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#REDIRECT [[Polygon#Naming]]
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{{Article for deletion/dated|page=Triacontatetragon|timestamp=20170812013545|year=2017|month=August|day=12|substed=yes|help=off}}
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{{notability|date=August 2017}}
{{Regular polygon db|Regular polygon stat table|p34}}
In [[geometry]], a '''triacontatetragon'''<!-- or '''tessaracontagon'''--> is a thirty-four-sided [[polygon]] or 34-gon. The sum of any triacontatetragon's interior angles is 5760 degrees.


{{Rcat shell|
==Regular triacontatetragon==
{{R to related topic}}
A ''[[regular polygon|regular]] triacontatetragon'' is represented by [[Schläfli symbol]] {34} and can also be constructed as a [[Truncation (geometry)|truncated]] [[17-gon]], t{17}, which alternates two types of edges.
}}

One interior angle in a regular triacontatetragon is (2880/17)°, meaning that one exterior angle would be (180/17)°.

The [[area]] of a regular triacontatetragon is (with {{nowrap|''t'' {{=}} edge length}})
:<math>A = \frac{17}{2}t^2 \cot \frac{\pi}{34}</math>

and its [[inradius]] is
:<math>r = \frac{1}{2}t \cot \frac{\pi}{34}</math>

The factor <math>\cot \frac{\pi}{34}</math> is a root of the [[equation]] <math>x^{16} - 136x^{14} + 2 380x^{12} - 12 376x^{10} + 24 310x^{8} - 19 448x^{6} + 6 188x^{4} - 680x^{2} + 17</math>.

The [[circumradius]] of a regular tetracontagon is
:<math>R = \frac{1}{2}t \csc \frac{\pi}{34}</math>

As 34 = 2<sup></sup> × 17, a regular triacontatetragon is [[constructible polygon|constructible]] using a [[compass and straightedge]].<ref>[http://mathworld.wolfram.com/ConstructiblePolygon.html Constructible Polygon]</ref> As a [[truncation (geometry)|truncated]] [[17-gon]], it can be constructed by an edge-[[bisection]] of a regular 17-gon. This means that the values of <math>\sin \frac{\pi}{34}</math> and <math>\cos \frac{\pi}{34}</math><!-- may be expressed in radicals as follows:

:<math>\sin \frac{\pi}{34} = \frac{1}{16}\sqrt{+\sqrt{17}+\sqrt{+\sqrt{17}}+\sqrt{+\sqrt{17}+\sqrt{+\sqrt{17}}}}</math>
:<math>\cos \frac{\pi}{34} = \frac{1+\sqrt{17}+\sqrt{17+3\sqrt{17}}+\sqrt{+\sqrt{17}+2\sqrt{170+38\sqrt{17}}}}{16}</math>-->

== The golden ratio ==
::<math> \frac{\overline{HM}}{\overline{BH}} = \frac{\overline{BM}}{\overline{HM}} = \frac{1 + \sqrt{5}}{2}= \varphi \approx 1.618 </math>
{{clear}}

[[File:01-Vierunddreißigeck-Seitenlänge gegeben.gif|thumb|500px|Regular tetracontagon with given side length, animation<br />
(The construction is very similar to that of [[Icosagon#The_golden_ratio_in_decagon| icosagon with given side length]])]]

== Symmetry==
[[File:Symmetries_of_tetracontagon.png|thumb|240px|The symmetries of a regular tetracontagon. Light blue lines show subgroups of index 2. The left and right subgraphs are positionally related by index 5 subgroups.]]
The ''regular tetracontagon'' has Dih<sub>40</sub> [[dihedral symmetry]], order 80, represented by 40 lines of reflection. Dih<sub>40</sub> has 7 dihedral subgroups: (Dih<sub>20</sub>, Dih<sub>10</sub>, Dih<sub>5</sub>), and (Dih<sub>8</sub>, Dih<sub>4</sub>, Dih<sub>2</sub>, Dih<sub>1</sub>). It also has eight more [[cyclic group|cyclic]] symmetries as subgroups: (Z<sub>40</sub>, Z<sub>20</sub>, Z<sub>10</sub>, Z<sub>5</sub>), and (Z<sub>8</sub>, Z<sub>4</sub>, Z<sub>2</sub>, Z<sub>1</sub>), with Z<sub>n</sub> representing &pi;/''n'' radian rotational symmetry.

[[John Horton Conway|John Conway]] labels these lower symmetries with a letter and order of the symmetry follows the letter.<ref>'''The Symmetries of Things''', Chapter 20</ref> 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 tetracontagons. Only the '''g40''' subgroup has no degrees of freedom but can seen as [[directed edge]]s.

