Triacontagon
Regular triacontagon  

A regular triacontagon  
Type  Regular polygon 
Edges and vertices  30 
Schläfli symbol  {30}, t{15} 
Coxeter diagram 

Symmetry group  Dihedral (D_{30}), order 2×30 
Internal angle (degrees)  168° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a triacontagon or 30gon is a thirtysided polygon. The sum of any triacontagon's interior angles is 5040 degrees.
Regular triacontagon
The regular triacontagon is a constructible polygon, by an edgebisection of a regular pentadecagon, and can also be constructed as a truncated pentadecagon, t{15}. A truncated triacontagon, t{30}, is a hexacontagon, {60}.
One interior angle in a regular triacontagon is 168°, 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 polygons: 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 t = edge length)
The inradius of a regular triacontagon is
The circumradius of a regular triacontagon is
Construction
As 30 = 2 × 3 × 5, a regular triacontagon is constructible using a compass and straightedge.^{[1]}
Symmetry
The regular triacontagon has Dih_{30} dihedral symmetry, order 60, represented by 30 lines of reflection. Dih_{30} has 7 dihedral subgroups: Dih_{15}, (Dih_{10}, Dih_{5}), (Dih_{6}, Dih_{3}), and (Dih_{2}, Dih_{1}). It also has eight more cyclic symmetries as subgroups: (Z_{30}, Z_{15}), (Z_{10}, Z_{5}), (Z_{6}, Z_{3}), and (Z_{2}, Z_{1}), with Z_{n} representing π/n radian rotational symmetry.
John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[2]} 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 seen as directed edges.
Dissection
Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/2 parallelograms.^{[3]} 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 15cube.
Triacontagram
A triacontagram is a 30sided star polygon. There are 3 regular forms given by Schläfli symbols {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same vertex configuration.
Compounds and stars  

Form  Compounds  Star polygon  Compound  
Picture  {30/2}=2{15} 
{30/3}=3{10} 
{30/4}=2{15/2} 
{30/5}=5{6} 
{30/6}=6{5} 
{30/7} 
{30/8}=2{15/4} 
Interior angle  156°  144°  132°  120°  108°  96°  84° 
Form  Compounds  Star polygon  Compound  Star polygon  Compounds  
Picture  {30/9}=3{10/3} 
{30/10}=10{3} 
{30/11} 
{30/12}=6{5/2} 
{30/13} 
{30/14}=2{15/7} 
{30/15}=15{2} 
Interior angle  72°  60°  48°  36°  24°  12°  0° 
There are also 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}.^{[4]}
Compounds and stars  

Quasiregular  Isogonal  Quasiregular Double coverings  
t{15} = {30} 
t{15/14}=2{15/7}  
t{15/7}={30/7} 
t{15/8}=2{15/4}  
t{15/11}={30/11} 
t{15/4}=2{15/2}  
t{15/13}={30/13} 
t{15/2}=2{15} 
Petrie polygons
The regular triacontagon is the Petrie polygon for three 8dimensional polytopes with E_{8} symmetry, shown in orthogonal projections in the E_{8} Coxeter plane. It is also the Petrie polygon for two 4dimensional polytopes, shown in the H_{4} Coxeter plane.
E_{8}  H_{4}  

4_{21} 
2_{41} 
1_{42} 
120cell 
600cell 
The regular triacontagram {30/7} is also the Petrie polygon for the great grand stellated 120cell and grand 600cell.
References
 ↑ Constructible Polygon
 ↑ The Symmetries of Things, Chapter 20
 ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
 ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum