Triacontagon

Regular triacontagon
A regular triacontagon
Type	Regular polygon
Edgesandvertices	30
Schläfli symbol	{30}, t{15}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral(D30), order 2×30
Internal angle(degrees)	168°
Properties	Convex,cyclic,equilateral,isogonal,isotoxal
Dual polygon	Self

Ingeometry,atriacontagonor 30-gon is a thirty-sidedpolygon.The sum of any triacontagon's interior angles is5040degrees.

Regular triacontagon

Theregulartriacontagonis aconstructible polygon,by an edge-bisectionof a regularpentadecagon,and can also be constructed as atruncatedpentadecagon,t{15}. Atruncatedtriacontagon, t{30}, is ahexacontagon,{60}.

One interior angle in aregulartriacontagon 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 theinterior anglesof smallerpolygons:168° is the sum of the interior angles of theequilateral triangle(60°) and theregular pentagon(108°).

Theareaof a regular triacontagon is (witht= edge length)

$A={\frac {15}{2}}t^{2}\cot {\frac {\pi }{30}}={\frac {15}{4}}t^{2}\left({\sqrt {15}}+3{\sqrt {3}}+{\sqrt {2}}{\sqrt {25+11{\sqrt {5}}}}\right)$

Theinradiusof a regular triacontagon is

$r={\frac {1}{2}}t\cot {\frac {\pi }{30}}={\frac {1}{4}}t\left({\sqrt {15}}+3{\sqrt {3}}+{\sqrt {2}}{\sqrt {25+11{\sqrt {5}}}}\right)$

Thecircumradiusof a regular triacontagon is

$R={\frac {1}{2}}t\csc {\frac {\pi }{30}}={\frac {1}{2}}t\left(2+{\sqrt {5}}+{\sqrt {15+6{\sqrt {5}}}}\right)$

Construction

Regular triacontagon with given circumcircle. D is the midpoint of AM, DC = DF, and CF, which is the side length of the regularpentagon,is E₂₅E₁.Since 1/30 = 1/5 - 1/6, the difference between the arcs subtended by the sides of a regular pentagon and hexagon (E₂₅E₁and E₂₅A) is that of the regular triacontagon, AE₁.

As 30 = 2 × 3 × 5, a regular triacontagon isconstructibleusing acompass and straightedge.

Symmetry

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.

Theregular triacontagonhas Dih₃₀dihedral symmetry,order 60, represented by 30 lines of reflection. Dih₃₀has 7 dihedral subgroups: Dih₁₅,(Dih₁₀,Dih₅), (Dih₆,Dih₃), and (Dih₂,Dih₁). It also has eight morecyclicsymmetries as subgroups: (Z₃₀,Z₁₅), (Z₁₀,Z₅), (Z₆,Z₃), and (Z₂,Z₁), with Z_nrepresenting π/nradian rotational symmetry.

John Conwaylabels these lower symmetries with a letter and order of the symmetry follows the letter.He givesd(diagonal) with mirror lines through vertices,pwith mirror lines through edges (perpendicular),iwith mirror lines through both vertices and edges, andgfor rotational symmetry.a1labels no symmetry.

These lower symmetries allows degrees of freedoms in defining irregular triacontagons. Only theg30subgroup has no degrees of freedom but can be seen asdirected edges.

Dissection

30-gon with 420 rhombs

Coxeterstates that everyzonogon(a 2m-gon whose opposite sides are parallel and of equal length) can be dissected intom(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For theregular triacontagon,m=15, it can be divided into 105: 7 sets of 15 rhombs. This decomposition is based on aPetrie polygonprojection of a15-cube.

Examples

Triacontagram

A triacontagram is a 30-sidedstar polygon(though the word is extremely rare). There are 3 regular forms given bySchläfli symbols{30/7}, {30/11}, and {30/13}, and 11 compound star figures with the samevertex 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 alsoisogonaltriacontagrams constructed as deeper truncations of the regularpentadecagon{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}.

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 thePetrie polygonfor three 8-dimensional polytopes with E₈symmetry, shown inorthogonal projectionsin the E₈Coxeter plane.It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H₄Coxeter plane.

E8			H4
421	241	142	120-cell	600-cell

The regular triacontagram {30/7} is also the Petrie polygon for thegreat grand stellated 120-cellandgrand 600-cell.

References

• Naming Polygons and Polyhedra
• triacontagon