--><!--== Tetracontagram==
A tetracontagram is a 34-sided [[star polygon]]. There are seven regular forms given by [[Schläfli symbol]]s {34/3}, {34/5}, {34/7}, {34/9}, {34/11}, {34/13}, and {34/15}, and 12 compound [[star figure]]s with the same [[vertex configuration]].

{| class=wikitable
|+ Regular [[star polygon]]s {34/k}
|- align=center
!Picture
|[[File:Star polygon 34-3.svg|80px]]<br>{40/3}
|[[File:Star polygon 34-7.svg|80px]]<br>{40/7}
|[[File:Star polygon 34-9.svg|80px]]<br>{40/9}
|[[File:Star polygon 34-11.svg|80px]]<br>{40/11}
|[[File:Star polygon 34-13.svg|80px]]<br>{40/13}
|[[File:Star polygon 40-17.svg|80px]]<br>{40/17}
|[[File:Star polygon 40-19.svg|80px]]<br>{40/19}
|- align=center
![[Interior angle]]
|153°
|117°
|99°
|81°
|63°
|27°
|9°
|}

{| class=wikitable
|+ Regular compound polygons
|- align=center
!Picture
|[[File:Star polygon 40-2.png|60px]]<br>{40/2}=2{20}
|[[File:Star polygon 40-4.png|60px]]<br>{40/4}=4{10}
|[[File:Star polygon 40-5.png|60px]]<br>{40/5}=5{8}
|[[File:Star polygon 40-6.png|60px]]<br>{40/6}=2{20/3}
|[[File:Star polygon 40-8.png|60px]]<br>{40/8}=8{5}
|[[File:Star polygon 40-10.png|60px]]<br>{40/10}=10{4}
|- align=center
!Interior angle
|162°
|144°
|135°
|126°
|108°
|90°
|- align=center
!Picture
|[[File:Star polygon 40-12.png|60px]]<br>{40/12}=4{10/3}
|[[File:Star polygon 40-14.png|60px]]<br>{40/14}=2{20/7}
|[[File:Star polygon 40-15.png|60px]]<br>{40/15}=5{8/3}
|[[File:Star polygon 40-16.png|60px]]<br>{40/16}=8{5/2}
|[[File:Star polygon 40-18.png|60px]]<br>{40/18}=2{20/9}
|[[File:Star polygon 40-20.png|60px]]<br>{40/20}=20{2}
|- align=center
!Interior angle
|72°
|54°
|45°
|36°
|18°
|0°
|}

Many [[17-gonal figure|17-gonal]] tetracontagrams can also be constructed as deeper truncations of the regular [[17-gon]] {17} and 17-grams {17/3}, {17/5}, and {17/7}. These also create many quasitruncations: t{17/11}={34/11}, t{17/13}={34/13}, t{17/5}={34/5}, and t{17/3}={34/3}. Some of the 17-gonal tetracontagrams are depicted below, as a truncation sequence with endpoints t{17}={34} and t{17/15}={34/15}.

{| class=wikitable
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_polygon_truncation_20_1.svg|60px]]<BR>t{20}={40}<BR>{{CDD|node_1|20|node_1}}
|[[File:regular_polygon_truncation_20_2.svg|100px]]
|[[File:regular_polygon_truncation_20_3.svg|100px]]
|[[File:regular_polygon_truncation_20_4.svg|100px]]
|[[File:regular_polygon_truncation_20_5.svg|100px]]
|[[File:regular_polygon_truncation_20_6.svg|100px]]
|- align=center valign=top
|[[File:regular_polygon_truncation_20_7.svg|100px]]
|[[File:regular_polygon_truncation_20_8.svg|100px]]
|[[File:regular_polygon_truncation_20_9.svg|100px]]
|[[File:regular_polygon_truncation_20_10.svg|100px]]
|BGCOLOR="#e0e0ff"|[[File:regular_polygon_truncation_20_11.svg|100px]]<BR>t{20/19}={40/19}<BR>{{CDD|node_1|20|rat|19|node_1}}
|}

-->==References==
{{reflist}}
*{{MathWorld|title=Triacontatetragon|urlname=Triacontatetragon}}
*[http://mathforum.org/dr.math/faq/faq.polygon.names.html Naming Polygons and Polyhedra]
*[https://books.google.com/books?id=ugBDAAAAIAAJ&lpg=PA194&ots=eu5YMWxvXz&dq=tessaracontagon&pg=PA194#v=onepage&q=tessaracontagon&f=false tessaracontagon]

{{Polygons}}
[[Category:Polygons]]
[[Category:Constructible polygons]]

Latest revision as of 07:40, 21 December 2021

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