View source for Triacontadigon

{{short description|Polygon with 32 edges}}
{{Regular polygon db|Regular polygon stat table|p32}}
In [[Geometry|geometry]], a '''triacontadigon''' (or '''triacontakaidigon''') or '''32-gon''' is a thirty-two-sided polygon. In Greek, the prefix triaconta- means 30 and di- means 2. The sum of any triacontadigon's interior angles is 5400 degrees.

An older name is '''tricontadoagon'''.<ref>[https://books.google.com/books?id=ugBDAAAAIAAJ&lpg=PA194&ots=eu5YMWxvXz&dq=pentecontagon&pg=PA194&redir_esc=y#v=onepage&q=pentecontagon&f=false A Mathematical Solution Book Containing Systematic Solutions to Many of the Most Difficult Problems] by Benjamin Franklin Finkel</ref> Another name is '''icosidodecagon''', suggesting a (20 and 12)-gon, in parallel to the 32-faced [[Icosidodecahedron|icosidodecahedron]], which has 20 triangles and 12 pentagons.<ref>{{MathWorld|Icosidodecagon|Icosidodecagon}}</ref>

==Regular triacontadigon==
The ''[[Regular polygon|regular]] triacontadigon'' can be constructed as a [[Truncation (geometry)|truncated]] [[Hexadecagon|hexadecagon]], t{16}, a twice-truncated [[Octagon|octagon]], tt{8}, and a thrice-truncated [[Square|square]]. A truncated triacontadigon, t{32}, is a [[Hexacontatetragon|hexacontatetragon]], {64}.

One interior angle in a [[Regular polygon|regular]] triacontadigon is 168{{frac|3|4}}°, meaning that one exterior angle would be 11{{frac|1|4}}°.

The [[Area|area]] of a regular triacontadigon is (with {{nowrap|''t'' {{=}} edge length}})
:<math>\begin{align}
A = &8t^2 \cot \frac{\pi}{32}\\
= &8t^2 \left(1+ \sqrt{2} + \sqrt{4 +2 \sqrt{2} } +\sqrt{8 +4 \sqrt{2} +2 \sqrt{20 +14 \sqrt{2} } } \right)
\end{align} </math>

and its inradius is
:<math>r = \frac{1}{2}t \cot \frac{\pi}{32}</math>

The circumradius of a regular triacontadigon is
:<math>\begin{align}
R = &\frac{1}{2}t \csc \frac{\pi}{32}\\
= &\frac{1}{2}t \left(\sqrt{16+8 \sqrt{2} +4 \sqrt{20 +14 \sqrt{2} } +2 \sqrt{168 +116 \sqrt{2} +2 \sqrt{13780 +9742 \sqrt{2} } } }\right)
\end{align}</math>
{{See|Trigonometric constants expressed in real radicals#5.625°: regular triacontadigon (32-sided polygon)}}

===Construction===
As 32 = 2<sup>5</sup> (a [[Power of two|power of two]]), the regular triacontadigon is a [[Constructible polygon|constructible polygon]]. It can be constructed by an edge-[[Bisection|bisection]] of a regular [[Hexadecagon|hexadecagon]].<ref>[http://mathworld.wolfram.com/ConstructiblePolygon.html Constructible Polygon]</ref>

{{-}}

== Symmetry==
{| class=wikitable align=right width=340
|- valign=top
|[[File:Symmetries_of_triacontadigon.png|180px]]
|The symmetries of a regular triacontadigon. Lines of reflections are blue through vertices, and purple through edges. Gyrations are given as numbers in the center. Vertices are colored by their symmetry positions.
|}
The ''regular triacontadigon'' has Dih<sub>32</sub> dihedral symmetry, order 64, represented by 32 lines of reflection. Dih<sub>32</sub> has 5 dihedral subgroups: Dih<sub>16</sub>, Dih<sub>8</sub>, Dih<sub>4</sub>, Dih<sub>2</sub> and Dih<sub>1</sub> and 6 more [[Cyclic group|cyclic]] symmetries: 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.

On the regular triacontadigon, there are 17 distinct symmetries. [[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, [[Biography:Chaim Goodman-Strauss|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 '''r64''' for the full reflective symmetry, Dih<sub>16</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 triacontadigons. Only the '''g32''' subgroup has no degrees of freedom but can seen as directed edges.
{{-}}
== Dissection==
{| class=wikitable align=right
|+ 32-gon with 480 rhombs
|- align=center
|[[File:32-gon rhombic dissection-size2.svg|160px]]<BR>regular
|[[File:Isotoxal 32-gon rhombic dissection-size2.svg|160px]]<BR>Isotoxal
|}

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 polygons with evenly many sides, in which case the parallelograms are all rhombi.  For the ''regular triacontadigon'', ''m''=16, and it can be divided into 120: 8 squares and 7 sets of 16 rhombs. This decomposition is based on a [[Petrie polygon]] projection of a 16-cube.

