# Triacontatetragon

Regular triacontatetragon
A regular triacontatetragon
Type Regular polygon
Edges and vertices 34
Schläfli symbol {34}, t{17}
Coxeter diagram
Symmetry group Dihedral (D34), order 2×34
Internal angle (degrees) ≈169.41°
Dual polygon Self
Properties Convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon.[1] The sum of any triacontatetragon's interior angles is 5760 degrees.

## Regular triacontatetragon

A regular triacontatetragon is represented by Schläfli symbol {34} and can also be constructed as a truncated 17-gon, t{17}, which alternates two types of edges.

One interior angle in a regular triacontatetragon is (2880/17)°, meaning that one exterior angle would be (180/17)°.

The area of a regular triacontatetragon is (with t = edge length)

${\displaystyle A={\frac {17}{2}}t^{2}\cot {\frac {\pi }{34}}}$

${\displaystyle r={\frac {1}{2}}t\cot {\frac {\pi }{34}}}$

The factor ${\displaystyle \cot {\frac {\pi }{34}}}$ is a root of the equation ${\displaystyle x^{16}-136x^{14}+2380x^{12}-12376x^{10}+24310x^{8}-19448x^{6}+6188x^{4}-680x^{2}+17}$.

The circumradius of a regular triacontatetragon is

${\displaystyle R={\frac {1}{2}}t\csc {\frac {\pi }{34}}}$

As 34 = 2 × 17 and 17 is a Fermat prime, a regular triacontatetragon is constructible using a compass and straightedge.[2][3][4] As a truncated 17-gon, it can be constructed by an edge-bisection of a regular 17-gon. This means that the values of ${\displaystyle \sin {\frac {\pi }{34}}}$ and ${\displaystyle \cos {\frac {\pi }{34}}}$ may be expressed in terms of nested radicals.

## Triacontatetragram

A triacontatetragram is a 34-sided star polygon. There are seven regular forms given by Schläfli symbols {34/3}, {34/5}, {34/7}, {34/9}, {34/11}, {34/13}, and {34/15}, and nine compound star figures with the same vertex configuration.

 {34/3} {34/5} {34/7} {34/9} {34/11} {34/13} {34/15}

Many isogonal triacontatetragrams can also be constructed as deeper truncations of the regular heptadecagon {17} and heptadecagrams {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. These also create eight quasitruncations: t{17/9} = {34/9}, t{17/10} = {34/10}, t{17/11} = {34/11}, t{17/12} = {34/12}, t{17/13} = {34/13}, t{17/14} = {34/14}, t{17/15} = {34/15}, and t{17/16} = {34/16}. Some of the isogonal triacontatetragrams are depicted below, as a truncation sequence with endpoints t{17}={34} and t{17/16}={34/16}.[5]

 t{17}={34} t{17/16}={34/16}

## References

1. ^ "Ask Dr. Math: Naming Polygons and Polyhedra". mathforum.org. Retrieved 2017-09-05.
2. ^ W., Weisstein, Eric. "Constructible Polygon". mathworld.wolfram.com. Retrieved 2017-09-01.
3. ^ Chepmell, C. H. (1913-03-01). "A construction of the regular polygon of 34 sides". Mathematische Annalen. 74 (1): 150–151. doi:10.1007/bf01455349. ISSN 0025-5831.
4. ^ White, Charles Edgar (1913). Theory of Irreducible Cases of Equations and Its Applications in Algebra, Geometry, and Trigonometry. p. 79.
5. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum