View source for Triacontagon

{{short description|Polygon with 30 edges}}
{{Regular polygon db|Regular polygon stat table|p30}}
In [[geometry]], a '''triacontagon''' or 30-gon  is a thirty-sided [[polygon]]. The sum of any triacontagon's interior angles is [[5040 (number)|5040]] degrees.

==Regular triacontagon ==

The ''[[regular polygon|regular]] triacontagon'' is a [[constructible polygon]], by an edge-[[bisection]] of a regular [[pentadecagon]], and can also be constructed as a [[Truncation (geometry)|truncated]] [[pentadecagon]], t{15}. A [[Truncation (geometry)|truncated]] triacontagon, t{30}, is a [[hexacontagon]], {60}.

One interior angle in a [[regular polygon|regular]] triacontagon is 168 degrees, meaning that one exterior angle would be 12°. The triacontagon is the largest regular polygon whose interior angle is the sum of the [[interior angles]] of smaller [[polygon]]s: 168° is the sum of the interior angles of the [[equilateral triangle]] (60°) and the [[regular pentagon]] (108°).

The [[area]] of a regular triacontagon is (with {{nowrap|''t'' {{=}} edge length}})<ref>{{MathWorld|title=Triacontagon|urlname=Triacontagon}}</ref>
:<math>A = \frac{15}{2} t^2 \cot \frac{\pi}{30} = \frac{15}{4} t^2 \left(\sqrt{15} + 3\sqrt{3} + \sqrt{2}\sqrt{25+11\sqrt{5}}\right)</math>

The [[inradius]] of a regular triacontagon is
:<math>r = \frac{1}{2} t \cot \frac{\pi}{30} = \frac{1}{4} t \left(\sqrt{15} + 3\sqrt{3} + \sqrt{2}\sqrt{25+11\sqrt{5}}\right)</math>

The [[circumradius]] of a regular triacontagon is
:<math>R = \frac{1}{2} t \csc \frac{\pi}{30} = \frac{1}{2} t \left(2 + \sqrt{5} + \sqrt{15+6\sqrt{5}}\right)</math>

===Construction===
[[File:01-Dreißigeck.svg|thumb|500px|Regular triacontagon with given circumcircle. D is the midpoint of AM, DC = DF, and CF, which is the side length of the regular [[pentagon]], is E<sub>25</sub>E<sub>1</sub>.  Since 1/30 = 1/5 - 1/6, the difference between the arcs subtended by the sides of a regular pentagon and hexagon (E<sub>25</sub>E<sub>1</sub> and E<sub>25</sub>A) is that of the regular triacontagon, AE<sub>1</sub>.]]
As 30 = 2 × 3 × 5 , a regular triacontagon is [[constructible polygon|constructible]] using a [[compass and straightedge]].<ref>[http://mathworld.wolfram.com/ConstructiblePolygon.html Constructible Polygon]</ref>
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== Symmetry==
[[File:Symmetries of triacontagon.png|thumb|480px|The symmetries of a regular triacontagon as shown with colors on edges and vertices. 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. Subgroup symmetries are connected by colored lines, index 2, 3, and 5.]]
The ''regular triacontagon'' has Dih<sub>30</sub> [[dihedral symmetry]], order 60, represented by 30 lines of reflection. Dih<sub>30</sub> has 7 dihedral subgroups: Dih<sub>15</sub>, (Dih<sub>10</sub>, Dih<sub>5</sub>), (Dih<sub>6</sub>, Dih<sub>3</sub>), and (Dih<sub>2</sub>, Dih<sub>1</sub>). It also has eight more [[cyclic group|cyclic]] symmetries as subgroups: (Z<sub>30</sub>, Z<sub>15</sub>), (Z<sub>10</sub>, Z<sub>5</sub>), (Z<sub>6</sub>, Z<sub>3</sub>), and (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 triacontagons. Only the '''g30''' subgroup has no degrees of freedom but can be seen as [[directed edge]]s.
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== Dissection==
[[File:30-gon rhombic dissection-size2.svg|thumb|30-gon with 420 rhombs]]
[[Coxeter]] states that every [[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 triacontagon'', ''m''=15, it can be divided into 105: 7 sets of 15 rhombs. This decomposition is based on a [[Petrie polygon]] projection of a [[15-cube]].
{| class=wikitable
|+ Examples
|- valign=top align=center
|[[File:30-gon rhombic dissection.svg|160px]]
|[[File:30-gon-dissection-star.svg|160px]]
|[[File:30-gon rhombic dissection2.svg|160px]]
|[[File:30-gon rhombic dissectionx.svg|160px]]
|[[File:30-gon-dissection-random.svg|160px]]
|}

