geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a triacontagon or 30-gon is a thirty-sided

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

. The sum of any triacontagon's interior angles is 5040 degrees.

Regular triacontagon

The '' regular triacontagon'' is a

constructible polygon In mathematics, a constructible polygon is a regular polygon that can be Compass and straightedge constructions, constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regu ...

, by an edge-

bisection In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''s ...

of a regular

pentadecagon In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. Regular pentadecagon A '' regular pentadecagon'' is represented by Schläfli symbol . A regular pentadecagon has interior angles of 156 °, and with a side ...

, and can also be constructed as a truncated

, t. A truncated triacontagon, t, is a hexacontagon, . One interior angle in a 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

s: 168° is the sum of the interior angles of the

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

(60°) and the

regular pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

(108°). The

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of a regular triacontagon is (with ) :

A = \frac t^2 \cot \frac = \frac t^2 \left(\sqrt + 3\sqrt + \sqrt\sqrt\right)

The

inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

of a regular triacontagon is :

r = \frac t \cot \frac = \frac t \left(\sqrt + 3\sqrt + \sqrt\sqrt\right)

The circumradius of a regular triacontagon is :

R = \frac t \csc \frac = \frac t \left(2 + \sqrt + \sqrt\right)

Construction

As 30 = 2 × 3 × 5 , a regular triacontagon is constructible using a

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...

Symmetry

The ''regular triacontagon'' has Dih₃₀

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

, order 60, represented by 30 lines of reflection. Dih₃₀ has 7 dihedral subgroups: Dih₁₅, (Dih₁₀, Dih₅), (Dih₆, Dih₃), and (Dih₂, Dih₁). It also has eight more cyclic symmetries as subgroups: (Z₃₀, Z₁₅), (Z₁₀, Z₅), (Z₆, Z₃), and (Z₂, Z₁), 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. 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 edges.

Dissection

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

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. 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 In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

projection of a 15-cube.

Triacontagram

A triacontagram is a 30-sided

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...

(though the word is extremely rare). There are 3 regular forms given by

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

s , , and , and 11 compound star figures with the same

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

. There are also isogonal triacontagrams constructed as deeper truncations of the regular

and pentadecagram , and inverted pentadecagrams , and . Other truncations form double coverings: t

2, t

2, and t

2.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'',

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentPetrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

for three 8-dimensional polytopes with E₈ symmetry, shown in

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...

s in the E₈ Coxeter plane. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H₄ Coxeter plane. The regular triacontagram is also the Petrie polygon for the great grand stellated 120-cell and grand 600-cell.

References

Naming Polygons and Polyhedratriacontagon
{{polygons Constructible polygons Polygons by the number of sides