View source for Hexacontatetragon

{{short description|Polygon with 64 edges}}
{{Regular polygon db|Regular polygon stat table|p64}}
In [[Geometry|geometry]], a '''hexacontatetragon''' (or '''hexacontakaitetragon''') or 64-gon is a [[Polygon#Constructing higher names|sixty-four-sided]] [[Polygon|polygon]]. (In Greek, the prefix hexaconta- means 60 and tetra- means 4.) The sum of any hexacontatetragon's interior angles is 11160 degrees.

==Regular hexacontatetragon==
The ''[[Regular polygon|regular]] hexacontatetragon'' can be constructed as a [[Truncation (geometry)|truncated]] [[Triacontadigon|triacontadigon]], t{32}, a twice-truncated [[Hexadecagon|hexadecagon]], tt{16}, a thrice-truncated [[Octagon|octagon]], ttt{8}, a fourfold-truncated [[Square|square]], tttt{4}, and a fivefold-truncated [[Digon|digon]], {{not a typo|ttttt{2}}}.

One interior angle in a [[Regular polygon|regular]] hexacontatetragon is 174{{frac|3|8}}°, meaning that one exterior angle would be 5{{frac|5|8}}°.

The [[Area|area]] of a regular hexacontatetragon is (with {{nowrap|''t'' {{=}} edge length}})
:<math>A = 16t^2 \cot \frac{\pi}{64}</math>

and its inradius is
:<math>r = \frac{t}{2} \cot \frac{\pi}{64}</math>

The circumradius of a regular hexacontatetragon is
:<math>R = \frac{t}{2} \csc \frac{\pi}{64}</math>

{{See|Trigonometric constants expressed in real radicals#2.8125°: regular hexacontatetragon (64-sided polygon)}}

===Construction===
Since 64 = 2<sup>6</sup> (a [[Power of two|power of two]]), a regular hexacontatetragon is [[Constructible polygon|constructible]] using a compass and straightedge.<ref>[http://mathworld.wolfram.com/ConstructiblePolygon.html Constructible Polygon]</ref> As a truncated [[Triacontadigon|triacontadigon]], it can be constructed by an edge-[[Bisection|bisection]] of a regular triacontadigon.

== Symmetry==
[[File:Symmetries_of_hexacontatetragon.png|150px|thumb|Symmetries of hexacontatetragons]]
The ''regular hexacontatetragon'' has Dih<sub>64</sub> dihedral symmetry, order 128, represented by 64 lines of reflection. Dih<sub>64</sub> has 6 dihedral subgroups: 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 7 more [[Cyclic group|cyclic]] symmetries: Z<sub>64</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>, with Z<sub>n</sub> representing &pi;/''n'' radian rotational symmetry.

These 13 symmetries generate 20 unique symmetries on the regular hexacontatetragon. [[Biography:John Horton Conway|John Conway]] labels these lower symmetries with a letter and order of the symmetry follows the letter.<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> He gives '''r128''' for the full reflective symmetry, Dih<sub>64</sub>, 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 hexacontatetragons. Only the '''g64''' subgroup has no degrees of freedom but can seen as directed edges.

{{-}}
== Dissection==
[[File:64-gon rhombic dissection-size2.svg|thumb|64-gon with 1740 rhombs]] 
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 hexacontatetragon'', ''m''=32, and it can be divided into 496: 16 squares and 15 sets of 32 rhombs. This decomposition is based on a [[Petrie polygon]] projection of a 32-cube.
{| class=wikitable
|+ Examples
|[[File:64-gon rhombic dissection.svg|160px]]
|[[File:64-gon rhombic dissection2.svg|160px]]
|[[File:64-gon rhombic dissectionx.svg|160px]]
|}
{{-}}

==Hexacontatetragram==
A hexacontatetragram is a 64-sided [[Star polygon|star polygon]]. There are 15 regular forms given by [[Schläfli symbol]]s {64/3}, {64/5}, {64/7}, {64/9}, {64/11}, {64/13}, {64/15}, {64/17}, {64/19}, {64/21}, {64/23}, {64/25}, {64/27}, {64/29}, {64/31}, as well as 16 compound star figures with the same [[Vertex configuration|vertex configuration]].

{| class=wikitable
|+ Regular [[Star polygon|star polygon]]s {64/k}
|- align=center
!Picture
|[[File:Star polygon 64-3.svg|100px]]<br>{64/3}
|[[File:Star polygon 64-5.svg|100px]]<br>{64/5}
|[[File:Star polygon 64-7.svg|100px]]<br>{64/7}
|[[File:Star polygon 64-9.svg|100px]]<br>{64/9}
|[[File:Star polygon 64-11.svg|100px]]<br>{64/11}
|[[File:Star polygon 64-13.svg|100px]]<br>{64/13}
|[[File:Star polygon 64-15.svg|100px]]<br>{64/15}
|[[File:Star polygon 64-17.svg|100px]]<br>{64/17}
|- align=center
!Interior angle
|163.125°
|151.875°
|140.625°
|129.375°
|118.125°
|106.875°
|95.625°
|84.375°
|- align=center
!Picture
|[[File:Star polygon 64-19.svg|100px]]<br>{64/19}
|[[File:Star polygon 64-21.svg|100px]]<br>{64/21}
|[[File:Star polygon 64-23.svg|100px]]<br>{64/23}
|[[File:Star polygon 64-25.svg|100px]]<br>{64/25}
|[[File:Star polygon 64-27.svg|100px]]<br>{64/27}
|[[File:Star polygon 64-29.svg|100px]]<br>{64/29}
|[[File:Star polygon 64-31.svg|100px]]<br>{64/31}
|&nbsp;
|- align=center
!Interior angle
|73.125°
|61.875°
|50.625°
|39.375°
|28.125°
|16.875°
|5.625°
|&nbsp;
|}

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

[[Category:Constructible polygons]]

{{Sourceattribution|Hexacontatetragon|1}}