Triacontagon

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Regular triacontagon
Regular polygon 30.svg
A regular triacontagon
Type Regular polygon
Edges and vertices 30
Schläfli symbol {30}
t{15}
Coxeter diagram CDel node 1.pngCDel 3x.pngCDel 0x.pngCDel node.png
CDel node 1.pngCDel 15.pngCDel node 1.png
Symmetry group Dihedral (D30), order 2×30
Internal angle (degrees) 168°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a triacontagon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

The regular triacontagon is a constructible polygon, by an edge-bisection of a regular pentadecagon, and can also be constructed as a truncated pentadecagon, t{15}.

Regular triacontagon properties[edit]

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)

A = \frac{15}{2} t^2 \cot \frac{\pi}{30} = \frac{15}{2} t^2 (\sqrt{23 + 10 \sqrt{5} + 2 \sqrt{3(85 + 38 \sqrt{5})}} = \frac{15}{4} t^2 (\sqrt{15} + 3\sqrt{3} + \sqrt{2}\sqrt{25+11\sqrt{5}})

The inradius of a regular triacontagon is

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

The circumradius of a regular triacontagon is

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

Construction[edit]

A regular triacontagon is constructible using a compass and straightedge.[1]

Triacontagram[edit]

A triacontagram is a 30-sided 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.

Form Compounds Star polygon Compound
Picture Regular star figure 2(15,1).svg
{30/2}=2{15}
Regular star figure 3(10,1).svg
{30/3}=3{10}
Regular star figure 2(15,2).svg
{30/4}=2{15/2}
Regular star figure 5(6,1).svg
{30/5}=5{6}
Regular star figure 6(5,1).svg
{30/6}=6{5}
Regular star polygon 30-7.svg
{30/7}
Regular star figure 2(15,4).svg
{30/8}=2{15/4}
Interior angle 156° 144° 132° 120° 108° 96° 84°
Form Compounds Star polygon Compound Star polygon Compounds
Picture Regular star figure 3(10,3).svg
{30/9}=3{10/3}
Regular star figure 10(3,1).svg
{30/10}=10{3}
Regular star polygon 30-11.svg
{30/11}
Regular star figure 6(5,2).svg
{30/12}=6{5/2}
Regular star polygon 30-13.svg
{30/13}
Regular star figure 2(15,7).svg
{30/14}=2{15/7}
Regular star figure 15(2,1).svg
{30/15}=15{2}
Interior angle 72° 60° 48° 36° 24° 12°

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}.[2]

Quasiregular Isogonal Quasiregular
Double coverings
Regular polygon truncation 15 1.svg
t{15} = {30}
Regular polygon truncation 15 2.svg Regular polygon truncation 15 3.svg Regular polygon truncation 15 4.svg Regular polygon truncation 15 5.svg Regular polygon truncation 15 6.svg Regular polygon truncation 15 7.svg Regular polygon truncation 15 8.svg Regular star polygon 15-7.svg
t{15/14}=2{15/7}
Regular star truncation 15-7 1.svg
t{15/7}={30/7}
Regular star truncation 15-7 2.svg Regular star truncation 15-7 3.svg Regular star truncation 15-7 4.svg Regular star truncation 15-7 5.svg Regular star truncation 15-7 6.svg Regular star truncation 15-7 7.svg Regular star truncation 15-7 8.svg Regular star polygon 15-4.svg
t{15/8}=2{15/4}
Regular star truncation 15-11 1.svg
t{15/11}={30/11}
Regular star truncation 15-11 2.svg Regular star truncation 15-11 3.svg Regular star truncation 15-11 4.svg Regular star truncation 15-11 5.svg Regular star truncation 15-11 6.svg Regular star truncation 15-11 7.svg Regular star truncation 15-11 8.svg Regular star polygon 15-2.svg
t{15/4}=2{15/2}
Regular star truncation 15-13 1.svg
t{15/13}={30/13}
Regular star truncation 15-13 2.svg Regular star truncation 15-13 3.svg Regular star truncation 15-13 4.svg Regular star truncation 15-13 5.svg Regular star truncation 15-13 6.svg Regular star truncation 15-13 7.svg Regular star truncation 15-13 8.svg Regular polygon 15.svg
t{15/2}=2{15}

Petrie polygons[edit]

The regular triacontagon is the Petrie polygon for three 8-dimensional polytopes with E8 symmetry, shown in orthogonal projections in the E8 Coxeter plane. It is also the Petrie polygon for two 4-dimensional polytopes, shown in the H4 Coxeter plane.

E8 H4
E8Petrie.svg
421
2 41 t0 E8.svg
241
Gosset 1 42 polytope petrie.svg
142
120-cell graph H4.svg
120-cell
600-cell graph H4.svg
600-cell

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

References[edit]

  1. ^ Constructible Polygon
  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