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Regular tetracontadigon  

A regular tetracontadigon


Type  Regular polygon 
Edges and vertices  42 
Schläfli symbol  {42}, t{21} 
Coxeter diagram  
Symmetry group  Dihedral (D_{42}), order 2×42 
Internal angle (degrees)  171+3/7° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a tetracontadigon (or tetracontakaidigon) or 42gon is a fortytwosided polygon. (In Greek, the prefix tetraconta means 40 and di means 2.) The sum of any tetracontadigon's interior angles is 7200 degrees.
The regular tetracontadigon can be constructed as a truncated icosihenagon, t{21}.
One interior angle in a regular tetracontadigon is 171^{3}⁄_{7}°, meaning that one exterior angle would be 8^{4}⁄_{7}°.
The area of a regular tetracontadigon is (with t = edge length)
and its inradius is
The circumradius of a regular tetracontadigon is
Since 42 = 2 × 3 × 7, a regular tetracontadigon is not constructible using a compass and straightedge,^{[1]} but is constructible if the use of an angle trisector is allowed.^{[2]}
The regular tetracontadigon has Dih_{42} dihedral symmetry, order 84, represented by 42 lines of reflection. Dih_{42} has 7 dihedral subgroups: Dih_{21}, (Dih_{14}, Dih_{7}), (Dih_{6}, Dih_{3}), and (Dih_{2}, Dih_{1}) and 8 more cyclic symmetries: (Z_{42}, Z_{21}), (Z_{14}, Z_{7}), (Z_{6}, Z_{3}), and (Z_{2}, Z_{1}), with Z_{n} representing π/n radian rotational symmetry.
These 16 symmetries generate 20 unique symmetries on the regular tetracontadigon. John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[3]} He gives r84 for the full reflective symmetry, Dih_{42}, and a1 for no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular tetracontadigons. Only the g42 subgroup has no degrees of freedom but can seen as directed edges.
Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/2 parallelograms.^{[4]} In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetracontatetragon, m=21, it can be divided into 210: 10 sets of 21 rhombs. This decomposition is based on a Petrie polygon projection of a 21cube.
An equilateral triangle, a regular heptagon, and a regular tetracontadigon can completely fill a plane vertex. However the entire plane cannot be tiled with regular polygons while including this vertex figure.^{[5]} However it can be used in a tiling with equilateral polygons including rhombi.^{[6]}
A tetracontadigram is a 42sided star polygon. There are five regular forms given by Schläfli symbols {42/5}, {42/11}, {42/13}, {42/17}, and {42/19}, as well as 15 compound star figures with the same vertex configuration.
Picture  {42/5} 
{42/11} 
{42/13} 
{42/17} 
{42/19} 

Interior angle  ≈137.143°  ≈85.7143°  ≈68.5714°  ≈34.2857°  ≈17.1429° 