Triacontagon

[2] The regular triacontagon has Dih30 dihedral symmetry, order 60, represented by 30 lines of reflection.

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.

[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi.

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.

[5] The regular triacontagon is the Petrie polygon for three 8-dimensional polytopes with E8 symmetry, shown in orthogonal projections in the E8 Coxeter plane.

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 25 E 1 . Since 1/30 = 1/5 - 1/6, the difference between the arcs subtended by the sides of a regular pentagon and hexagon (E 25 E 1 and E 25 A) is that of the regular triacontagon, AE 1 .
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.
30-gon with 420 rhombs