geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

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

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

. 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 constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...

, by an edge- bisection of a regular

pentadecagon In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon. Regular pentadecagon A ''regular polygon, regular pentadecagon'' is represented by Schläfli symbol . A Regular polygon, regular pentadecagon has interior angl ...

, and can also be constructed as a truncated

, t. A truncated triacontagon, t, is a

hexacontagon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two tog ...

, . 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 In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

(60°) and the

regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

(108°). The

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

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.

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 Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...

symmetries as subgroups: (Z₃₀, Z₁₅), (Z₁₀, Z₅), (Z₆, Z₃), and (Z₂, Z₁), with Z_n representing π/''n'' radian rotational symmetry.

John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

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 seen as

directed edge In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...

Dissection

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

states that every

zonogon In geometry, a zonogon is a centrally-symmetric, convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations. Examples A regular polygon is a zonogon if and ...

(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 In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square () and a cube (). It is a Closed set, closed, Compact space, compact, Convex polytope, convex figure whose 1-N-skeleton, skeleton consis ...

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, certain notable ones can arise through truncation operations ...

. There are 3 regular forms given by

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

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

vertex configuration In geometry, a vertex configurationCrystallography ...

. 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 ( he, ברנקו גרונבאום; 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 wer ...

s in the E₈

Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H₄

. The regular triacontagram is also the Petrie polygon for the

great grand stellated 120-cell In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol , one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and ...

and

grand 600-cell In geometry, the grand 600-cell or grand polytetrahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells. It is one of four ''regular star 4-polytopes'' dis ...

References

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