== Triacontagram ==
A triacontagram is a 30-sided [[star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[vertex configuration]].
{| class="wikitable collapsible collapsed"
!colspan=8|Compounds and stars
|-
!Form
!colspan=5|Compounds
!Star polygon
!Compound
|- align=center
!Picture
|[[File:regular star figure 2(15,1).svg|100px]]<BR>{30/2}=2{15}
|[[File:regular star figure 3(10,1).svg|100px]]<br>{30/3}=3{10}
|[[File:regular star figure 2(15,2).svg|100px]]<BR>{30/4}=2{15/2}
|[[File:regular star figure 5(6,1).svg|100px]]<BR>{30/5}=5{6}
|[[File:regular star figure 6(5,1).svg|100px]]<br>{30/6}=6{5}
|BGCOLOR="#ffe0e0"|[[File:Regular star polygon 30-7.svg|100px]]<BR>{30/7}
|[[File:regular star figure 2(15,4).svg|100px]]<BR>{30/8}=2{15/4}
|- align=center
! [[Interior angle]]
| 156°
| 144°
| 132°
| 120°
| 108°
| 96°
| 84°
|-
!Form
!colspan=2|Compounds
!Star polygon
!Compound
!Star polygon
!colspan=2|Compounds
|- align=center
!Picture
|[[File:regular star figure 3(10,3).svg|100px]]<BR>{30/9}=3{10/3}
|[[File:regular star figure 10(3,1).svg|100px]]<BR>{30/10}=10{3}
|BGCOLOR="#ffe0e0"|[[File:Regular star polygon 30-11.svg|100px]]<BR>{30/11}
|[[File:regular star figure 6(5,2).svg|100px]]<BR>{30/12}=6{5/2}
|BGCOLOR="#ffe0e0"|[[File:Regular star polygon 30-13.svg|100px]]<BR>{30/13}
|[[File:regular star figure 2(15,7).svg|100px]]<BR>{30/14}=2{15/7}
|[[File:regular star figure 15(2,1).svg|100px]]<BR>{30/15}=15{2}
|- align=center
! Interior angle
| 72°
| 60°
| 48°
| 36°
| 24°
| 12°
| 0°
|}

There are also [[isogonal figure|isogonal]] triacontagrams constructed as deeper truncations of the regular [[pentadecagon]] {15} and pentadecagram {15/7}, and inverted pentadecagrams {15/11}, and {15/13}. Other truncations form double coverings: t{15/14}={30/14}=2{15/7}, t{15/8}={30/8}=2{15/4}, t{15/4}={30/4}=2{15/4}, and t{15/2}={30/2}=2{15}.<ref>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', [[Branko Grünbaum]]</ref>