{| class=wikitable
|+ Examples
|[[File:32-gon rhombic dissection.svg|160px]]
|[[File:32-gon rhombic dissection2.svg|160px]]
|[[File:32-gon rhombic dissectionx.svg|160px]]
|[[File:32-gon-dissection-random.svg|160px]]
|}

==Triacontadigram==
A triacontadigram is a 32-sided [[Star polygon|star polygon]]. There are seven regular forms given by [[Schläfli symbol]]s {32/3}, {32/5}, {32/7}, {32/9}, {32/11}, {32/13}, and {32/15}, and eight compound star figures with the same [[Vertex configuration|vertex configuration]].
{| class="wikitable collapsible collapsed"
!colspan=7|Regular star polygons {32/k}
|- align=center
! Picture
|[[File:Regular star polygon 32-3.svg|120px]]<br>{32/3}
|[[File:Regular star polygon 32-5.svg|120px]]<br>{32/5}
|[[File:Regular star polygon 32-7.svg|120px]]<br>{32/7}
|[[File:Regular star polygon 32-9.svg|120px]]<br>{32/9}
|[[File:Regular star polygon 32-11.svg|120px]]<br>{32/11}
|[[File:Regular star polygon 32-13.svg|120px]]<br>{32/13}
|[[File:Regular star polygon 32-15.svg|120px]]<br>{32/15}
|- align=center
! Interior angle
| 146.25°
| 123.75°
| 101.25°
| 78.75°
| 56.25°
| 33.75°
| 11.25°
|}

Many [[Isogonal figure|isogonal]] triacontadigrams can also be constructed as deeper truncations of the regular [[Hexadecagon|hexadecagon]] {16} and hexadecagrams {16/3}, {16/5}, and {16/7}. These also create four quasitruncations: t{16/9} = {32/9}, t{16/11} = {32/11}, t{16/13} = {32/13}, and t{16/15} = {32/15}. Some of the isogonal triacontadigrams are depicted below as part of the aforementioned truncation sequences.<ref>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', [[Biography:Branko Grünbaum|Branko Grünbaum]]</ref>

{| class="wikitable collapsible collapsed"
!colspan=8|isogonal triacontadigrams
|- align="center"
|BGCOLOR="#ffe0e0"|[[File:regular_polygon_truncation_16_1.svg|100px]]<BR>t{16} = {32}<BR>{{CDD|node_1|16|node_1}}
|[[File:regular_polygon_truncation_16_2.svg|100px]]
|[[File:regular_polygon_truncation_16_3.svg|100px]]
|[[File:regular_polygon_truncation_16_4.svg|100px]]
|[[File:regular_polygon_truncation_16_5.svg|100px]]
|[[File:regular_polygon_truncation_16_6.svg|100px]]
|[[File:regular_polygon_truncation_16_7.svg|100px]]
|[[File:regular_polygon_truncation_16_8.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:regular_polygon_truncation_16_9.svg|100px]]<BR>t{16/15}={32/15}<BR>{{CDD|node_1|16|rat|15|node_1}}
|-
|- align=center
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_16-3_1.svg|100px]]<BR>t{16/3} = {32/3}<BR>{{CDD|node_1|16|rat|3x|node_1}}
|[[File:regular_star_truncation_16-3_2.svg|100px]]
|[[File:regular_star_truncation_16-3_3.svg|100px]]
|[[File:regular_star_truncation_16-3_4.svg|100px]]
|[[File:regular_star_truncation_16-3_5.svg|100px]]
|[[File:regular_star_truncation_16-3_6.svg|100px]]
|[[File:regular_star_truncation_16-3_7.svg|100px]]
|[[File:regular_star_truncation_16-3_8.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:regular_star_truncation_16-3_9.svg|100px]]<BR>t{16/13}={32/13}<BR>{{CDD|node_1|16|rat|13|node_1}}
|- align=center
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_16-5_1.svg|100px]]<BR>t{16/5} = {32/5}<BR>{{CDD|node_1|16|rat|5|node_1}}
|[[File:regular_star_truncation_16-5_2.svg|100px]]
|[[File:regular_star_truncation_16-5_3.svg|100px]]
|[[File:regular_star_truncation_16-5_4.svg|100px]]
|[[File:regular_star_truncation_16-5_5.svg|100px]]
|[[File:regular_star_truncation_16-5_6.svg|100px]]
|[[File:regular_star_truncation_16-5_7.svg|100px]]
|[[File:regular_star_truncation_16-5_8.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:regular_star_truncation_16-5_9.svg|100px]]<BR>t{16/11}={32/11}<BR>{{CDD|node_1|16|rat|11|node_1}}
|- align=center
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_16-7_1.svg|100px]]<BR>t{16/7} = {32/7}<BR>{{CDD|node_1|16|rat|7|node_1}}
|[[File:regular_star_truncation_16-7_2.svg|100px]]
|[[File:regular_star_truncation_16-7_3.svg|100px]]
|[[File:regular_star_truncation_16-7_4.svg|100px]]
|[[File:regular_star_truncation_16-7_5.svg|100px]]
|[[File:regular_star_truncation_16-7_6.svg|100px]]
|[[File:regular_star_truncation_16-7_7.svg|100px]]
|[[File:regular_star_truncation_16-7_8.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:regular_star_truncation_16-7_9.svg|100px]]<BR>t{16/9}={32/9}<BR>{{CDD|node_1|16|rat|9|node_1}}
|}

==References==
{{Reflist}}

* [http://mathforum.org/dr.math/faq/faq.polygon.names.html Naming Polygons and Polyhedra]
* CRC Concise Encyclopedia of Mathematics, Second Edition, [[Biography:Eric W. Weisstein|Eric W. Weisstein]] [https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA1458&lpg=PA1458&dq=Icosidodecagon+-mathworld&source=bl&ots=Cbm0OWxeUB&sig=sx23ssW_2iAARQgzRJqtGXa7fcs&hl=en&sa=X&ved=0CCsQ6AEwAmoVChMI1dztl7rayAIVwbYeCh0HBwc3#v=onepage&q=Icosidodecagon%20-mathworld&f=false icosidodecagon]
{{Clear}}
{{polygons}}

[[Category:Constructible polygons]]

{{Sourceattribution|Triacontadigon|1}}