</sub>, Z<sub>48</sub>, Z<sub>24</sub>, Z<sub>12</sub>, Z<sub>6</sub>, Z<sub>3</sub>), (Z<sub>32</sub>, Z<sub>16</sub>, Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>, and Z<sub>1</sub>).

These 24 symmetries can be seen in 34 distinct symmetries on the enneacontahexagon. [[Biography:John Horton Conway|John Conway]] labels these by a letter and group order.<ref>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)</ref> 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|zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into ''m''(''m''-1)/2 parallelograms.<ref>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular this is true for [[Regular polygon|regular polygon]]s 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 48-cube.
{| class=wikitable
|+ Examples
|[[File:96-gon rhombic dissection.svg|160px]]
|[[File:96-gon rhombic dissection2.svg|160px]]
|}

==Enneacontahexagram==
An enneacontahexagram is a 96-sided [[Star polygon|star polygon]]. There are 15 regular forms given by [[Schläfli symbol]]s {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|vertex configuration]].

{| class=wikitable
|+ Regular [[Star polygon|star polygon]]s {96/k}
|- align=center
!Picture
|[[File:Star polygon 96-5.svg|100px]]<br>{96/5}
|[[File:Star polygon 96-7.svg|100px]]<br>{96/7}
|[[File:Star polygon 96-11.svg|100px]]<br>{96/11}
|[[File:Star polygon 96-13.svg|100px]]<br>{96/13}
|[[File:Star polygon 96-17.svg|100px]]<br>{96/17}
|[[File:Star polygon 96-19.svg|100px]]<br>{96/19}
|[[File:Star polygon 96-23.svg|100px]]<br>{96/23}
|[[File:Star polygon 96-25.svg|100px]]<br>{96/25}
|- align=center
!Interior angle
|161.25°
|153.75°
|138.75°
|131.25°
|116.25°
|108.75°
|93.75°
|86.25°
|- align=center
!Picture
|[[File:Star polygon 96-29.svg|100px]]<br>{96/29}
|[[File:Star polygon 96-31.svg|100px]]<br>{96/31}
|[[File:Star polygon 96-35.svg|100px]]<br>{96/35}
|[[File:Star polygon 96-37.svg|100px]]<br>{96/37}
|[[File:Star polygon 96-41.svg|100px]]<br>{96/41}
|[[File:Star polygon 96-43.svg|100px]]<br>{96/43}
|[[File:Star polygon 96-47.svg|100px]]<br>{96/47}
|&nbsp;
|- align=center
!Interior angle
|71.25°
|63.75°
|48.75°
|41.25°
|26.25°
|18.75°
|3.75°
|&nbsp;
|}

==References==
{{Reflist}}
*[http://mathforum.org/dr.math/faq/faq.polygon.names.html Naming Polygons and Polyhedra]

{{Clear}}
{{polygons}}

[[Category:Constructible polygons]]

{{Sourceattribution|Enneacontahexagon|1}}