{| class="wikitable collapsible collapsed"
!colspan=12|Compounds and stars
|-
!Quasiregular
!colspan=7|Isogonal
!Quasiregular<BR>Double coverings
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular polygon truncation 15 1.svg|80px]]<BR>t{15} = {30}
|[[File:regular polygon truncation 15 2.svg|80px]]
|[[File:regular polygon truncation 15 3.svg|80px]]
|[[File:regular polygon truncation 15 4.svg|80px]]
|[[File:regular polygon truncation 15 5.svg|80px]]
|[[File:regular polygon truncation 15 6.svg|80px]]
|[[File:regular polygon truncation 15 7.svg|80px]]
|[[File:regular polygon truncation 15 8.svg|80px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star polygon 15-7.svg|80px]]<BR>t{15/14}=2{15/7}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular star truncation 15-7 1.svg|80px]]<BR>t{15/7}={30/7}
|[[File:regular star truncation 15-7 2.svg|80px]]
|[[File:regular star truncation 15-7 3.svg|80px]]
|[[File:regular star truncation 15-7 4.svg|80px]]
|[[File:regular star truncation 15-7 5.svg|80px]]
|[[File:regular star truncation 15-7 6.svg|80px]]
|[[File:regular star truncation 15-7 7.svg|80px]]
|[[File:regular star truncation 15-7 8.svg|80px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star polygon 15-4.svg|80px]]<BR>t{15/8}=2{15/4}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular star truncation 15-11 1.svg|80px]]<BR>t{15/11}={30/11}
|[[File:regular star truncation 15-11 2.svg|80px]]
|[[File:regular star truncation 15-11 3.svg|80px]]
|[[File:regular star truncation 15-11 4.svg|80px]]
|[[File:regular star truncation 15-11 5.svg|80px]]
|[[File:regular star truncation 15-11 6.svg|80px]]
|[[File:regular star truncation 15-11 7.svg|80px]]
|[[File:regular star truncation 15-11 8.svg|80px]]
|BGCOLOR="#e0e0ff"|[[File:Regular star polygon 15-2.svg|80px]]<BR>t{15/4}=2{15/2}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular star truncation 15-13 1.svg|80px]]<BR>t{15/13}={30/13}
|[[File:regular star truncation 15-13 2.svg|80px]]
|[[File:regular star truncation 15-13 3.svg|80px]]
|[[File:regular star truncation 15-13 4.svg|80px]]
|[[File:regular star truncation 15-13 5.svg|80px]]
|[[File:regular star truncation 15-13 6.svg|80px]]
|[[File:regular star truncation 15-13 7.svg|80px]]
|[[File:regular star truncation 15-13 8.svg|80px]]
|BGCOLOR="#e0e0ff"|[[File:Regular polygon 15.svg|80px]]<BR>t{15/2}=2{15}
|}

==Petrie polygons==
{{Disputed section|date=May 2025}}
The regular triacontagon is the [[Petrie polygon]] for three 8-dimensional polytopes with E<sub>8</sub> symmetry, shown in [[orthogonal projection]]s in the E<sub>8</sub> [[Coxeter plane]]. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H<sub>4</sub> [[Coxeter plane]].

{| class=wikitable
|- align=center valign=top 
!colspan=3 valign=center|E<sub>8</sub>
!colspan=2 valign=center|H<sub>4</sub>
|- align=center
|[[File:E8Petrie.svg|150px]]<BR>[[4 21 polytope|4<sub>21</sub>]]
|[[File:2 41 t0 E8.svg|150px]]<BR>[[2 41 polytope|2<sub>41</sub>]]
|[[File:Gosset 1 42 polytope petrie.svg|150px]]<BR>[[1 42 polytope|1<sub>42</sub>]]
|[[File:120-cell graph H4.svg|150px]]<BR>[[120-cell]]
|[[File:600-cell graph H4.svg|150px]]<BR>[[600-cell]]
|}
The regular triacontagram {30/7} is also the Petrie polygon for the [[great grand stellated 120-cell]] and [[grand 600-cell]].

==References==
{{Reflist}}
*[http://mathforum.org/dr.math/faq/faq.polygon.names.html Naming Polygons and Polyhedra]
*[https://books.google.com/books?id=ugBDAAAAIAAJ&dq=triacontagon&pg=PA194 triacontagon]

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{{polygons}}

[[Category:Constructible polygons]]
[[Category:Polygons by the number of sides]]