This page uses content from Wikipedia and is licensed under CC BYSA.
Regular triacontatetragon  

A regular triacontatetragon


Type  Regular polygon 
Edges and vertices  34 
Schläfli symbol  {34}, t{17} 
Coxeter diagram  
Symmetry group  Dihedral (D_{34}), order 2×34 
Internal angle (degrees)  160+7/17° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a triacontatetragon or triacontakaitetragon is a thirtyfoursided polygon or 34gon.^{[1]} The sum of any triacontatetragon's interior angles is 5760 degrees.
A regular triacontatetragon is represented by Schläfli symbol {34} and can also be constructed as a truncated 17gon, 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)
and its inradius is
The factor is a root of the equation .
The circumradius of a regular triacontatetragon is
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 17gon, it can be constructed by an edgebisection of a regular 17gon. This means that the values of and may be expressed in terms of nested radicals.
Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/2 parallelograms.^{[5]} 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 17cube.
A triacontatetragram is a 34sided 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}.^{[6]}
t{17}={34} 
t{17/16}={34/16} 