Triacontatetragon

Regular triacontatetragon
A regular triacontatetragon
Type	Regular polygon
Edges and vertices	34
Schläfli symbol	{34}, t{17}
Coxeter–Dynkin diagrams
Symmetry group	Dihedral (D34), order 2×34
Internal angle (degrees)	169.412°
Properties	Convex, cyclic, equilateral, isogonal, isotoxal
Dual polygon	Self

In geometry, a triacontatetragon or triacontakaitetragon is a thirty-four-sided polygon or 34-gon. 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)

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

and its inradius is

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

The factor $\cot {\frac {\pi }{34}}$ is a root of the equation $x^{16}-136x^{14}+2380x^{12}-12376x^{10}+24310x^{8}-19448x^{6}+6188x^{4}-680x^{2}+17=0$.

The circumradius of a regular triacontatetragon is

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

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

Dissection

34-gon with 544 rhombs

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. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular triacontatetragon, m=17, it can be divided into 136: 8 sets of 17 rhombs. This decomposition is based on a Petrie polygon projection of a 17-cube.

Examples

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}.

t{17}={34}

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